Abstract
We study long time dynamics of combustive processes in random media, modeled by reaction–diffusion equations with random ignition reactions. One expects that under reasonable hypotheses on the randomness, the large space-time scale dynamics of solutions to these equations is almost surely governed by a different effective PDE, which should be a homogeneous Hamilton–Jacobi equation. While this was previously proved in one dimension as well as for isotropic reactions in several dimensions (i.e., with radially symmetric laws), we provide here the first proof of this phenomenon in the general non-isotropic multidimensional setting. Our results hold for reactions that have finite ranges of dependence (i.e., their values are independent at sufficiently distant points in space) as well as for some with infinite ranges of dependence, and are based on proving existence of deterministic front (propagation) speeds in all directions for these reactions.
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1 Introduction
The reaction–diffusion equation
with \((t,x)\in (0,\infty )\times {\mathbb {R}}^d\) and \(\omega \) an element of some probability space \((\Omega ,{{\mathcal {F}}},{\mathbb {P}})\), models a host of physical phenomena occurring in random media. These phenomena all exhibit diffusion, modeled by the Laplacian, as well as some reactive process, modeled by the non-linear reaction function f. The nature of the latter process determines the behavior of f in the variable u, which models the property under study and will take values between its minimum and maximum, customarily normalized to be 0 and 1.
When \(u=0\) is an unstable equilibrium for the \((x,\omega )\)-dependent ODE \(\dot{u}=f(x,u,\omega )\) and \(u=1\) a stable one (e.g., when \(f>0\) for \(u\in (0,1)\)), the reaction is of the monostable type. A special case of this are the Kolmogorov–Petrovskii–Piskunov (KPP) or Fisher-KPP type reactions [9, 13], for which the growth rate \(u^{-1}f(x,u,\omega )\) of the reactive process is largest near \(u=0\) for each \((x,\omega )\) (e.g., in the case of logistic growth functions \(f(x,u,\omega )=g(x,\omega )u(1-u)\)). These reactions are used in, for instance, population dynamics models, with u being the normalized population density and \(u^{-1}f(x,u,\omega )\) the sum of the birth and death rates. When both \(u=0\) and \(u=1\) are asymptotically stable equilibria for \(\dot{u}=f(x,u,\omega )\) (e.g., when \(f(x,u,\omega )=g(x,\omega )u(1-u)(u-h(x,\omega ))\) with \(h(x,\omega )\in (0,1)\)), the reaction is of the bistable type, used in modeling phase transition processes.
In this paper we will consider the third main type of reactions, modeling various combustive processes, including forest fires. Here u is the normalized temperature and f vanishes for all u below some possibly \((x,\omega )\)-dependent ignition temperature (so \(u=0\) is typically a stable but not asymptotically stable equilibrium), which is why these reactions are of the ignition type. Our interest is in the long term dynamics of solutions to (1.1). The PDE typically exhibits ballistic propagation of solutions, which means that the state \(u\sim 1\) invades the region where initially \(u\sim 0\) at a linear-in-time rate. If the medium is sufficiently random, one expects this invasion to acquire a deterministic asymptotic speed as \(t\rightarrow \infty \), which may depend on the invading direction but not on the position (or \(\omega \)), due to averaging of the variations in the medium over long distances.
This phenomenon is called homogenization, because over large space-time scales, solutions behave as if the medium were possibly non-isotropic but homogeneous (i.e., direction- but not position-dependent). One can study solutions on these scales by rescaling them via the transformation
with \(\varepsilon >0\) small, which turns (1.1) into
If we now take \(\varepsilon \rightarrow 0\), the hope is to recover some (almost surely) \(\omega \)-independent limit \(u_\varepsilon \rightarrow {\bar{u}}\), in an appropriate sense and for appropriate initial data \(u_\varepsilon (0,\cdot ,\omega )\), that should ideally also satisfy some limiting effective PDE.
However, unlike in typical homogenization scenarios, the limiting PDE for reaction–diffusion equations cannot be another reaction–diffusion equation, or even another second order parabolic PDE. The reason for this is that one expects solutions to exhibit uniformly bounded in time width of the regions where transition between values \(u\sim 0\) and \(u\sim 1\) happens, which means that this width becomes zero in the scaling from (1.2) as \(\varepsilon \rightarrow 0\), and any limiting function \({\bar{u}}\) takes only values 0 and 1. For instance, in the homogeneous deterministic reaction case \(f(x,u,\omega )=f(u)\), the simplest solutions are traveling fronts, which are of the form \(u(t,x)=U(x\cdot e-ct)\) for some vector \(e\in {\mathbb {S}}^{d-1}\), where the front profile and speed (U, c) solve the ODE \(U''+cU'+f(U)=0\) with boundary values \(U(-\infty )=1\) and \(U(\infty )=0\). Clearly, the region where \(u(t,\cdot )\in [\eta ,1-\eta ]\) for any fixed \(\eta >0\) is a slab of a constant-in-t width that shrinks to zero as we take \(\varepsilon \rightarrow 0\) in (1.2), but then the limiting solution will be the (discontinuous) characteristic function of the half-space-time \(\{x\cdot e<ct\}\), which does not solve a second order parabolic PDE.
This suggests that any effective equation should be of the first order, with any limiting function \({\bar{u}}\) being its discontinuous solution, taking only values 0 and 1. The expectation of the effective (asymptotic) propagation speeds being direction- but not position-dependent then suggests that the effective PDE should be the Hamilton–Jacobi equation
with \(c^*(e)\) being the \((x,\omega )\)-independent effective propagation speed in direction \(e\in {\mathbb {S}}^{d-1}\). Moreover, the traveling front solutions above suggest that in the deterministic homogeneous reaction case, the speed \(c^*(e)\) should be precisely the traveling front speed c (which is also direction-independent in that case). One may therefore hope that in the general random case, it is also possible to find some front-like solutions in all directions \(e\in {\mathbb {S}}^{d-1}\), and that each of these has an associated speed \(c^*(e)\) in some sense.
Unfortunately, there are some serious obstacles to realizing this hope. The first is that its basic premise, that the width of the transition region where \(u(t,\cdot ,\omega )\in [\eta ,1-\eta ]\) stays uniformly bounded in time (or at least o(t)) for any fixed \(\eta >0\), may not be true in some media. The second author in fact showed that this need not happen for bistable reactions, even for periodic ones in one dimension [28], where solutions can develop linearly-in-time growing intervals on which they are close to periodic functions with values strictly away from 0 and 1. As a result, there may be no analog of a traveling front for such reactions, and hence no homogenization as described above.
Recalling pictures of forest fires, which are usually actively burning only along the margins of the already burnt area, one may hope that such issues do not occur for ignition reactions. The second author showed that this is indeed the case in dimensions \(d\leqq 3\) [29], where the widths of the transition regions (properly defined, as these regions may have complicated geometries in heterogeneous media; see (1.7) below) indeed remain uniformly bounded in time, by constants depending on \(\eta \) above and some bounds on the reaction. However, he also showed in [29] that this need not be the case in dimensions \(d\geqq 4\), where these widths may grow linearly in time as in the above bistable example. Nevertheless, the relevant examples have a special structure and it is not clear to what extent they indicate possible almost sure behaviors of solutions for various stochastic reactions (in particular, those with finite ranges of spatial dependence).
All of this demonstrates the difficulties associated with even the question whether solutions to (1.1) have some basic properties required for one to be able to initiate the study of homogenization for (1.1). This is the reason for relatively little progress in this area, until recently, particularly in the multi-dimensional case \(d\geqq 2\). In the one-dimensional setting, there are only two directions of propagation of solutions, and homogenization simply refers to showing that solutions starting from large enough compactly supported initial data propagate almost surely with some deterministic asymptotic speeds \(c_+\) (to the right) and \(c_-\) (to the left). Moreover, the transition regions (which are intervals) have trivial geometries. This allowed several authors to obtain such “homogenization” results in this setting for all three types of stationary ergodic reactions—KPP [10], ignition [19, 27], and bistable [19, 22, 28]—although with some non-trivial limitations in the latter case, due to the counterexamples from [28] mentioned above. There are also a number of 1D and quasi-1D results concerning related models and/or periodic reactions, which we do not discuss here.
Once we move to higher dimensions, the geometry of the level sets of solutions becomes much more complicated, and relatively little is known. One previous result appears in the paper [17] by Lions and Souganidis, which studies homogenization for viscous Hamilton–Jacobi equations. Their Theorem 9.3 states that homogenization also holds for general stationary ergodic KPP reactions in any dimension. (While it is indicated in [17] that its proof can be obtained via methods from [17, 18] and two other papers, a proof is not provided there.) The reason why Hamilton–Jacobi homogenization techniques should be applicable to KPP reaction–diffusion equations is that the dynamics of solutions for these reactions is determined, to the leading order, by the linearization of (1.1) (i.e., of f) at \(u=0\). This linear PDE can then be turned into a viscous Hamilton–Jacobi equation with a convex Hamiltonian via the Hopf-Cole transformation.
This linearization approach can only work for KPP reactions, and is not applicable to other types, including other monostable ones. In particular, it cannot be used in ignition–reaction–based models of combustion, where one has to work with the original non-linear PDE. Because of this complication, so far there has only been a single result proving homogenization for (non-KPP) stationary ergodic reactions in several dimensions. This is a conditional result by Lin and the second author [16], who proved homogenization for ignition reactions whose Wulff shapes exist and have no corners (a Wulff shape for (1.1), if it exists, is an open set \({{\mathcal {S}}}\subseteq {\mathbb {R}}^d\) such that solutions starting from any large enough compactly supported initial data converge to \(\chi _{{\mathcal {S}}}\) as \(t\rightarrow \infty \), after being scaled down by t in space). They also showed that these properties hold for isotropic ignition reactions in dimensions \(d\leqq 3\), with the dimension limitation being used to show that the Wulff shape exists (recall the above-mentioned examples of solutions with linearly growing widths of transition regions in dimensions \(d\geqq 4\) from [29]) and isotropy then guaranteeing that the Wulff shape is a (corner-less) ball centered at the origin. We also note that it follows from a result of Caffarelli, Lee, and Mellet [6] that Wulff shapes can have corners, even for periodic ignition reactions in two dimensions.
In fact, even homogenization for periodic reactions in several dimensions has seen fairly limited progress until recently, despite many results concerning existence of pulsating fronts and Wulff shapes for such reactions (see [5, 16, 23, 24] and references therein). While Theorem 9.3 in [17] applies to periodic KPP reactions (and is based in part on methods from [18], applicable to KPP reactions in periodic media), homogenization for periodic non-KPP reactions in several dimensions has only recently been obtained for ignition reactions as a byproduct of the method in [16], as well as for monostable reactions by Alfaro and Giletti [1] (for initial data with smooth convex supports, later extended to general convex supports in [16]).
In this paper we prove for the first time unconditional stochastic homogenization for ignition reactions, without assuming the reaction to be isotropic. Our Theorems 1.3 and 1.4 below are valid for random pure ignition reactions (see Definition 1.2) in dimensions \(d\leqq 3\) that either have a finite range of dependence (see Definition 1.1) or can be uniformly approximated by such reactions. We also extend these results in Theorems 1.7 and 1.8 to ignition reactions in any dimension, provided some a priori assumptions on the dynamics of certain special solutions to (1.1) are satisfied.
Our proof uses a result from [16], which shows that to prove homogenization, it suffices to show that the above-mentioned propagation speeds \(c^*(e)\) (called front speeds) exist for all directions \(e\in {\mathbb {S}}^{d-1}\), are almost-surely \(\omega \)-independent, and also exclusive (see Definition 6.1). This is, however, a difficult problem in general, and [16] was only able to show existence of a deterministic front speed in direction e when the reaction has a Wulff shape with outer normal vector e at some point (this is where the absence of corners is needed), because then the expanding Wulff shape can be used at large times to locally approximate a front-like solution propagating in direction e.
To show the existence of deterministic front speeds, we apply a method modeled on the one employed by Armstrong and Cardaliaguet [2] in their proof of homogenization for Hamilton–Jacobi equations with \(\alpha \)-homogeneous (for \(\alpha \geqq 1\)) non-convex (in \(\nabla u\)) Hamiltonians with finite ranges of dependence. This was the first proof of stochastic Hamilton–Jacobi homogenization for non-convex Hamiltonians in several dimensions without special structural hypotheses (such as \(H(x,\nabla u,\omega )=H(\nabla u)+V(x,\omega )\)). While there are many homogenization results for convex and level-set-convex Hamiltonians, including in the paper [3] by Armstrong, Cardaliaguet, and Souganidis where the method from [2] was first used to study Hamilton-Jacobi equations (it is based on ideas from the study of first passage percolation [12, 25]), non-convexity of the Hamiltonian presents serious issues. In fact, similarly to our reaction–diffusion setting, there are examples when homogenization does not happen for Hamilton–Jacobi equations with stationary ergodic non-convex Hamiltonians [8, 26], even in one dimension. The approach in [2] overcomes these problems by leveraging the finite range of dependence hypothesis (which is akin to an i.i.d. medium setting) and the resulting mixing properties of the environment to obtain strong quantitative estimates on the solutions where a soft approach via ergodic theorems does not appear to work. These estimates involve fluctuations of the values of solutions to the so-called metric problem for any compact set \(S\subseteq {\mathbb {R}}^d\) (which is an appropriate time-independent Hamilton–Jacobi PDE on \({\mathbb {R}}^d{\setminus } S\)) with a smooth enough boundary. These estimates improve at an exponential rate as the distance from S increases, and were then upgraded to similar estimates for S being any half-space.
Here we apply this strategy to reaction–diffusion equations, with the relevant estimates involving fluctuations of “arrival times” at any point \(x\in {\mathbb {R}}^d\) for solutions initially approximating \(\chi _S\) (we only need to consider \(S=B_k(0)\) for any \(k\in {\mathbb {N}}\)). We still obtain an exponentially-in-d(x, S) decaying estimate (see Proposition 3.8 below), albeit at a slower rate. However, we are also able to extend it to some reactions with infinite ranges of dependence (see Proposition 4.2) by carefully tracking the dependence of this rate on the range of dependence of f when the latter is finite, something that was described in [2] as completely open in the Hamilton–Jacobi setting (and appears to remain such at this time)!
After we upgrade this estimate from balls to half-spaces, we are able to prove existence of deterministic exclusive front speeds in all directions, and thus homogenization after using results from [16]. We note that while the effective equations in Hamilton–Jacobi homogenization are still Hamilton–Jacobi PDE (although some of their terms can disappear in the homogenization process), and the limiting functions are their continuous solutions, our limiting functions are discontinuous viscosity solutions to (1.4), which causes extra difficulties in the analysis. For a more thorough discussion of similarities and differences between Hamilton–Jacobi homogenization for non-convex Hamiltonians and reaction–diffusion homogenization, as well as for further references, we refer the reader to the introduction of [16].
1.1 Hypotheses and Main Results
Let us now turn to our main results. Our goal is to show that as \(\varepsilon \rightarrow 0\), solutions to (1.3) with initial data approximating \(\chi _A\) for any open set \(A\subseteq {\mathbb {R}}^d\) converge to the unique (discontinuous viscosity) solution to (1.4) with initial data \(\chi _A\). Here, of course, \(c^*(e)\) are the deterministic front speeds discussed above, and establishing their existence forms the bulk of our work.
One can show that if \(c^*:{\mathbb {S}}^{d-1}\rightarrow (0,\infty )\) is Lipschitz (which will be our case), then for any open \(A\subseteq {\mathbb {R}}^d\), there is an open set \(\Theta ^{A, c^*}\subseteq (0,\infty )\times {\mathbb {R}}^d\) such that the unique solution to (1.4) with initial data \(\chi _A\) is \({\bar{u}}:=\chi _{\Theta ^{A, c^*}}\). In fact, this set can also be found from the formula
where \(v_{0}:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) is any Lipschitz function satisfying \(v_0>0\) on A and \(v_0<0\) on \({\mathbb {R}}^d{\setminus } {\overline{A}}\), and v is the unique (continuous) viscosity solution to (1.4) with \(v(0,\cdot )=v_{0}\). The open set \(\Theta ^{A, c^{*}}\) is then independent of the choice of \(v_0\) as above, and \(\partial \Theta ^{A, c^{*}}\) has zero measure.
All these claims are contained in Theorem 5.3 in [16], which is a combination of results by Barles, Soner, and Souganidis [4], Crandall, Ishii, and Lions [7], Souganidis [21], and Soravia [20]. The reader can also consult Definition 5.1 in [16] for the definition of viscosity solutions to initial value problems for (1.4).
We also note that it was shown in the proof of Theorem 1.4(iii) in [16] that for any convex open \(A\subseteq {\mathbb {R}}^d\) we have the explicit formula
In particular, if \(A=\{x\in {\mathbb {R}}^d\,|\,x\cdot e<0\}\) is the half-space with outer normal e, then we obviously have \(\Theta ^{A, c^*}=\{(t,x)\in (0,\infty )\times {\mathbb {R}}^d\,|\,x\cdot e< c^*(e)t\}\). This also shows that if we let \((\Theta ^{A, c^*})_t\) be the spatial slice of \(\Theta ^{A, c^*}\) at the time \(t>0\), then for any open bounded A we have
(e.g., in the sense of Hausdorff distances of boundaries of sets). Hence the set on the right-hand side is the Wulff shape for (1.4), and therefore also for (1.1) if homogenization holds.
We will consider here stationary ignition reactions that either have finite ranges of dependence, or can be uniformly approximated by such reactions (see Example 1.5 below for a simple example of the latter). These properties are summarized in the following definition and in hypothesis (H1) below.
Definition 1.1
Consider a probability space \((\Omega ,{{\mathcal {F}}},{\mathbb {P}})\) that is endowed with a group of measure-preserving bijections \(\{{\Upsilon _y:\Omega \rightarrow \Omega }\}_{y\in {\mathbb {R}}^d}\) such that for all \(y,z\in {\mathbb {R}}^d\) we have
A reaction function \(f:{\mathbb {R}}^d\times [0,1]\times \Omega \rightarrow [0,\infty )\), uniformly continuous in the first two arguments and with the random variables \(X_{x,u} :=f(x,u,\cdot )\) being \({{\mathcal {F}}}\)-measurable for all \((x,u)\in {\mathbb {R}}^d\times [0,1]\), is called stationary if for each \((x,y,u,\omega )\in {\mathbb {R}}^{2d}\times [0,1]\times \Omega \) we have
The range of dependence of such f is the infimum of all \(r\in {\mathbb {R}}^+\cup \{\infty \}\) such that
for any \(U,V\subseteq {\mathbb {R}}^d\) with \(d(U,V)\geqq r\), where \({{\mathcal {E}}}(U)\) is the \(\sigma \)-algebra generated by the family of random variables \( \{X_{x,u} \,|\, (x,u)\in U\times [0,1]\}. \)
Remark
While stationary reactions with finite ranges of dependence are also stationary ergodic, we will not need to use this property here due to our quantitative approach. We note that although the main results in [16] apply to stationary ergodic reactions, that assumption is only needed to prove that all the deterministic (exclusive) front speeds for (1.1) exist and are strong (see Definition 6.1 below), which we instead prove in Sects. 3–6.
We will consider here stationary reaction functions \(f:{\mathbb {R}}^d\times [0,1]\times \Omega \rightarrow [0,\infty )\), and extend them to \({\mathbb {R}}^d\times {\mathbb {R}}\times \Omega \) by 0 whenever we need to evaluate them with \(u\notin [0,1]\). Additionally, our reactions will be of the ignition type. That is, we will assume the following hypothesis:
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(H1) The reaction f is stationary, Lipschitz in both x and u with constant \({M}\geqq 1\), and there are \(\theta _1\in (0,\frac{1}{2})\), \(m_1>1\), and \(\alpha _1>0\) such that \(f(\cdot ,u,\cdot )\equiv 0\) for \(u\in [0,\theta _1]\cup \{1\}\), \(f(\cdot ,u,\cdot )\geqq \alpha _1 (1-u)^{m_1}\) for \(u\in [1-\theta _1,1)\), and f is non-increasing in \(u\in [1-\theta _1,1)\).
It is not difficult to see that one cannot hope for general reactions satisfying (H1) to lead to homogenization for (1.1), even if f is independent of \((x,\omega )\) (see, e.g., [28, 29]). Indeed, if f is allowed to vanish at some intermediate value \(\theta '\in (\theta _1,1-\theta _1)\) and is also sufficiently large for some \(u\in (\theta _1,\theta ')\), solutions could easily form “plateaus” with values near \(\theta '\) (or another intermediate value) whose widths grow linearly in time. And if that happens, the widths of these plateaus will not vanish even after the scaling from (1.2) is applied.
To avoid this scenario, one should assume that as the argument u grows from 0 to 1 (for any fixed \((x,\omega )\)), the reaction f cannot become arbitrarily small (except near \(u=1\)) once it has become large enough. This is expressed in Definition 2.3 below, which was used in [29] to show that not only solutions to (1.1) do not develop such plateaus, but the transition from values \(u\sim 0\) to values \(u\sim 1\) in fact occurs over uniformly-in-time bounded distances in space (see, e.g., Lemma 2.4 below). Our most general results apply in this setting, as well as when one instead only assumes at most \(O(t^\alpha )\) growth of the above transition distances, with \(\alpha <1\) (see hypothesis (H2’) below).
However, for the sake of simplicity, in our first two results we will consider the case where the reaction does not become arbitrarily small (except near \(u=1\)) after it has become just positive. That is, once u has exceeded the ignition temperature
Of course, this is the case for any realistic model of combustion, where the reaction rate is positive at all temperatures above the ignition temperature (its vanishing at \(u=1\) is due to fuel exhaustion in systems of equations for temperature and concentration of the reactant, which in certain regimes simplify to (1.1) with \(f(\cdot ,1,\cdot )\equiv 0\)).
Definition 1.2
A reaction f satisfying (H1) is a stationary pure ignition reaction if for each \(\eta >0\) we have
Remark. This definition (with the bound for \(u\in [1-\theta _1,1)\) being \(\inf _{(x,\omega )} f(x,u,\omega ) >0\)) is from [28]. Note that it is trivially satisfied, for instance, when \(f(x,u,\omega )=g(x,\omega )F_0(u)\), with g bounded away from 0 and \(\infty \), Lipschitz in x, and stationary in \(\omega \), and with Lipschitz \(F_0:[0,1]\rightarrow [0,\infty )\) such that \(F_0=0\) on \([0,\theta _0]\cup \{1\}\) and \(F_0>0\) on \((\theta _0,1)\) for some \(\theta _0\in (0,1)\), and \(F_0\) is non-increasing and bounded below by \(\alpha _1 (1-u)^{m_1}\) near 1 (for some \(m_1,\alpha _1\)).
We will therefore start by assuming the following hypothesis:
-
(H2) f is a stationary pure ignition reaction and \(d \leqq 3\).
The additional restriction \(d\leqq 3\) is necessitated by the above-mentioned surprising result from [29], where the second author showed that even for pure ignition reactions, transition from values \(u= \eta \) to values \(u= 1-\eta \) may only occur over linearly-in-time growing distances for solutions to (1.1) and all small \(\eta >0\) in dimensions \(d\geqq 4\) (while these distances remain bounded in dimensions \(d\leqq 3\)).
We are now ready to state our first main homogenization result. In it and later we use the notation \(B_r(A):=A+(B_r(0)\cup \{0\})\) and \(A^0_r:=A\backslash \overline{B_r(\partial A)}\) for \(A\subseteq {\mathbb {R}}^d\) and \(r\geqq 0\). For the sake of generality, we also allow O(1) shifts and o(1) errors in initial data as \(\varepsilon \rightarrow 0\) in (1.3).
Theorem 1.3
If f satisfying (H2) has a finite range of dependence, then there is Lipschitz \(c^*:{\mathbb {S}}^{d-1}\rightarrow (0,\infty )\) such that the following holds for any open \(A\subseteq {\mathbb {R}}^d\) and \(\Theta ^{A, c^*}\) from (1.5). If \(\Lambda >0\), and for all \(\omega \in \Omega \) and \(\varepsilon >0\), the function \(u_\varepsilon (\cdot ,\cdot ,\omega )\) solves (1.3) and satisfies
for some \(y_\varepsilon \in B_{\Lambda }(0)\) and some \(\psi \) with \(\lim _{\varepsilon \rightarrow 0} \psi (\varepsilon )=0\) (when \(y_\varepsilon =0\) and \(\psi (\varepsilon )= 0\), this becomes just \((1-\theta _1)\chi _{A}\leqq {u_\varepsilon }(0,\cdot ,\omega ) \leqq \chi _{A}\)), then for almost all \(\omega \in \Omega \) we have
locally uniformly on \(([0,\infty )\times {\mathbb {R}}^d){\setminus } \partial \Theta ^{A, c^*}\).
Remark
Our proofs use results from [16] which in fact show that in all our main results, \(1-\theta _1\) in (1.6) can be replaced by any \(\theta \) satisfying \(\inf _{(x,u,\omega )\in {\mathbb {R}}^d\times [\theta ,1-\theta _1]\times \Omega } f(x,u,\omega )>0\).
We next extend this to the case of reactions with infinite ranges of dependence that are uniform limits of reactions with finite ranges of dependence. Here we will also require some uniform decay of f near \(u=1\). This is the content of the next two hypotheses.
-
(H3) There are \(m_3\geqq 1\) and \(\alpha _3>0\) such that for all \(\eta \in (0,\frac{1}{2}\theta _1]\) we have
$$\begin{aligned} \inf _{\begin{array}{c} (x,\omega )\in {\mathbb {R}}^d\times \Omega \\ u\in [1- \theta _1/2,1] \end{array}}\left( f(x,u-\eta ,\omega )-f(x,u,\omega )\right) \geqq \alpha _3 \eta ^{m_3}. \end{aligned}$$ -
(H4) There are \(m_4,n_4,\alpha _4>0\) such that for each \(n\geqq n_4\), there exists a stationary reaction \(f_n\) with range of dependence \(\leqq n\) and \(\Vert f_n-f\Vert _\infty \leqq \alpha _4 {n^{-m_4}}\).
Theorem 1.4
Theorem 1.3 holds for any f satisfying (H2)–(H4).
While this result does not cover all interesting random pure ignition reactions with correlations of \(f(x,u,\cdot )\) and \(f(y,v,\cdot )\) decreasing as \(|x-y|\rightarrow \infty \) (for all \(u,v\in [0,1]\)), it does apply to many of them. Here is a simple such example.
Example 1.5
Let \(d\leqq 3\), \(F_0:[0,1]\rightarrow [0,\infty )\) be Lipschitz with \(F_0=0\) on \([0,\theta _0]\cup \{1\}\) and \(F_0>0\) on \((\theta _0,1)\) for some \(\theta _0\in (0,1)\), and \(F_0'(u)\leqq -(1-u)^m\) near \(u=1\) for some m. Also pick some Lipschitz \(g:{\mathbb {R}}^d\rightarrow [0,\infty )\) with \(\sup _{x\in {\mathbb {R}}^d} |x|^{m'}g(x)<\infty \) for some \(m'>0\), and some Lebesgue measurable \(a:[0,1]\rightarrow [0,1]\). Consider the product probability space \(\Omega =[0,1]^{{\mathbb {Z}}^d}\), with the Lebesgue measure on each copy of [0, 1], and for any \(k\in {\mathbb {Z}}^d\), denote by \(\omega _k\) the \(k^\mathrm{th}\) coordinate of \(\omega \in \Omega \) (note that these are i.i.d. random variables). Let \(\Upsilon _y:\Omega \rightarrow \Omega \) for \(y\in {\mathbb {Z}}^d\) be given by \((\Upsilon _y\omega )_k:=\omega _{y+k}\) for all \(k\in {\mathbb {Z}}^d\). Then the random reaction strength model
satisfies (H2)–(H4) (see next paragraph for stationarity), with \(f_n\) defined as f but with g replaced by \(g_n(x):=g(x)\min \{1,d(x,{\mathbb {R}}^d{\setminus } B_{n/2}(0))\}\). Hence Theorem 1.4 applies. Note that f may have infinite range of dependence when g is not compactly supported.
Note also that while this f is stationary only with respect to integer shifts (i.e., \(y\in {\mathbb {Z}}^d\) in Definition 1.1), such settings can be easily transformed to the case considered in the present paper by letting \({\tilde{\Omega }}:=\Omega \times [0,1)^d\) with the product measure, \({\tilde{f}}(x,u,(\omega ,z)):= f(x+z,u,\omega )\), and \({\tilde{\Upsilon }}_y(\omega ,z):=(\Upsilon _{\lfloor y+z \rfloor }\omega , \{y+z\})\) for \(y\in {\mathbb {R}}^d\). Since inclusion of \(y_\varepsilon \) in (1.6) shows that all our main results continue to hold if we replace the identified full-measure set \(\tilde{\Omega }'\subseteq {\tilde{\Omega }}\) by \(\bigcup _{y\in {\mathbb {R}}^d} \tilde{\Upsilon }_y{\tilde{\Omega }}'\), which is of the form \(\Omega '\times [0,1)^d\), they then also apply in integer-shift settings.
In the above example and in Theorem 1.4, reactions f with infinite ranges of dependence are uniform limits of those with finite ranges of dependence. The next example is a natural situation when this need not be the case, a reaction–diffusion analog of the setting where sticks of random unbounded lengths are randomly positioned in \({\mathbb {R}}^d\). While Theorem 1.4 does not apply here, one can instead use its generalization, Theorem 1.7 below, which allows this.
Example 1.6
Consider the setting from Example 1.5, without the functions g and a. Instead pick some uniformly bounded and uniformly Lipschitz \(g_j:{\mathbb {R}}^d\rightarrow [0,\infty )\) (\(j\in {\mathbb {N}}\)) that satisfy \(\sup _{j\in {\mathbb {N}}} \sup _{|x|> j} |x|^{m'}g_j(x)<\infty \) for some \(m'>0\), and some Lebesgue measurable \(a:[0,1]\rightarrow {\mathbb {N}}\) with \(|a^{-1}(j)|\leqq j^{-\gamma }\) for some \(\gamma >3d+2\) and all \(j\in {\mathbb {N}}\). Then
satisfies (H3) and (H4’) below (see Example 1.5 for stationarity), with \(f_n\) defined as f but with \(g_j\) replaced by \(g_{j,n}(x):=g_j(x) \min \{1,2d(x,{\mathbb {R}}^d{\setminus } B_{n/2}(0))\}\). This uses the fact that
for some \(m_4'>0\) (because \(\gamma >3d+2\)), with the left-hand side being (up to a constant factor) an upper bound on the probability that \(a(\omega _k)\geqq \max \{\frac{n}{2}, |k|-n^{1+m_4'}\}\) for at least one \(k\in {\mathbb {Z}}^d\). Hence Theorem 1.7 below applies. Note that f need not be a uniform limit of reactions with finite ranges of dependence when the functions \(g_j\) do not decay uniformly to 0 as \(x\rightarrow \infty \).
1.2 Generalizations
As we indicated above, it is not clear whether the limitation on the dimension in (H2) is necessary to obtain a sufficiently general result. However, since both conditions in (H2) are only needed to guarantee certain estimates for some special solutions to (1.1) (see Lemma 2.4 below), including that the transition from values \(u\sim 0\) to values \(u\sim 1\) occurs over spatial distances that grow only sub-linearly in time, as we mentioned above (Lemma 2.4 shows that in the case of (H2) these distances are in fact uniformly bounded), we can extend our results to more general settings as long as these estimates still hold there. In particular, this might be the case for stationary ignition reactions in dimensions \(d\geqq 4\).
In order to state this alternative to hypothesis (H2), let us define for any \(0<\eta<\theta <1\) the width of the transition zone from \(\eta \) to \(\theta \) for a solution \(u:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) to (1.1) at some time \(t\geqq 0\) to be (see [29])
The special solutions for which we need to assume certain bounds on these quantities will be essentially those evolving from characteristic functions of the balls \(B_k(0)\subseteq {\mathbb {R}}^d\), with \(k\in {\mathbb {N}}\).
It will however be more convenient to work with approximations \(u_{0,k}\) of these characteristic functions that have two useful properties. First, they are close to 1 on \(B_k(0)\) but are strictly below 1 (which will allow us to treat general initial data from (1.6)), and are supported on \(B_{k+R_0}(0)\) for some fixed \(R_0\). Specifically, we will require that
holds with \(\theta ^*>0\) from (2.1) below. We note that we could in fact replace \(1-\theta ^*\) in (1.8) by any \(\theta <1\) satisfying \(\inf _{(x,u,\omega )\in {\mathbb {R}}^d\times [\theta ,1-\theta _1]\times \Omega } f(x,u,\omega )>0\), but we make our choice for the sake of convenience (Lemma 2.1 shows that solutions \(u:(0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) with \(u(t,x)\geqq 1-\theta ^*\) for some \((t,x)\in [1,\infty )\times {\mathbb {R}}^d\) converge locally uniformly to 1). The second property is that the corresponding solutions to (1.1) satisfy \(u_t>0\). For this, it suffices to have
with \(F(u):=\inf _{(x,\omega )\in {\mathbb {R}}^d\times \Omega } f(x,u,\omega )\), which yields \(\Delta u_{0,k}+f(\cdot , u_{0,k},\omega )\geqq 0\) for any \(\omega \in \Omega \). Then \(u_t> 0\) follows for the corresponding solution u at all positive times because \(v:=u_t\) solves the linear equation \(v_t=\Delta v+f_u(x,u(t,x),\omega )v\) with \(v(0,\cdot )\geqq 0\) and \(v(0,\cdot )\not \equiv 0\) (due to (1.8) and \(F(1-\theta ^*)>0\)).
It is easy to construct radial functions satisfying (1.8) and (1.9), since then (1.9) becomes a simple ordinary differential inequality. (This is in fact possible for any set \(S\subseteq {\mathbb {R}}^d\), without radial symmetry but still with a uniform \(R_0\), and we do so in Lemma 2.2 below.) Let us now pick one such \(u_{0,k}\) for each \(k\in {\mathbb {N}}\) (any one can be chosen), and denote by \({{\mathcal {U}}}_f\) the set of all solutions u to (1.1) obtained by choosing any \(\omega \in \Omega \) and initial data \(u(0,\cdot )=u_{0,k}\) for any \(k\in {\mathbb {N}}\). We can now replace (H2) by
-
(H2’) f satisfies (H1) and there are \(\alpha _2<1\) and \({m_2}>0\) such that
$$\begin{aligned} \begin{aligned}&\limsup _{t\rightarrow \infty }\sup _{u\in {{\mathcal {U}}}_{f}}\sup _{\eta>0} \,\frac{L_{u,\eta ,1-\theta ^*}(t)}{t^{\alpha _2} \eta ^{-{m_2}}} <\infty ,\\&\liminf _{t\rightarrow \infty } \inf _{u\in {{\mathcal {U}}}_{f}} \inf _{ u(t,x)\in [\theta ^*,1-\theta ^*]} \, u_t(t,x) >0. \end{aligned} \end{aligned}$$(1.10)Here \({{\mathcal {U}}}_f\) is as above, with some \(u_{0,k}\) satisfying (1.8) and (1.9) for each \(k\in {\mathbb {N}}\), and \(\theta ^*=\theta ^*(M,\theta _1,m_1,\alpha _1)\) from (2.1) and \(R_0=R_0(M,\theta _1,m_1,\alpha _1)\) are independent of k.
Remark
1. The first statement in (1.10) allows \(L_{u,\eta ,1-\theta ^*}(t)\) to grow algebraically in both \(\eta \rightarrow 0\) and \(t\rightarrow \infty \) (note that \(\alpha _2<1\) is critical here because the scaling from (1.2) yields \(L_{u_{\varepsilon },\eta ,1-\theta ^*}(t)=\varepsilon L_{u,\eta ,1-\theta ^*}\left( \varepsilon ^{-1} {t}\right) \), which will then vanish on any bounded time interval as we take \(\varepsilon \rightarrow 0\)). We note that Lemma 2.4 below shows that in the case of (H2), the former growth is only logarithmic while the latter is non-existent.
2. Lemma 2.4 shows that the second statement in (1.10) holds as well if one assumes (H2) (recall also that all \(u\in {{\mathcal {U}}}_f\) satisfy \(u_t> 0\)). Nevertheless, we will further weaken this hypothesis in Theorem 1.8 below.
3. We could also replace \(B_k(0)\) and \(B_{k+R_0}(0)\) in (1.8) by \(B_{r_k}(0)\) and \(B_{r_k+R_0}(0)\) for any sequence \(r_k\rightarrow \infty \), without any change to our results.
After replacing (H2) by (H2’), we must also adjust (H4) in the extension of Theorem 1.4, in order to ensure that the reactions \(f_n\) will satisfy (H2’) with uniform constants. Note that when (H2) holds, we will show in Corollary 2.7 that \(f_n\) from (H4) can be perturbed so that this is the case, but we do not know whether this remains true when we only assume (H2’).
In addition, we also state this new version of (H4) so that it applies to some f that are not uniform limits of reactions with finite ranges of dependence (see Example 1.6 above).
-
(H4’) There are \(m_4,m_4',n_4,\alpha _4>0\) such that for each \(n\geqq n_4\), there exists a stationary reaction \(f_n\) with range of dependence \(\leqq n\) and
$$\begin{aligned} {\mathbb {P}}\left( \sup _{|x|< n^{1+m_4'}} \, \sup _{u\in [0,1]} |f_n(x,u,\cdot ) - f(x,u,\cdot ) | > \alpha _4 {n^{-m_4}}\right) \leqq n^{-(2d+1+m_4')}. \end{aligned}$$Moreover, (H2’) holds uniformly in n (i.e., reactions \(f_n\) satisfy (H1) with the same \(M,\theta _1,m_1,\alpha _1\), and (1.10) with \({{\mathcal {U}}}_{f}\) replaced by \(\bigcup _{n\geqq n_4}{{\mathcal {U}}}_{f_n}\)).
We note that the initial data \(u_{0,k}\) used in the definition of \({{\mathcal {U}}}_{f_n}\) are in principle allowed to be different for distinct n (but \(\theta ^*\) and \(R_0\) are uniform in n; also \(r_k\) in Remark 3 above).
Theorem 1.7
Theorem 1.3 holds for any f that either satisfies (H2’) and has a finite range of dependence, or satisfies (H3) and (H4’).
Finally, we show that one can also allow a power decay in time in the second statement in (1.10), at the expense of either having to extend this assumption to a slightly larger family of special solutions or obtaining the result for a smaller family of initial data.
For each \(a\in [0,\frac{1}{2}\theta ^*]\), let \({{\mathcal {U}}}_{f,a}\) be defined as \({{\mathcal {U}}}_f\) above, but with (1.8) replaced by
for initial data denoted \(u_{0,k,a}\) instead of \(u_{0,k}\) (so now \(u_{0,k,a}-a\) is supported in \(B_{k+R_0}(0)\)). Obviously \({{\mathcal {U}}}_{f,0}={{\mathcal {U}}}_f\), and one can find such initial data (for any \(S\subseteq {\mathbb {R}}^d\) and with \(R_0\) uniform in a) via Lemma 2.2 with \(\alpha _1\) replaced by \(\alpha _1(1-\frac{1}{8}{\theta _1})^{m_1-1}\) (since \(a\leqq \frac{1}{8}\theta _1\)) and then applying the scaling \(u_{0,S,a}:=(1-a)u_{0,S}+a\).
We can now replace (H2’) and (H4’) by
-
(H2”) f satisfies (H1) and there are \(\alpha _2<1\), \(m_2> 0\), \(a_2\in [0,\frac{1}{2}\theta ^*]\), and \(\alpha _2'<\min \{ \frac{1}{m_1-1}, \frac{1-\alpha _2}{m_2} \}\) such that
$$\begin{aligned} \begin{aligned}&\limsup _{t\rightarrow \infty } \sup _{a\in [0,a_2]} \sup _{u\in {{\mathcal {U}}}_{f,a}} \sup _{\eta>0}\frac{L_{u,\eta +a,1-\theta ^*}(t)}{t^{\alpha _2} \eta ^{-{m_2}}}<\infty , \\&\liminf _{t\rightarrow \infty } \inf _{a\in [0,a_2]} \inf _{u\in {{\mathcal {U}}}_{f,a}} \inf _{ u(t,x)\in [\theta ^*,1-\theta ^*]} \, u_t(t,x)\,t^{\alpha _2'}>0. \end{aligned} \end{aligned}$$(1.12)Here \({{\mathcal {U}}}_{f,a}\) is as above, with some \(u_{0,k,a}\) satisfying (1.11) and (1.9) for each \(k\in {\mathbb {N}}\), and \(\theta ^*=\theta ^*(M,\theta _1,m_1,\alpha _1)\) from (2.1) and \(R_0=R_0(M,\theta _1,m_1,\alpha _1)\) are independent of k.
-
(H4”) f satisfies (H4’) with (H2”) in place of (H2’), and also \(\alpha _2'<\frac{m_4}{m_3}\).
Of course, these hypotheses coincide with (H2’) and (H4’) when \(a_2=0=\alpha _2'\). With them, we can now state our second generalization of Theorems 1.3 and 1.4.
Theorem 1.8
Assume that f either satisfies (H2”) and has a finite range of dependence, or satisfies (H3) and (H4”).
-
(i)
If \(a_2>0\), then Theorem 1.3 holds for such f.
-
(ii)
If \(a_2=0\), then Theorem 1.3 holds for such f with A convex and (1.6) replaced by
$$\begin{aligned} (1-\theta _1)\chi _{A^0_{\psi (\varepsilon )}}\leqq {u_\varepsilon }(0,\cdot +y_\varepsilon ,\omega )\leqq (1-\Lambda ^{-1})\chi _{B_{\psi (\varepsilon )}(A)}\,. \end{aligned}$$
1.3 Organization of the Paper and Acknowledgements
In Sect. 2 we collect most important notation and prove several preliminary results. These include Corollary 2.7, which shows that Theorems 1.3 and 1.4 follow from Theorem 1.7. It will therefore suffice to prove Theorems 1.7 and 1.8. We prove the first one in Sect. 6, after obtaining crucial quantitative estimates on long-time dynamics of solutions to (1.1) in Sects. 3–5 (specifically, Propositions 3.8, 4.2, and 5.1, with the first two of these being essentially the same result but assuming (H2’)+finite range in the first and (H3)+(H4’) in the second). In Sect. 7 we then show how to extend all these results to the cases considered in Theorem 1.8.
2 Preliminaries and Notation
In this section we collect some previous results and preliminary lemmas, all of which hold uniformly in \(\omega \) and without needing to assume stationarity of the reaction. We will therefore use the following hypothesis.
-
(H1’) f satisfies (H1) except possibly the stationarity hypothesis.
At the end of the section we also collect all the important notations in one place.
Let us start with a basic lower bound (see, e.g., [29, Lemma 3.1]), which shows that general solutions to (1.1) propagate with speed no less than some \(c_0=c_0(M,\theta _1,m_1,\alpha _1)>0\). We will choose this to be the unique front speed for the homogeneous reaction \(F_0:[0,1]\rightarrow [0,\infty )\) defined to be the largest M-Lipschitz function with \(F_0(u)\leqq \alpha _1(1-u)^{m_1}\chi _{[1-\theta _1,1]}(u)\) (so clearly \(F_0\leqq F\)).
Hence \(c_0\) is the unique number such that the PDE \(u_t=u_{xx} + F_0(u)\) in one space dimension has a traveling front solution \(u(t,x)=U(x-c_0t)\) with \(U(-\infty )=1\) and \(U(\infty )=0\).
Lemma 2.1
There exists \(\theta _2=\theta _2({M},\theta _1,m_1,\alpha _1)<1\) such that for each \(c<c_0\) and \(\theta <1\), there is \(\kappa _0=\kappa _0({M},\theta _1,m_1,\alpha _1,c,\theta )\geqq 1\) such that the following holds. If \(u:(0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) is a solution to (1.1) with f satisfying (H1’) and with some \(\omega \in \Omega \), and if \(u(t_0,y)\geqq \theta _2\) for some \(t_0\geqq 1\) and \(y\in {\mathbb {R}}^d\), then for all \(t\geqq t_0+\kappa _0\),
If we also have that \(u_t\geqq 0\), then this clearly holds with any \(t_0\geqq 0\) (and \(\kappa _0\) increased by 1).
Now let
where \(\theta _2=\theta _2({M},\frac{1}{2}\theta _1,m_1,\alpha _1(1-\frac{1}{8}{\theta _1})^{m_1-1})<1\).
Remark
Addition of the factors \(\frac{1}{2}\) and \((1-\frac{1}{8}{\theta _1})^{m_1-1}\) here is due to the scaling \(u\mapsto (1-a)u+a\) mentioned before (H2”), as we shall see in Sect. 7. All arguments before Proposition 7.4 will only require \(\theta _2=\theta _2({M},\theta _1,m_1,\alpha _1)\) here, and also only that \(\theta ^*\leqq \frac{1}{2}\min \{1-\theta _2,\,\theta _1\}\). So we could define \(\theta ^*\) this way in Theorem 1.7 and its proof.
In the rest of the paper we will primarily use Lemma 2.1 with \(c=\frac{c_0}{2}\) and \(\theta =1-\theta ^*\), and we will therefore define
Having defined this \(\theta ^*\), let us next construct the initial data \(u_{0,S}\) from the introduction, which are perturbations of the functions \(\chi _S\) that also satisfy (1.9).
Lemma 2.2
There is \(R_0=R_0(M,\theta _1,m_1,\alpha _1)\geqq 1\) such that for any f satisfying (H1’) and \(S\subseteq {\mathbb {R}}^d\), there is a smooth function \(u_{0,S}\) satisfying (1.9) and
Remark
Recall that (1.9) implies that the relevant solutions to (1.1) satisfy \(u_t>0\). Moreover, (2.3) yields the uniform bound \(L_{u,\eta ,1-\theta ^*}(0)\leqq R_0\) for all \(\eta \in (0,1-\theta ^*)\) and \(S\subseteq {\mathbb {R}}^d\), which is relevant for the next result.
Lemma 2.2 is proved in “Appendix A”.
Let us now turn to the consequences of (H2) obtained in [29]. In fact, these results hold for the following more general classes of functions.
Definition 2.3
For \(M,\theta _1,m_1,\alpha _1\) from (H1) (and \(F_0\) defined above), and for any \(\zeta ,\xi >0\), let \({{\mathcal {F}}}(F_0,M,\theta _1,\zeta ,\xi )\) be the class of all f satisfying (H1’) such that
where (with the convention \(\inf \emptyset =\infty \))
Remark
1. Although we could instead write \({{\mathcal {F}}}(M,\theta _1,m_1,\alpha _1,\zeta ,\xi )\), we use notation from [29].
2. Note that pure ignition reactions belong to \(\bigcup _{\xi >0} {{\mathcal {F}}}(F_0,M,\theta _1,\zeta ,\xi )\) for any \(\zeta >0\).
It was shown in [29] that if \(d\leqq 3\) and f is from the class \({{\mathcal {F}}}(F_0,M,\theta _1,\zeta ,\xi )\) for some \(F_0,M,\theta _1,\xi \) and \(\zeta <c_0^2/4\), then transitions from values \(u\sim 0\) to values \(u\sim 1\) for fairly general solutions to (1.1) occur over uniformly-in-time bounded distances. In view of our interest in solutions from \({{\mathcal {U}}}_f\), with initial data satisfying Lemma 2.2, the following result will be relevant:
Lemma 2.4
Let \(d\leqq 3\), let \(F_0,M,\theta _1\) be as in Definition 2.3 and \(\theta ^*\) from (2.1), and consider any \(\xi >0\) and \(\zeta <c_0^2/4\). There is \(\Lambda >0\) and for any \(\eta \in (0,\frac{1}{2})\) there are \(\mu _\eta ,\kappa _\eta >0\) such that if \(f\in {{\mathcal {F}}}(F_0,M,\theta _1,\zeta ,\xi )\) and u solves (1.1) with some \(\omega \in \Omega \) and initial data satisfying Lemma 2.2 for some \(S\subseteq {\mathbb {R}}^d\), then
and for any \(\eta >0\) we have
Proof
Recall that we have \(L_{u,\eta ,1-\theta ^*}(0)\leqq R_0\) for all \(\eta \in (0,1-\theta ^*)\). We then obtain (2.5) from [29, Theorem 2.5(i)] with \((\varepsilon ',\varepsilon )=(1-\theta ^*,\eta )\) (we can choose there \(\varepsilon _0=\theta ^*\), and then \(\varepsilon '=1-\varepsilon _0\)), with independence of \(\kappa _\eta \) on u due to Remark 1 after the theorem.
To obtain (2.4), we instead use (4.14) in [29] with \((\varepsilon _0,h,t_0)=(\theta ^*,0,0)\) (where \(\varepsilon _0\) was used to define the function \(Z_y\) in (4.14)). Then the bound \(Z_y(t_0)-Y_y^h(t_0)\leqq R_0\) for all \(y\in {\mathbb {R}}^d\) follows from \(L_{u,\eta ,1-\theta ^*}(0)\leqq R_0\) for all \(\eta \in (0,1-\theta ^*)\), so (4.14) yields \(Z_y(t)-Y_y^h(t)\leqq \lambda \) for all \((t,y)\in [0,\infty )\times {\mathbb {R}}^d\) and some \(\lambda =\lambda (F_0,M,\theta _1,\zeta ,\xi )\). The definition of \(Y_y^h(t)\) shows that if \(u(t,y)\geqq \eta \) for some (t, y), then \(Y_y^h(t)\leqq \psi ^{-1}(\frac{1}{\eta })\), with \(\psi ''(r)+\frac{d-1}{r} \psi '(r)=\zeta \psi (r)\) on \((0,\infty )\) and \(\psi (0)=1\) (hence we have \(\lim _{r\rightarrow \infty } r^{(d-1)/2} e^{-\sqrt{\zeta }r} \psi (r)\in (0,\infty )\)). Therefore \(\psi (r)\geqq e^{\sqrt{\zeta }r/2}\) for all large enough r, so \(Z_y(t)\leqq \lambda + \frac{2}{\sqrt{\zeta }} |\ln \eta |\) for all small enough \(\eta \). But since \(Z_y(t)\) is the distance from y to the nearest point x with \(u(t,x)\geqq 1-\varepsilon _0=1-\theta ^*\), we obtain \(L_{u,\eta ,1-\theta ^*}(t)\leqq \lambda + \frac{2}{\sqrt{\zeta }} |\ln \eta |\) for all \(t\geqq 0\) and all small enough \(\eta >0\). This yields (2.4) with some \(\Lambda =\Lambda (F_0,M,\theta _1,\zeta ,\xi )\). \(\quad \square \)
The next result is a counterpart of Lemma 2.1 (see [16, Lemma 2.2]). It shows that the speed of propagation of perturbations of solutions to (1.1) is bounded above by \(2 \sqrt{Md\,}\) (in fact, the bound \(2\sqrt{M}\) works as well, but we will not need it here).
Lemma 2.5
Let \(r>0\) and \(y\in {\mathbb {R}}^d\), and let \(u_1:[0,\infty )\times B_r(y)\rightarrow (-\infty ,1]\) be a subsolution and \(u_2:[0,\infty )\times B_r(y)\rightarrow [0,\infty )\) a supersolution to (1.1) with some f satisfying (H1’) and some \(\omega \in \Omega \). If \(u_1(0,\cdot )\leqq u_2(0,\cdot )\) on \(B_r(y)\), then for all \((t,x)\in [0,\infty )\times B_r(y)\) we have
Remark
This was stated in [16] with \(u_1,u_2\) having values in [0, 1] only, but the proof applies to our case without change.
This yields the following corollary (as above, \(c_1\) in this result could be just \(2\sqrt{M\,}\)):
Corollary 2.6
If \(u:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solves (1.1) with some f satisfying (H1’) and some \(\omega \in \Omega \), then for any \(t\geqq 0\) we have
where
Proof
The claim \(c_1>c_0\) follows from the well-known estimate \(c_0<2\sqrt{M}\) for any M-Lipschitz ignition reaction \(F_0\). If \(t\geqq 0\) and \(y\notin B_{c_1t+\kappa _1} ( \{ x\,|\, u(0,x)\geqq \theta _1\} )\), then Lemma 2.5 with \(u_1=u\), \(u_2 \equiv \theta _1\), and \(r=c_1t+\kappa _1\) yields
finishing the proof. \(\quad \square \)
We can now use these results to show that Theorems 1.3 and 1.4 follow from Theorem 1.7, so it will suffice to prove the latter (and then Theorem 1.8). This also means that we will assume either (H2’) or (H4’) in the rest of Sects. 2–6. Hence, then there will be \(\mu _*,\kappa _*>0\) such that for any \(u\in {{\mathcal {U}}}_f\) or \(u\in \bigcup _{n\geqq n_4}{{\mathcal {U}}}_{f_n}\), respectively, we have
Notice that while (1.10) only allows us to state the first of these claims for \(t\ge \kappa _*\), one can extend this to all \(t\geqq 0\) (with a different \(\mu _*\)). This is because Lemma 2.5 with \(x=y\) shows that if \(u(t,y)\ge \eta \) for some \(u\in {{\mathcal {U}}}_f\) (or \(u\in \bigcup _{n\geqq n_4}{{\mathcal {U}}}_{f_n}\)), \((t,y)\in [0,\kappa ^*]\times {\mathbb {R}}^d\), and \(\eta >0\), then
(when \(u(0,\cdot )=u_{0,k}\)), so \(L_{u,\eta ,1-\theta ^*}(t)\) satisfies the same upper bound because \(u_t\geqq 0\). We note that we could also include \(t\in [0,\kappa _*)\) in the second claim of (2.6), at the expense of some extra work, but this would not be as useful.
Corollary 2.7
(H2) implies (H2’). Also, (H2) and (H4) imply (H4’) for some sequence of reactions \(g_n\) in place of \(f_n\), with possibly different \(M,\theta _1,n_4,\alpha _4\) and with \(m_4'=\infty \).
Proof
Remark 2 after Definition 2.3 and Lemma 2.4 show that (H2) implies (H2’) with \(\alpha _2=0\) and any \(m_2>0\).
Let us now assume (H2) and (H4), and with the convention that \([a,b]=\emptyset \) if \(a>b\), let
These reactions are non-increasing in \(u\in [1-\theta _1,1]\), and still satisfy \(\Vert {\tilde{f}}_n-f\Vert _\infty \leqq {\alpha _4 n^{-m_4}}\), because f is non-increasing in \(u\in [1-\theta _1,1]\). Also, each \({\tilde{f}}_n\) is obviously stationary with range of dependence no greater than that of \(f_n\).
Next, let \(\phi :{\mathbb {R}}^{d+1}\rightarrow [0,\infty )\) be a smooth function supported in \(B_1(0)\) and with integral over \(B_1(0)\) equal to 1. Then let \(\phi _n(x,u):=n^{(d+1)m_4}\phi (n^{m_4} x,n^{m_4} u)\), and consider \(\tilde{f}_n*\phi _n\) (with the convolution in (x, u); recall that all reactions are extended by 0 to \(u\notin [0,1]\)). With \(\nabla \) being either \(\nabla _x\) or \(\partial _u\), we then have
Finally, recall F from (1.9) and let \(\bar{F}(u):=\sup _{(x,\omega )\in {\mathbb {R}}^d\times \Omega } f(x,u,\omega )\), and
It is not hard to see that for all large enough n, these functions satisfy (H1) with \((M,\theta _1)\) replaced by \((M+ \alpha _4 \Vert \nabla _x\phi \Vert _1,\frac{1}{2} {\theta _1})\). We also have \(\Vert \tilde{g}_n-f\Vert _\infty \leqq (2M+\alpha _4) n^{-m_4}\) for all large enough n because \(\Vert {\tilde{f}}_n-f\Vert _\infty \leqq \alpha _4 {n^{-m_4}}\) and \(F\leqq f(x,\cdot ,\omega )\leqq {\bar{F}}\) for all \((x,\omega )\), and the range of dependence of \({\tilde{g}}_n\) is at most \(n+2n^{-m_4}\leqq n+2\).
Since \(f\in {{\mathcal {F}}}(F_0,M,\theta _1,\zeta ,\xi )\) for some \(\zeta <{c_0^2}/{4}\) and \(\xi >0\), \(\Vert {\tilde{g}}_n-f\Vert _\infty \leqq (2M+\alpha _4) n^{-m_4}\) yields \({\tilde{g}}_n\in {{\mathcal {F}}}(F_0,M+\alpha _4 \Vert \nabla _x\phi \Vert _1,\frac{1}{2}\theta _1,\frac{1}{8}(4\zeta +c_0^2),\frac{1}{2}\xi )\) for all large enough n. Therefore, as at the start of this proof, we obtain that the \({\tilde{g}}_n\) satisfy (H2’) uniformly in n, for all large enough n. Hence, the reactions \(g_n:={\tilde{g}}_{n-2}\) satisfy (H4’) with possibly different \(M,\theta _1,n_4,\alpha _4\) and with \(m_4'=\infty \). \(\quad \square \)
The next result uses Lemma 2.5 and the bound \(f(\cdot ,u,\cdot )\geqq \alpha _1(1-u)^{m_1}\) for u near 1 to essentially obtain an upper bound on \(\kappa _0(M,\theta _1,m_1,\alpha _1,\frac{c_0}{4},\theta )\) from Lemma 2.1 as \(\theta \rightarrow 1\) (we could similarly do this for any \(c<c_0\) in place of \(\frac{c_0}{4}\)).
Lemma 2.8
Let \(u:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solve (1.1) with f satisfying (H1’) and some \(\omega \in \Omega \). There is \(D_1=D_1(M,\theta _1,m_1,\alpha _1)\) such that if \(u(t_0,y)\geqq 1-\theta ^*\) for some \(t_0\geqq 1\) and \(y\in {\mathbb {R}}^d\), then for any \(\theta \in [1-\theta ^*,1)\) and \(t\geqq t_0 + D_1 (1-\theta )^{1-m_1}\) we have
Proof
Lemma 2.1 shows that with \(\kappa _0\) from (2.2) we have
for any \(t\geqq t_0+\kappa _0\). Since
solves the ODE \(U'= \alpha _1 (1-U)^{m_1}\) with \(U(0)=1-\theta ^*\), it follows from Lemma 2.5 that
for all \((s,x)\in [0,\infty )\times {\mathbb {R}}^d\). Picking \(s:=\frac{1}{(m_1-1)\alpha _1} [(\frac{1-\theta }{2})^{1-m_1} - (\theta ^*)^{1-m_1}]\) makes \(U(s)\geqq \frac{1+\theta }{2}\). The second term on the right will be no more than \(\frac{1-\theta }{2}\) when \(|x-y|\leqq \frac{c_0}{4} (t+s-t_0)\), provided
which holds as long as \(t\geqq t_0 +D_1(1-\theta )^{1-m_1}\) for some \(D_1=D_1(M,\theta _1,m_1,\alpha _1)\geqq \kappa _0\). Replacing \(D_1\) by \(D_1+\frac{1}{(m_1-1)\alpha _1} 2^{m_1-1}\) now yields the claim for \(t+s\) in place of t whenever \(t+s\geqq t_0 +D_1(1-\theta )^{1-m_1}\). \(\quad \square \)
Finally, we will need two results providing estimates on how much solutions to (1.1) may change when the reaction f is perturbed. The first one concerns the case when the reaction can change only where the solution is initially close to 1. In it, we will also use the definition
Lemma 2.9
Let \(f_1\) satisfy (H2’) and \(f_2\) satisfy (H1’), and let \(M_*:=\frac{1+M}{\mu _*}\), with \(\mu _*,\kappa _*\) from (2.6) for all \(u\in {{\mathcal {U}}}_{f_1}\). Fix some \(\omega \in \Omega \) and let \(u_1,u_2:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solve (1.1) with \(f_1,f_2\) in place of f, respectively. If \(u_1\in {{\mathcal {U}}}_{f_1}\), \(t_0\geqq 0\), and for some \(\eta \in [0,\frac{1}{2}\min \{{\theta ^*},M_*^{-1}\} ]\) we have
then
is a supersolution to (1.1) with \(f_2\) in place of f on \((\kappa _*,\infty )\times {\mathbb {R}}^d\), and
is a subsolution to (1.1) with \(f_2\) in place of f on \((2\kappa _*,\infty )\times \{ x\in {\mathbb {R}}^d \,|\,u_1(t_0,x)<1-\eta \}\).
Moreover, there is \(D_2=D_2(M,\theta _1,m_1,\alpha _1)\geqq 1\) such that if also
for some \(y\in {\mathbb {R}}^d\) and \(R\geqq D_2(1+{T}_{u_2}(y)),\) then
The proof of Lemma 2.9 appears in “Appendix B”.
Our last preliminary lemma concerns the case when the reaction may be perturbed anywhere, although not by a lot.
Lemma 2.10
Let \(f_1\) satisfy (H2’) and \(f_2\) satisfy (H1’), with at least one of them satisfying (H3) with \(\alpha _3\leqq 1\), and let \(M_*,D_2\) be from Lemma 2.9. Fix some \(\omega \in \Omega \) and let \(u_1,u_2:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solve (1.1) with \(f_1,f_2\) in place of f, respectively. If \(u_1\in {{\mathcal {U}}}_{f_1}\), for some \(y\in {\mathbb {R}}^d\), \(R\geqq D_2(1+{T}_{u_2}(y))\), and \(\eta \in [0,\frac{1}{2}\min \{{\theta ^*},M_*^{-1}\} ]\) we have
and \(u_2(0,x)\leqq u_1(t_0,x)\) for some \(t_0\geqq 0\) and all \(x\in B_R(y)\), then (2.8) holds.
The proof of Lemma 2.10 appears in “Appendix C”.
2.1 Notations
Since Sects. 3–6 are just the proof of Theorem 1.7, we will assume either (H2’) or (H3)+(H4’) in them. Any constants \(C,C_0,C_\delta ,C(\delta ),\dots \) may depend on
(except for \(m_3,\alpha _3,m_4,m_4',n_4,\alpha _4\) when (H2’) is assumed; dependence on d is implicit in the whole paper). Any other dependence will be explicitly declared, for instance, “for some \(C_\delta \)” or “for some \(C=C(\delta )\)” will mean that this constant depends on \(\delta \) as well as (2.10). These constants may also vary from line to line. We recall that \(M,\theta _1,m_1,\alpha _1\) are from (H1); \(m_2,\alpha _2\) from (H2’); \(m_3,\alpha _3\) from (H3); \(m_4,m_4',n_4,\alpha _4\) from (H4’); and \(\mu _*,\kappa _*\) from (2.6).
The constants
also only depend on subsets of (2.10), with \(\theta _2,c_0,\kappa _0\) from Lemma 2.1; \(\theta ^*\) from (2.1); \(R_0\) from Lemma 2.2; \(c_1,\kappa _1\) from Corollary 2.6; \(D_1\) from Lemma 2.8; and \(M_*,D_2\) from Lemma 2.9.
For \(A\subseteq {\mathbb {R}}^d\) and \(r\geqq 0\) we let \(B_r(A):=A+(B_r(0)\cup \{0\})\) and \(A^0_r:=A\backslash \overline{B_r(\partial A)}\) (so \(A^0_0\) is the interior of A). For sets \(U,V\subseteq {\mathbb {R}}^d\), we let \( d(U,V):=\inf _{x\in U\, \& \,y\in V} |x-y|\) and
be their standard and Hausdorff distances.
Widths \(L_{u,\eta ,\theta }(t)\) of transition zones of solutions are defined in (1.7), and the special sets \({{\mathcal {U}}}_f\) of solutions evolving from approximate characteristic functions of balls are defined in (H2’).
Finally, we recall that \({{\mathcal {E}}}(U)\) is the \(\sigma \)-algebra generated by the family of random variables \( \{f(x,u,\cdot ) \,|\, (x,u)\in U\times [0,1]\}. \) Further important notation related to the dependence of reactions and solutions on \(\omega \) appears below, particularly early in Sects. 3 and 5.
3 Fluctuations for Reactions with Finite Ranges of Dependence
The proof of Theorem 1.7 will be based on the analysis of the dynamics of special solutions, starting from the approximate characteristic functions \(u_{0,S}\) of sets \(S\subseteq {\mathbb {R}}^d\) satisfying Lemma 2.2. For each S and \(\omega \in \Omega \), we therefore let \(u(\cdot ,\cdot ,\omega ;S)\) be the solution to
Recall that \(u_t(\cdot ,\cdot ,\omega ;S)\geqq 0\) because \(u_{0,S}\) satisfies (1.9). Let us also define for any \(x\in {\mathbb {R}}^d\),
with \(\theta ^*\) from (2.1), which can be thought of as the time when this solution reaches x.
Our goal is now to estimate the stochastic fluctuations of \(T(x,\omega ;S)\). In this section we will consider the first case in Theorem 1.7, when the reaction satisfies (H2’) and has a finite range of dependence. We will only need to treat the cases when S is either a ball or a half-space. We start with S being a ball, when the solution in (3.1) will be precisely the one from \({{\mathcal {U}}}_f\) corresponding to \((S,\omega )\). Hence, below always \(u(\cdot ,\cdot ,\omega ;S)\in {{\mathcal {U}}}_f\) and (2.6) holds for it.
Remark
We note that if we enlarge \({{\mathcal {U}}}_f\) (or each \({{\mathcal {U}}}_{f_n}\)) to include solutions with initial data \(u_{0,S}\) from Lemma 2.2 for all S from some family \({{\mathcal {S}}}\) of bounded subsets of \({\mathbb {R}}^d\), and (2.6) still continues to hold with some \(\mu _*,\kappa _*>0\), then everything in this section and the next holds without change (and with uniform constants) for either all \(S\in {{\mathcal {S}}}\) (the results involving balls only) or for all local limits in Hausdorff distance of translations of sets from \({{\mathcal {S}}}\) (the results involving half-spaces). In particular, Remark 2 after Definition 2.3 and Lemma 2.4 show that if we assume (H2), then we can let \({{\mathcal {S}}}\) be the family of all bounded subsets of \({\mathbb {R}}^d\).
Proposition 3.1
Let f satisfy (H2’) and let
There is \(C_0\geqq 1\) such that if f has range of dependence at most \(\rho \in [1,\infty )\) and \(S=B_k(0)\) for some \(k\in {\mathbb {N}}\), then for all \(x\in {\mathbb {R}}^d\) and \(\lambda >0\) we have
Remark
1. The point here is that by choosing \(\lambda \sim d(x,S)^\gamma \) for some \(\gamma \in (\frac{1+\beta _1}{2},1)\), one obtains a fast-decreasing bound on the probability of \(O(d(x,S)^\gamma )\) fluctuations of \(T(x,\cdot \,;S)\) from its mean (the latter is of course \(\sim d(x,S)\) by Lemma 2.1 and Corollary 2.6).
2. Note that Remark 2 after Definition 2.3 and Lemma 2.4 show that (H2) implies (H2’) with \(\alpha _2=0\) and any \(m_2>0\), so then \(\beta _1=\frac{m_1-1}{m_1}\). If there is also \(\theta <1\) such that \(\inf _{(x,u,\omega )\in {\mathbb {R}}^d\times (\theta ,1)\times \Omega } (1-u)^{-1} f(x,u,\omega )>0\), then this means that \(\beta _1>0\) can be made arbitrarily small.
The rest of this section is devoted to proving Proposition 3.1 (and then extending it from balls to half-spaces in Proposition 3.8). We will therefore assume (H2’) and the range of dependence of f being at most \(\rho \in [1,\infty )\). We will also fix \(S=B_k(0)\) and drop it from the notation (so the functions from (3.1) and (3.2) are \(u(t,x,\omega )\) and \(T(x,\omega )\), respectively) but all estimates will be independent of S (i.e., of k). Recall that all constants with C in them depend on (2.10), with any other dependence explicitly stated, and may vary from line to line.
3.1 Construction of a Martingale
Let \({{\mathcal {K}}}\) be the set of all non-empty compact subsets of \({\mathbb {R}}^d\), and endow it with Hausdorff distance \({d}_H\). For each \(a=(a_1,\ldots ,a_d)\in {\mathbb {Z}}^d\), let
and for each finite \(\emptyset \ne A\subseteq {\mathbb {Z}}^d\), let
Let us label all such A as \(A_0,A_1,\dots \), and denote \(P_i:=P_{A_i}\). Then \(\{{P}_i\}_{i\in {\mathbb {N}}_0}\) is a (pairwise disjoint) partition of the metric space \(({{\mathcal {K}}},{d}_H)\), and \({{\,\mathrm{diam}\,}}_H({P}_i)= 1\) for each \(i\in {\mathbb {N}}_0\).
For each i, let
(note that \(K_i\) is not compact), so that for each \(K\in P_i\) we have
Notice also that for each \(i,j\in {\mathbb {N}}_0\) we have
Next, for any \((t,\omega )\in [0,\infty )\times \Omega \) and \(\theta \in (0,1)\), we let
Since S is bounded, Lemma 2.5 shows that all these sets are compact. Then for all \(i\in {\mathbb {N}}_0\) and \(t\geqq 0\) we let
with \({{\theta ^*}}\) from (2.1). Then \(\{E_i(t)\}_{i\in {\mathbb {N}}_0}\) is a pairwise disjoint partition of \(\Omega \) for each \(t\geqq 0\) because \(\Gamma _{u,{{\theta ^*}}}(t,\omega )\ne \emptyset \) (due to \(\theta ^*\leqq 1-\theta ^*\)). Also note that \(\alpha _2<1\) allows us to only track the evolution of one of the sets \(\Gamma _{u,\theta }\) (see the remark after (H2’)), and in this section we choose it to be \(\Gamma _{u,\theta ^*}\). Finally, for \((t,\omega )\in [0,\infty )\times \Omega \), let
and for each \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) let
The latter is a slightly different version of the set of \(\omega \) for which the solution u has reached x by time t (we have \(F_{t,x} \supseteq \{\omega \in \Omega \,|\, T(x,\omega )\leqq t\} \supseteq F_{t-C,x}\) for some \(C>0\) and all large enough t, due to (2.6)). Let us also pick \(A_0\) so that \({\iota }(0,\omega )=0\) for each \(\omega \in \Omega \) (note that \({\iota }(0,\omega )\) does not depend on \(\omega \)).
Since \(u_t\geqq 0\) and \(f(\cdot ,u,\cdot )\equiv 0\) for \(u\in [0,\theta ^*]\), the solution \(u(\cdot ,\cdot ,\omega )\) only depends on the reaction inside \(K_{{\iota }(t,\omega )}\) up to time t. Hence, we have the following lemma.
Lemma 3.2
For any \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) and \(s\in [0,t]\), the set \(E_i(t)\) and the function \(u(s,x,\cdot )\chi _{E_i(t)}(\cdot )\) are \({{\mathcal {E}}}(K_i)\)-measurable for each \(i\in {\mathbb {N}}_0\).
Proof
Fix any \(i\in {\mathbb {N}}_0\), and let g be any reaction satisfying (H1’) and
Then for each \(\omega \in \Omega \), let v be the solution to
with the same initial data \(v(0,\cdot ,\omega )=u_{0,S}\) as u, and fix any \(t\geqq 0\). Since \(u_t,v_t\geqq 0\) and \(f(\cdot ,u,\cdot )\equiv 0\) for \(u\in [0,\theta ^*]\), it follows that \(v(s,\cdot ,\omega )=u(s,\cdot ,\omega )\) whenever \(\omega \in E_i(t)\) and \(s\leqq t\). Similarly, we have \(v(s,\cdot ,\omega )=u(s,\cdot ,\omega )\) whenever \(\Gamma _{v,{{\theta ^*}}}(t,\omega )\in P_i\) and \(s\leqq t\) (with \(\Gamma _{v,\theta }\) defined analogously to \(\Gamma _{u,\theta }\)). In particular, \(\omega \in E_i(t)\) if and only if \(\Gamma _{v,{{\theta ^*}}}(t,\omega )\in P_i\).
Since this holds for each g as above, it follows that \(E_i(t)\in {{\mathcal {E}}}(K_i)\), and then also that \(u(s,x,\cdot )\chi _{E_i(t)}(\cdot )\) is \({{\mathcal {E}}}(K_i)\)-measurable for any \((s,x)\in [0,t]\times {\mathbb {R}}^d\). \(\quad \square \)
For each \(t\geqq 0\), let \({{\mathcal {G}}}_t\) be the \(\sigma \)-algebra on \(\Omega \) generated by
That is, \({{\mathcal {G}}}_t\) is generated by all events \(F\cap E_i(s)\) with \(i\in {\mathbb {N}}_0, F\in {{\mathcal {E}}}(K_i), s\in [0,t]\). Then \(\{{{\mathcal {G}}}_t\}_{t\geqq 0}\) is a filtration on \((\Omega ,{{\mathcal {F}}})\), and \({{\mathcal {G}}}_0={{\mathcal {E}}}(K_{0})\) because \(E_0(0)=\Omega \) and \(E_i(0)=\emptyset \) for \(i>0\). Lemma 3.2 then shows that \(u(s,x,\cdot )\) is \({{\mathcal {G}}}_t\)-measurable for any \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) and \(s\leqq t\). Since \(E_i(s)\in {{\mathcal {E}}}(K_i)\) for all \(i\in {\mathbb {N}}_0\) and \(s\geqq 0\) (so also \(F\cap E_i(s)\in {{\mathcal {E}}}(K_i)\) above), \({\iota }(s,\cdot )\) and \(F_{s,x}\) are also \({{\mathcal {G}}}_t\)-measurable for all \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) and \(s\leqq t\).
We note that \({{\mathcal {G}}}_t\) is actually simpler than its definition suggests, and for each \(t\geqq 0\) and \(j\in {\mathbb {N}}_0\) we in fact have
(recall also that \(\{E_j(t)\}_{j\in {\mathbb {N}}_0}\) is a partition of \(\Omega \) for each \(t\geqq 0\), and note that only finitely many of these sets are non-empty due to Lemma 2.5). Indeed, let us consider any \(i\in {\mathbb {N}}_0\) and \(s\in [0,t]\) such that \(E_i(s)\cap E_j(t)\ne \emptyset \). Then there is \(\omega \in \Omega \) such that
From (3.5) and
for any such \(\omega \), we obtain \(K_i\subseteq K_j\). But then \(E_i(s)\in {{\mathcal {E}}}(K_i)\subseteq {{\mathcal {E}}}(K_j)\) and
Since this clearly also holds when \(E_i(s)\cap E_j(t)= \emptyset \), the definition of \({{\mathcal {G}}}_t\) proves (3.6).
Similarly to [2, 3], we will prove Proposition 3.1 by studying the \({{\mathcal {G}}}_t\)-adapted martingale \(\{X_t:={\mathbb {E}}\left[ T(x,\cdot )\,|\,{{\mathcal {G}}}_t\right] \}_{t\geqq 0}\) for any \(x\in {\mathbb {R}}^d\) and estimating its increments, which will then allow us to apply Azuma’s inequality to bound the fluctuations of \(T(x,\cdot )\).
Lemma 3.3
(Azuma’s inequality) Let \(\left\{ X_{k}\right\} _{k\in {\mathbb {N}}_0}\) be a martingale on \((\Omega ,{{\mathcal {F}}},{\mathbb {P}})\). If for each \(k\in {\mathbb {N}}\) there is \(c_k\geqq 0\) such that \(|X_k-X_{k-1}|\leqq c_k\) almost surely, then for all \(\lambda >0\) and \(n\in {\mathbb {N}}\),
We first show that for any \(\omega \in F_{t,x}\) the difference \({\mathbb {E}}[T(x,\cdot )\,|\,{{\mathcal {G}}}_t](\omega )- T(x,\omega )\) is uniformly bounded, and 0 if \(\omega \in F_{s,x}\) when \(t-s\) is large enough (note that \(F_{s,x}\subseteq F_{t,x}\) if \(s\leqq t\) because \(u_t\geqq 0\)). This is due to \(F_{t,x}\in {{\mathcal {G}}}_t\) and the above-mentioned relationship of \(F_{t,x}\) and \(\{\omega \in \Omega \,|\, T(x,\omega )\leqq t\}\).
Lemma 3.4
There is \(C>0\) such that for each \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) we have
and if also \(s\in [0,t-C]\), then
Proof
For any \((x,\omega )\in {\mathbb {R}}^d\times \Omega \), let
If \(\omega \in F_{t,x}\) for some \(t\geqq 0\), then \(\tau (x,\omega )\leqq t\) due to \(K_{{\iota }(t,\omega )}\subseteq B_1(\Gamma _{u,\theta ^*}(t,\omega ))\). And since \(u(s,x,\cdot )\) is \({{\mathcal {G}}}_t\)-measurable for all \(s\leqq t\), we see that \(\tau (x,\cdot )\chi _{ F_{t,x}}(\cdot )\) is \({{\mathcal {G}}}_{t}\)-measurable.
For each \((x,\omega )\in {\mathbb {R}}^d\times \Omega \), there is \(y\in \overline{B_1(x)}\) with \(u(\tau (x,\omega ),y,\omega )\geqq \theta ^*\), so (2.6) shows that
and then Lemma 2.1 yields
Therefore there is C such that \(T(x,\omega )\leqq \tau (x,\omega )+C\), and hence \(|T(x,\omega )- \tau (x,\omega )|\leqq C\). Hence for any \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) we obtain using \({{\mathcal {G}}}_{t}\)-measurability of \(\tau (x,\cdot )\chi _{ F_{t,x}}\),
yielding the first claim. If also \(s\leqq t-C\), then for all \(\omega \in F_{s,x}\) we have
Since \(u(t,x,\cdot )\) is \({{\mathcal {G}}}_t\)-measurable, this shows that so is \(T(x,\cdot )\chi _{ F_{s,x}}\), and (3.7) follows. \(\quad \square \)
When \(\omega \in \Omega {\setminus } F_{t,x}\) (that is, essentially, when the solution u has not yet reached x by the time t), we will obtain a different kind of estimate.
Let \(\rho \geqq 1\) be from Proposition 3.1, and for each \(i\in {\mathbb {N}}_0\) let
where \(0\leqq \psi _i\leqq 1\) is Lipschitz with a uniform-in-i constant, with \(\psi _i\equiv 1\) on \(B_\rho (K_i)\) and \(\psi _i \equiv 0\) on \({\mathbb {R}}^d{\setminus } B_{\rho +1}(K_i)\). Then \(g_i\) is Lipschitz in (x, u) (with a uniform M-dependent constant \(\geqq M\), which we will call M from now on), and \(g_i(x,u,\cdot )\) is independent of \({{\mathcal {E}}}(K_i)\) for all \((x,u)\in {\mathbb {R}}^d\times [0,1]\) because f has range of dependence at most \(\rho \). Of course,
For each \(\omega \in \Omega \), let now \(v_i\) be the solution to
with \(u_{0,K_i}\) from Lemma 2.2. Then \(v_i(t,x,\cdot )\) is independent of \({{\mathcal {E}}}(K_i)\) for each (t, x), and so is
Proposition 3.5
There is \(C\geqq 1\) such that for each \((t,x)\in [0,\infty )\times {\mathbb {R}}^d\) and \(\omega \in \Omega {\setminus } F_{t,x}\) we have
Remark
This shows that the difference of the time it takes to reach x from S and the sum of any smaller time t and the time it takes to reach x from \(K_{{\iota }(t,\omega )}\) (which approximates \(\Gamma _{u,{{\theta ^*}}}(t,\omega )\)) is sublinear in d(x, S). Hence, this result yields a certain additive structure (up to lower order errors) for the arrival times of solutions with initial data from Lemma 2.2.
Proof
We will use \((t_0,x_0)\) in place of (t, x) in the proof. Fix any \(\omega \in \Omega {\setminus } F_{t_0,x_0}\) and let \(i:={\iota }(t_0,\omega )\). Note that since \(\omega \in E_i(t_0)\), we have
Moreover, since \(x_0 \notin K_i\), Lemma 2.1 shows that \(t_0\leqq \kappa _0+ \frac{2}{c_0}d(x_0,S)\).
Let us first show that
From (2.6) we know that with \(L_1:=\mu _*^{-1} (1+t_0^{\alpha _2}) (\theta ^*)^{-m_2}+1\) we have
Hence we obtain
This and Lemma 2.1 now show
for
hence
for any \(t\geqq t_0+t_1\). Since \(t_0\leqq \kappa _0+\frac{2}{c_0}d(x_0,S)\), there is \(C>0\) such that
Now take
where
Note that (3.3) shows that
It follows from (3.11) and Lemma 2.8 that for \(t_2:=\frac{4}{c_0}(\rho +1)+ D_1\eta ^{1-{m_1}}\) we have
Moreover, from (3.13) and (3.14) we see that there is \(C>0\) such that
We now apply Lemma 2.9 with \((f,g_i,u,v_i,t_0+t_3,\infty )\) in place of \((f_1, f_2,u_1, u_2,t_0,R)\). Its hypotheses are satisfied due to (3.9), (3.12) and (3.15), and it yields
This, (3.16), and \({T}_i(x_0,\omega )\leqq C(1+d(x_0,K_i))\leqq C(1+d(x_0,S))\) (which follows from Lemma 2.1, with some \(C>0\)) show that there is indeed some \(C>0\) such that
Let us now turn to
and let \(\eta \) be again from (3.13). We will now need to estimate u from above in terms of some time-shift of \(v_i\). It follows from Lemma 2.9 that
is a subsolution to (1.1) with reaction f on \((2\kappa _*,\infty )\times ({\mathbb {R}}^d{\setminus } \Gamma _{u,1-\eta }(t_0,\omega ))\). We also know that \(v_i\) is a solution of the same equation on \((0,\infty )\times ({\mathbb {R}}^d{\setminus } B_{\rho +1}(K_i))\). In order to be able to compare them, we need some more estimates involving these sets.
Since \(\theta ^*\leqq \theta _1\), Corollary 2.6 and (3.10) yield
This and (2.6) show that with \(L_2:= \mu _{*}^{-1}(1+(t_0+2\kappa _*)^{\alpha _2})\eta ^{-{m_2}}\) and
we have
We note that this also implies
Moreover, Lemma 2.8 shows that
with \(t_4:= \frac{4}{c_0}L_3 +D_1\eta ^{1-{m_1}}\). Then (3.13), (3.14), and \(t_0\leqq \kappa _0+\frac{2}{c_0}d(x_0,S)\) show that there is \(C>0\) such that
Using (3.20) and (3.18), we find that
From (3.9), (3.19), and Lemma 2.9 we also see that \(v_i\) and \(u_-\) are, respectively, a solution and a subsolution to (1.1) with reaction f on \((2\kappa _*,\infty )\times ({\mathbb {R}}^d{\setminus } B_{L_3}(K_i))\). So the second claim in (3.22) and the comparison principle yield
If \(x_0\notin B_{L_3}(K_i)\), then this shows that
Using again \({T}_i(x_0,\omega )\leqq C(1+d(x_0,S))\) (as we did above) and (3.21), we obtain
for some \(C>0\). If instead \(x_0\in B_{L_3}(K_i)\), the first claim in (3.22) and (3.21) again yield
because \(x_0 \notin K_i\) (and hence \(T(x_0,\omega )\geqq t_0\)). \(\quad \square \)
The last ingredient in the proof of Proposition 3.1 is an estimate on the difference of \({T}_{{\iota }(t,\omega )}(x,\omega )\) for two different times t.
Lemma 3.6
There is \(C>0\) such that for all \((x,\omega )\in {\mathbb {R}}^d\times \Omega \) and \(t_0,t_1\geqq 0\) we have
Proof
Let \(i_0:={\iota }(t_0,\omega )\) and \(i_1:={\iota }(t_1,\omega )\), and then \(\omega \in E_{i_0}(t_0)\cap E_{i_1}(t_1)\). Without loss of generality, let us assume \(t_1>t_0\). Then \(u_t\geqq 0\) and (3.5) show that \(K_{i_0}\subseteq K_{i_1}\).
If \(x\notin K_{i_1}\), Proposition 3.5 yields
If \(x\in K_{i_0}\), then \({T}_{i_1}(x,\omega )={T}_{i_0}(x,\omega )=0\). The result follows in either case.
Let us now assume that \(x\in K_{i_1}\backslash K_{i_0}\). Then \({T}_{i_1}(x,\omega )=0\), while Lemma 2.1 shows that
for some \(C>0\). From \(t_1>t_0\) and (2.6) we also have
and Corollary 2.6 yields
Hence there is \(C>0\) such that
Since also \( {d}_H(K_{{\iota }(t,\omega )},\Gamma _{u,{{\theta ^*}}}(t,\omega ))\leqq 1\) for all \(t\geqq 0\) (because \(\Gamma _{u,{{\theta ^*}}}(t,\omega )\in P_{{\iota }(t,\omega )}\)), this implies
This and (3.24) yields the claim. \(\quad \square \)
3.2 Proof of Proposition 3.1
If \(d(x,S)\leqq \rho \), then (3.4) holds for all \(\lambda >0\) as long as \(C_0\geqq 2 (\kappa _0+\frac{2}{c_0})^2 (\ln 2)^{-1}\). This is because \(T(x,\cdot )\leqq \kappa _0+\frac{2}{c_0}d(x,S)\) by Lemma 2.1, so one only needs to consider \(\lambda \leqq (\kappa _0+\frac{2}{c_0})(1+d(x,S))\), for which the right-hand side of (3.4) with this \(C_0\) is at least 1 due to \(\rho \geqq \frac{1}{2}(1+d(x,S))\).
It therefore suffices to consider the case \(d(x,S)> \rho \). In particular, we have \(x\notin K_0\) due to \(\rho \geqq 1\). Let us fix any such x and consider the \({{\mathcal {G}}}_t\)-adapted martingale \(\{X_t\}_{t\geqq 0}\) defined by
We want to apply Azuma’s inequality to it, which means that we need to obtain a suitable \(\omega \)-independent bound on \(|X_t-X_s|\) for any \(t> s>0\) (which we fix). Using
we find from Lemma 3.4 (recall that \(F_{s,x}\subseteq F_{t,x}\)) that there is \(C>0\) such that
Here we used that \(\omega \notin F_{s,x}\) precisely when \(x\notin K_{{\iota }(s,\omega )}\), and the sums are over all \(i\in {\mathbb {N}}_0\) such that \(x\notin K_i\). From Proposition 3.5 with s in place of t we have
The last equality holds because \({T}_i(x,\omega )=0\) when \(x\in K_i\). Since
(recall that \(E_i(s),E_j(t)\in {{\mathcal {G}}}_t\)) and Lemma 3.6 yields
it follows that with some \(C>0\) and \(C_{t,s,x}^\rho :=C(\rho +|t-s|+d(x,S)^{\beta _1})\), we have
We now claim that for any \(i\in {\mathbb {N}}_0\) we have
Since \( E_i(s)\in {{\mathcal {G}}}_s\), to prove this, we only need to show that
for each \(A\in {{\mathcal {G}}}_s\) such that \(A\subseteq E_i(s)\). But then \(A\in {{\mathcal {E}}}(K_i)\) by (3.6), so (3.28) follows from \({T}_i(x,\cdot )\) being independent of \({{\mathcal {E}}}(K_i)\).
Similarly to (3.27), we also have
for any \(j\in {\mathbb {N}}_0\). Then (3.26) becomes
Lemma 3.6 now shows that there is \(C>0\) such that for all \(\omega \in \Omega \) we have
By Lemma 2.1, we have \(F_{\tau _x,x}=\Omega \) when \(\tau _x:=\kappa _0+\frac{2}{c_0}d(x,S)\). So with C from Lemma 3.4,
Let \( \tau :=\rho +d(x,S)^{\beta _1}\) and let N be the smallest integer such that \(N\tau \geqq \tau _x+C\). Then there is \(C_1>0\) such that \(N\leqq C_1 d(x,S)(\rho +d(x,S)^{\beta _1})^{-1}\) (recall that \(d(x,S)>\rho \geqq 1\)). It follows from (3.29) that for \(k=0,\dots , N-1\) we have (uniformly in \(\omega \in \Omega \))
Now Azuma’s inequality (Lemma 3.3), \(X_0={\mathbb {E}}[T(x,\cdot \,)\,|\,{{\mathcal {G}}}_0]\), and (3.30) with \(t=N\tau \) yield for any \(\lambda \geqq 0\) (with C changing from line to line),
Since \(F_{0,x}=\emptyset \) (because \(x\notin K_0\)), Proposition 3.5 with \(t=0\) (and \({\iota }(0,\cdot )\equiv 0\)) yields
for some \(C'>0\) and all \(\omega \in \Omega \). This and \({T}_0(x,\cdot )\) being independent of \({{\mathcal {E}}}(K_0)={{\mathcal {G}}}_0\) yield
Hence, from (3.32) we obtain for any \(\lambda > 0\),
So for all \(\lambda \geqq C'(\rho +d(x,S)^{\beta _1})\) we have
which yields (3.4) for all \(\lambda \geqq 2C'(\rho +d(x,S)^{\beta _1})\) as long as \(C_0\geqq 4C\). But (3.4) also holds for all \(\lambda \leqq 2C'(\rho +d(x,S)^{\beta _1})\) as long as \(C_0\geqq 8(C')^2(\ln 2)^{-1}\) because \(d(x,S)>\rho \geqq 1>\beta _1\) (and so the right-hand side of (3.4) is \(\geqq 1\)). This finishes the proof.
3.3 Extension to Half-Spaces
We now extend Proposition 3.1 to half-spaces, denoting
for \(e\in {\mathbb {S}}^{d-1}\). This means that we need to enlarge \({{\mathcal {U}}}_f\) to include solutions initially approximating characteristic functions of half-spaces, with (H2’) extending as well.
Lemma 3.7
(H2’) implies (H2’) with \({{\mathcal {U}}}_f'\) in place of \({{\mathcal {U}}}_f\), with unchanged values of all the \(\sup _{u\in {{\mathcal {U}}}_f}\) and \(\inf _{u\in {{\mathcal {U}}}_f}\) in (1.10), and with \({{\mathcal {U}}}_f'\) defined as \({{\mathcal {U}}}_f\) but including the initial functions \(u_{0,k}\) from \({{\mathcal {U}}}_f\) as well as all locally uniform limits of their translations. (These are then functions \(u_{0,S}\) satisfying Lemma 2.2 for all balls \(S=B_k(y)\) and all half-spaces \(S={{\mathcal {H}}}_e^-+le\), due to well-known elliptic regularity estimates).
In particular, Lemmas 2.9 and 2.10 hold with \({{\mathcal {U}}}_{f_1}\) replaced by \({{\mathcal {U}}}_{f_1}'\).
Proof
Stationarity of f again shows that adding translations of the \(u_{0,k}\) to \({{\mathcal {U}}}_f\) does not change any of the \(\sup \) or \(\inf \). Well known parabolic regularity estimates now show that the \(\sup \) and \(\inf \) also remain unchanged when we add locally uniform limits of these translations to \({{\mathcal {U}}}_f\). The proofs of Lemmas 2.9 and 2.10 then extend to \({{\mathcal {U}}}_{f_1}'\) in place of \({{\mathcal {U}}}_{f_1}\) without change.
We note that the elliptic and parabolic regularity (Krylov-Safonov and Schauder) estimates used here can be found in [11, Theorem 4.6], [15, Theorem 4.1], and [14, Theorem 8.6.1]. \(\quad \square \)
Proposition 3.8
Proposition 3.1 holds for S being either any ball \(B_k(y)\) with \((k,y)\in {\mathbb {N}}\times {\mathbb {R}}^d\) or any half-space \({{\mathcal {H}}}_e^-+le\) with \((e,l)\in {\mathbb {S}}^{d-1}\times {\mathbb {R}}\) (with the functions \(u(\cdot ,\cdot ,\cdot ;S)\) from \({{\mathcal {U}}}_f'\)).
Proof
The claim for balls is immediate from stationarity. For the same reason, in the half-space case we only need to consider \(l=0\). Hence let \(S={{\mathcal {H}}}_e^-\) for some \(e\in {\mathbb {S}}^{d-1}\).
For each \(k\in {\mathbb {N}}\), let \(S_k:=B_k(-ke)\). Then \(\lim _{k\rightarrow \infty } d(x,S_k)= d(x,S)\) for each \(x\in {\mathbb {R}}^d\), so by Proposition 3.1 (with \(C_0\) independent of S), it suffices to show that
for some \(C>0\) (depending only on (2.10), as always) and any \(x\in {\mathbb {R}}^d\).
Let \(u_\omega :=u(\cdot ,\cdot ,\omega ;S)\) and \(u_{k,\omega }:=u(\cdot ,\cdot ,\omega ;S_k)\) for each \((k,\omega )\in {\mathbb {N}}\times \Omega \), and \(C:=\kappa _0+ \frac{2R_0}{ c_0}\). We then have \(u_\omega (C,\cdot )\geqq u_{k,\omega } (0,\cdot )\) by Lemma 2.1 and (2.3), so \(T(\cdot ,\omega ;S)\leqq T(\cdot ,\omega ;S_k) + C\).
Similarly, we have \(u_{k,\omega }(C+1,\cdot )\geqq u_{\omega } (0,\cdot )\) in \(B_{k^{1/2}}(0)\). Then from the last claim in Lemma 2.9 with \((\eta ,R)=(0,k^{1/2}-|x|)\) we obtain \(T(x,\omega ;S_k)\leqq T(x,\omega ;S) + C+1+2\kappa _*+\kappa _0\) whenever \(k^{1/2}\geqq |x|+ D_2(1+\kappa _0+ \frac{2}{ c_0} (d(x,S)+1))\), because then \(d(x,S_k)\leqq d(x,S)+1\) due to \(|x|\leqq k^{1/2}\), so \(k^{1/2}-|x|\geqq D_2(1+T(x,\omega ;S_n))\) by Lemma 2.1. \(\quad \square \)
4 Fluctuations for General Reactions
First, we show that (H4’) implies (H2’).
Lemma 4.1
(H4’) implies that f also satisfies (H2’) (possibly after removing from \(\Omega \) a measure-zero set that is invariant with respect to the group \(\{\Upsilon _y\}_{y\in {\mathbb {R}}^d}\), which we then do).
Proof
For each \(k\in {\mathbb {N}}\), let \(u_{0,k,n}\) be the initial datum for \(B_k(0)\) that enters in the definition of \({{\mathcal {U}}}_{f_n}\) (these might in principle be different for distinct n, as we do not assume them to be those from the proof of Lemma 2.2). By (1.9), (1.8), and elliptic regularity estimates, there is a subsequence \(\{n_j\}_{j\in {\mathbb {N}}}\) such that \(u_{0,k,n_j}\) converge uniformly to some \(u_{0,k}\) satisfying (1.9) and (1.8) as \(j\rightarrow \infty \). Since also (H2’) holds uniformly for \(f_{n}\), and the Borel-Cantelli Lemma shows that some subsequence of \(f_{n_j}\) converges to f locally uniformly on \({\mathbb {R}}^d\) for almost every \(\omega \in \Omega \) (and if this holds for some \(\omega \), then it obviously also holds for \(\Upsilon _y \omega \) with any \(y\in {\mathbb {R}}^d\)), it follows that (H2’) also holds for f, with the above \(u_{0,k}\) for each \(k\in {\mathbb {N}}\) and after removal of a \(\{\Upsilon _y\}\)-invariant measure-zero set from \(\Omega \). \(\quad \square \)
We now extend Proposition 3.8 to reactions f satisfying (H3) and (H4’). Recall (3.2) and that constants with C in them only depend on (2.10) unless explicitly stated otherwise.
Proposition 4.2
Let f satisfy (H3) and (H4’), and with \(\beta _1\) from (3.3) let
and
for all \(a\geqq 0\) (then \(\nu (a)\leqq \max \{ \lceil a \rceil ,n_4\}^{-(2d+1+m_4')}\) by (H4’)). There is \(C_0'\geqq 1\) such that for any S (and u) from Proposition 3.8, \(x\in {\mathbb {R}}^d\), and \(\lambda >0\) we have
Remark
Having this and Lemma 4.1, we will not need to use (H3) and (H4’) again.
Proof
Let us assume without loss that \(\alpha _3\leqq 1\) in (H3). Lemma 4.1 shows that f also satisfies (H2’), and then Lemma 3.7 shows that (H2’) holds with \({{\mathcal {U}}}_f'\cup \bigcup _{n\geqq n_4} {{\mathcal {U}}}_{f_n}'\) in place of \({{\mathcal {U}}}_f\).
For each \(\omega \in \Omega \) and S either a ball or a half-space, let \(u(\cdot ,\cdot ,\omega ,S)\) be the solution from \({{\mathcal {U}}}_f'\) corresponding to \((\omega ,S)\), and for each \(n\geqq n_4\), let \(u_n(\cdot ,\cdot ,\omega ,S)\) be the analogous solution from \({{\mathcal {U}}}_{f_n}'\). (Note that we do not require the initial data for u and \(u_n\) to be the same, although they may be.) Also let
Then (H4’) and Proposition 3.8 show, for all \(n\geqq n_4\),
Let \(C_1\geqq n_4\) be such that \((\frac{\alpha _4}{\alpha _3})^{1/m_3}C_1^{-m_4/m_3}\leqq \frac{1}{2}\min \{\theta ^*,M_*^{-1}\}\).
Stationarity of \(f,f_n\) shows that the definition of \(\nu (a)\) above is unchanged when in it we replace |z| by \(|z-x|\), for any \(x\in {\mathbb {R}}^d\). So for each x and \(n\geqq n_4\), there is \(\Omega _{n,x}\subseteq \Omega \) such that \({\mathbb {P}}(\Omega _{n,x})\geqq 1-\nu (n)\) and
Since f satisfies (H3), and \(u(\cdot ,\cdot ,\omega ;S)\in {{\mathcal {U}}}_f\) and \(u_n(\cdot ,\cdot ,\omega ;S)\in {{\mathcal {U}}}_{f_n}\) for each \(\omega \in \Omega _{n,x}\), Lemma 2.10 (see Lemma 3.7) applied twice with \(\eta =(\frac{\alpha _4}{\alpha _3} n^{-m_4})^{1/m_3}\), \(y=x\), \(R=n^{1+m_4'}\), \(t_0=\kappa _0+2c_0^{-1}R_0\), and \((f_1,f_2,u_1,u_2)\) being
respectively, yields for some \(C>0\) and all \(n\geqq \max \{C_1, [D_2(1+\kappa _0+\frac{2}{c_0}d(x,S))]^{1/(1+m_4')}\}\),
Here we also used Lemma 2.1 to show that \(T_n(x,\cdot ;S)\) and \(T(x,\cdot ;S)\) are at most \(\kappa _0 + \frac{2}{c_0}d(x,S)\).
Let now n be the smallest integer such that
Since \(\beta _3\ge \frac{1}{1+m_4'}\), there is \(C_2>0\) such that with C from (4.4) we have (uniformly in x, S)
If now \(\lambda \geqq C_2(1+d(x,S)^{(1+\beta _3)/2})\), then (4.3) and (4.4) imply
Hence the result holds with \(C_0':=\max \{ 4C_0C_2, 2C_2^2 (\ln 2)^{-1}\}\), because then it obviously also holds for any \(\lambda \leqq C_2(1+d(x,S)^{(1+\beta _3)/2})\). \(\quad \square \)
5 Convergence of the Mean Propagation Speeds
We now consider (3.1) with \(S={{\mathcal {H}}}_e^-+le\) for any \(e\in {\mathbb {S}}^{d-1}\) and \(l\in {\mathbb {R}}\), that is,
Here \(u_{0,{{\mathcal {H}}}_e^-+le}\) is the initial data used in the definition of \({{\mathcal {U}}}_f'\) in Lemma 3.7 (note that f satisfies (H2’) in both cases under consideration, due to Lemma 4.1). Hence, \(u(\cdot ,\cdot ,\omega ;{{\mathcal {H}}}_e^-+le)\in {{\mathcal {U}}}_f'\) for all \((e,l,\omega )\in {\mathbb {S}}^{d-1}\times {\mathbb {R}}\times \Omega \) and (2.6) holds for it.
We will now prove that \(\frac{1}{l} {\mathbb {E}}\left[ T(le,\cdot \,;{{\mathcal {H}}}_e^-)\right] \) converges as \(l\rightarrow \infty \), with \(T(le,\cdot \,;{{\mathcal {H}}}_e^-)\) from (3.2) (stationarity shows that the expectation is the same if le is replaced by any \(y_l\) with \(y_l\cdot e=l\)). Note that the reciprocal of this limit can then be considered the asymptotic mean speed of propagation of the solutions u in direction e (this mean is technically harmonic).
We also note that all constants in this section will be uniform in e, and recall that all constants with C in them depend on (2.10), with any other dependence explicitly indicated, and may vary from line to line.
Proposition 5.1
For each \(e\in {\mathbb {S}}^{d-1}\) there is \( \bar{T}(e)\in [\frac{1}{c_1},\frac{1}{c_0}]\) (depending also on f) and for each \(\delta >0\) there is \(C_\delta \geqq 1\) such that the following hold. If f satisfies (H2’) and has range of dependence at most \(\rho \in [1,\infty )\), then with \(\beta _1\) from (3.3) we have for all \(l\geqq 1\),
If instead f satisfies (H3) and (H4’), then with \(\beta _3<1\) from (4.1) we have for all \(l\geqq 1\),
We will fix e in the rest of this section and, for the sake of convenience, we will sometimes (but not always) drop \({{\mathcal {H}}}_e^-\) from the notation in (3.1) and (3.2) when \(S={{\mathcal {H}}}_e^-\). Hence we let
We will also prove both claims in Proposition 5.1 at the same time, with the notation
if f satisfies (H2’) and has range of dependence at most \(\rho \in [1,\infty )\), and
if f satisfies (H3) and (H4’), with \(\nu \) from (4.2). Here again, \(C\geqq 1\) will be a constant depending only on (2.10), which may vary from line to line. In particular, Propositions 3.8 and 4.2 show that in both cases we have for all \(e\in {\mathbb {S}}^{d-1}\), \(\lambda >0\), \(x\in {\mathbb {R}}^d\) with \(x\cdot e\geqq 1\) (and some \(C>0\) defining \({\bar{C}}_\rho \)),
We start with the following simple result.
Lemma 5.2
If \(x\cdot e\leqq l\) for some \((e,l,x)\in {\mathbb {S}}^{d-1}\times {\mathbb {R}}\times {\mathbb {R}}^d\), then
Proof
Since \(u(0,\cdot ,\omega ;{{\mathcal {H}}}_e^-)\leqq u(\tau _0,\cdot ,\omega ;{{\mathcal {H}}}_e^-+le-x)\) for \(\tau _0:=\kappa _0+\frac{2R_0}{c_0}\) by Lemma 2.1 and \(x\cdot e\leqq l\), we have
Therefore,
because f is stationary (if we assume (H4’), this follows from Lemma 4.1) \(\quad \square \)
The next result is an immediate consequence of Lemma 2.1 and Corollary 2.6.
Lemma 5.3
There is \(t_0\geqq 1\), depending only on (2.10), such that for all \(t\geqq t_0\) and \(\omega \in \Omega \),
In particular, for all \(x\in {\mathbb {R}}^d\) with \(x\cdot e\geqq l_0:=2c_1 t_0\) and \(\omega \in \Omega \) we have
In order to prove Proposition 5.1, it will be necessary to simultaneously prove it for \(T(x,\cdot \,;{{\mathcal {H}}}^-_e)\) with other points x satisfying \(x\cdot e=l\). The Infinite Monkey “Theorem” shows that in dimensions \(d\geqq 2\), this cannot involve all the points in the unbounded set \(\{x\cdot e=l\}\), but we will be able to include all such points with \(|x|\leqq O(l)\) (i.e., within a ball centered at le and with linearly-in-time growing radius due to (5.6)). This will be sufficient thanks to the speed of propagation of perturbations of solutions being finite (see Lemma 2.5).
This and Lemma 5.3 motivate the definitions of the cylinders
and of the corresponding times
Obviously \(T^+_e(R,l,\cdot )\leqq T^-_e(R,l,\cdot )\) because \(u_t(t,\cdot ,\cdot ;{{\mathcal {H}}}_e^-)> 0\) for all \(t>0\).
Our next result shows that means of these times are sufficiently close to \({\mathbb {E}}\left[ T(le,\cdot \,;{{\mathcal {H}}}_e^-)\right] \).
Lemma 5.4
There is \(C>0\), and for each \(\delta >0\) there is \(C_\delta \geqq 1\), such that with \(l_0\geqq 1\) from Lemma 5.3 we have for all \( l\geqq l_0\) and \(R\geqq 0\) (with \({{\bar{C}}}_\rho \) from (5.3) resp. (5.4)),
Proof
The definition of \(T^\pm _e(R,l,\omega )\) shows that it suffices to consider the case \(R\geqq l\geqq l_0\).
For any \(\lambda >0\) we have
Since Lemma 2.1 yields
for any \(\lambda \geqq 4\kappa _0+\frac{4(\sqrt{d}+R_0)}{c_0}=:C_1\) we obtain
Since \(x\in {{\mathcal {C}}}_e^-(R,l)\cap {\mathbb {Z}}^d\) implies \(x\cdot e\leqq l\) and f is stationary, Lemma 5.2 now yields
The number of terms in this sum is bounded by \(C_2 R^{d-1}l\) for some \(C_2>0\). Since each \(x\in {{\mathcal {C}}}_e^-(R,l)\) satisfies \(d(x,{{\mathcal {H}}}_e^-)\geqq \frac{c_0}{4c_1}l\), by (5.5) we have for each \(\lambda \geqq C_1\),
Moreover, this probability is clearly 0 when \(\lambda > \kappa _0 + \frac{2}{c_0}l\). Thus, for each \(\delta > 0\) we obtain
with some \(C_3>0\). Since
it follows from \(R\geqq l\) that there is \(C_{\delta }\geqq 1\) such that
This proves the first inequality in (5.7). The proof of the second is analogous. \(\quad \square \)
We can now show that \({\mathbb {E}}\left[ T(le,\cdot \,;{{\mathcal {H}}}_e^-) \right] \) is close to being linear in l.
Proposition 5.5
There is \(C>0\), and for each \(\delta >0\) there is \(C_\delta \geqq 1\), such that for all \(l, m \geqq 0\) we have (with \({{\bar{C}}}_\rho \) from (5.3) resp. (5.4))
Remark
We will in fact only need the weaker upper bound \({{\bar{C}}}_\rho C_{\delta }(l+m)^{\beta +\delta }\).
Proof
Without loss of generality, we can assume that \(l\geqq m \geqq 0\), and we also let \(l_0\) be from Lemma 5.3. By Lemma 2.1, for all \(m\leqq \frac{4c_1}{c_0}l_0\) we have \(T(me,\cdot ;{{\mathcal {H}}}_e^-)\leqq \kappa _0+8c_1l_0\) and
with \(\tau _0:= \kappa _0+8c_1l_0 +2c_0^{-1}R_0\). This and stationarity of f yield
All this and Lemma 5.2 with \((l+m,le)\) in place of (l, x) yield the claim for all \(m\leqq \frac{4c_1}{c_0}l_0\) and \(l\geqq m\), with any \(C_\delta \geqq 2\tau _0\).
Now assume that \(l\geqq m\geqq \frac{4c_1}{c_0}l_0\), and let us first prove the direction
Pick \(R_m:= D_2(1+ \frac{2m}{c_0})\), with \(D_2\) from Lemma 2.9, and denote \(T_1(\omega ):=T^-_e(R_m,l,\omega )\). Then Lemma 5.3 shows that for each \(\omega \in \Omega \) we have \({T_1(\omega )}\in [\frac{l}{2c_1},\frac{2l}{c_0}]\) and hence also
holds with \(u_1:=u(\cdot ,\cdot ,\omega ;{{\mathcal {H}}}_e^-)\). Therefore,
Hence Lemma 2.1 shows that, with \(u_2:=u(\cdot ,\cdot ,\omega ;{{\mathcal {H}}}^{-}_e+le)\) and \({\tau _0}:=\kappa _0+\frac{2R_0}{c_0}\), we have
Since the set above contains \( B_{R_m}((l+m)e)\) and we also have \(T((l+m)e,\omega ;{{\mathcal {H}}}^{-}_e+le)\leqq \frac{2m}{c_0}\) by Lemma 5.3, we can apply Lemma 2.9 (see Lemma 3.7) with \(f_1=f_2=f\) and
to obtain
Taking expectations on both sides of this inequality and using stationarity of f yields
Since \(R_m\leqq Cl\) and \( l^{d+1}\phi (\frac{c_0}{4c_1} l)\leqq C\) for some \(C>0\) due to (H4’) and \(\beta _3\ge \frac{d+1}{2d+1+m_4'}\), it now follows from Lemma 5.4 that for any \(\delta >0\), we indeed have
with some \(C,C_\delta \) (and \({{\bar{C}}}_\rho \) from (5.3) resp. (5.4)).
Let us now turn to the other direction (again assuming \(l\geqq m\geqq \frac{4c_1}{c_0}l_0\)),
the proof of which is a little more involved. With \(\beta _1\) from (3.3), let
and then (see Sect. 2.1 for the other constants)
For any \(\omega \in \Omega \), denote \(T_1'(\omega ):=T_e^{+}(R_{l}+R_m,l,\omega )\) and
Lemma 5.3 yields \(T_1'(\omega )\in \left[ \frac{l}{2c_1},\frac{2l}{c_0}\right] \) and hence also
where \({{\mathcal {H}}}_e^+:={\mathbb {R}}^d{\setminus } {{\mathcal {H}}}_e^-\). Therefore,
From (2.6) and \({T_1'(\omega )}\leqq \frac{2l}{c_0}\) we have \(R_l\geqq L_{u,\eta ,1-\theta ^*}({T_1'(\omega )})\), and so
As for \(u'_2\), Lemma 2.1 shows that
and Lemma 2.8 then shows that for \({\tau _0^l}:= \kappa _0+{2R_l}{c_0 }^{-1} + D_1\eta ^{1-{m_1}}\) we have
Note also that there is \(C>0\) such that
because (3.3) shows that
From (5.10) and (5.11) we now have that
Since the set above contains \( B_{R_m}((l+m)e)\), we can apply Lemma 2.9 (see Lemma 3.7) with \(f_1=f_2=f\) and
to obtain
provided we also have
(notice that \(T((l+m)e,\omega ;{{\mathcal {H}}}^{-}_e) \geqq T_1'(\omega )\) because \(l+m\leqq 2l\leqq \frac{4c_1}{c_0}l\)). However since Lemma 2.1 yields \(T((l+m)e,\omega ;{{\mathcal {H}}}^{-}_e+le)\leqq \kappa _0+ \frac{2m}{c_0}\), (5.13) obviously holds even if (5.14) fails. Since \(T((m+l)e,\omega ;{{\mathcal {H}}}^{-}_e)\leqq \frac{2(m+l)}{c_0}\leqq \frac{4l}{c_0}\) by Lemma 5.3, we get from (5.9) and (5.12),
for some \(C>0\). Taking expectations, using stationarity of f, \(\beta _1\leqq \beta \), as well as \(R_l+R_m\leqq Cl\) and \( l^{d+1}\phi (l)\leqq C\) for some \(C>0\) (due to (H4’), (3.3), and \(\beta _3\ge \frac{d+1}{2d+1+m_4'}\)), and applying Lemma 5.4 shows that for any \(\delta >0\) we indeed have
with some \(C,C_\delta \) (and \({{\bar{C}}}_\rho \) from (5.3) resp. (5.4)). \(\quad \square \)
Now we are ready to prove the main result of this section.
5.1 Proof of Proposition 5.1
Let \(G(l):=\frac{1}{l}{\mathbb {E}}[T(le,\cdot \,)]\geqq 0\) and \(\gamma :=\beta +\delta \). It follows from Lemma 2.1 and Corollary 2.6 that
It therefore suffices to show that there is \(C_\delta >0\) such that with either \({{\bar{C}}}_{\delta ,\rho }:=C_\delta \sqrt{\rho }\) (when f satisfies (H2’)) or \({{\bar{C}}}_{\delta ,\rho }:=C_\delta \) (when f satisfies (H3) and (H4’)), we have for all \(l\geqq m\geqq 2\),
Since \(\beta <1\), we only need to consider \(\delta >0\) such that \(\gamma <1\).
By Lemma 2.1, G(l) is no more than
for all \(l\geqq 1\), and we also have \(|T(le,\cdot )-T(me,\cdot )|\leqq \frac{4}{c_0}+\kappa _0\) when \(|l-m|\leqq 2\). Therefore, there exists \(C_0>0\) such that for all l, m satisfying \(m+2\geqq l\geqq m\geqq 2\) we have
Using Proposition 5.5, we also see that there is \(C_\delta \geqq 1\) such that for any \(l\geqq m\geqq 2\) (and with \({\bar{C}}_{\delta ,\rho } \geqq 1\) given above),
Now assume that for some \(p\in \{2,3,4,\ldots \}\), there is \(N_p\geqq 2C_0\) such that for all \(m\in [2,2^p]\) and \(l\geqq m\) we have
This is in fact true for \(p=2\), because (5.16) shows that (5.19) holds for all \(m\in [2,4]\) and \(l\geqq m\) with \(N_2=\max \{4(\frac{2}{c_0}+\kappa _0),2C_0\}\). We will then extend (5.19) to all \(m\in [2,2^{p+1}]\) and \(l\geqq m\), with a relevant new constant \(N_{p+1}\geqq N_p\).
First, note that for any \(m\in [2,2^{p+1}]\) and \(l\in [m, \frac{3}{2}m]\) we have
Indeed, this holds for \(l\in [m,m+2] \) due to (5.17) and \(\frac{C_0}{m}\leqq \frac{N_p}{m+2}\leqq 3^{-\gamma } N_p (m+2)^{\gamma -1}\). And if instead \(l\in [m+2,\frac{3}{2}m]\), then \(l-m\in [2,\min \{2^p,\frac{l}{3}\}]\), so it follows from (5.18) and the induction hypothesis (5.19) that
Let us now consider \(m\in (2^p,2^{p+1}]\) and \(l\geqq \frac{3}{2}m\). Let \(l_k:=2^{-k}l\) for \(k=0,1,\ldots ,j\), with j chosen so that \(l_j\in (\frac{3}{4}m,\frac{3}{2}m]\). Since \(l_k=l_{k-1}-l_k\), from (5.18) we obtain
where \({{\bar{C}}}_{\delta ,\rho }' :={\bar{C}}_{\delta ,\rho }(\frac{4}{3})^{1-\gamma }\frac{2^{\gamma -1}}{1-2^{\gamma -1}}<\infty \) (recall that \(\gamma <1\)).
If \(l_j\in [m,\frac{3}{2}m]\), then (5.20) yields
If instead \( l_j\in \left( \frac{3}{4}m,m\right) \), then \(l_j\in [2,2^{p+1}]\) and \(m\in [l_j,\frac{4}{3}l_j]\). Hence (5.20) again yields
In either case we obtain
This, (5.19), and (5.20) now prove (5.19) with \(p+1\) in place of p (so it holds for all \(m\in [2,2^{p+1}]\) and \(l\geqq m\)) and with
Since (5.19) holds for \(p=2\) with \(N_2=\max \{4(\frac{2}{c_0}+\kappa _0),2C_0\}\), it follows that it holds for any \(p=2,3,\dots \) with \(N_p= \max \left\{ 4(\frac{2}{c_0}+\kappa _0),2C_0,\frac{3^\gamma }{3^\gamma -1}({\bar{C}}_{\delta ,\rho }+{{\bar{C}}}_{\delta ,\rho }' ) \right\} =: {\bar{C}}_{\delta ,\rho }''\). This proves (5.15).
6 Deterministic Front Speeds and Proof of Theorem 1.7
We are now ready to prove Theorem 1.7. This is because it was shown in [16] that such homogenization results for reaction–diffusion equations and related models follow from appropriate estimates on the dynamics of the solutions to (5.1) for all vectors \(e\in {\mathbb {S}}^{d-1}\). We will be able to obtain these estimates using the main results from Sects. 3–5.
Specifically, we will use Proposition 5.1, and either Proposition 3.8 (when we assume (H2’)) or Proposition 4.2 (when we assume (H3) and (H4’)) in the proof. We will handle both cases at once, using that either of the latter two propositions yields (5.5) above, with the notation from either (5.3) in the first case or (5.4) in the second.
For us, the key result from [16] will be Theorem 5.4, which applies when for almost all \(\omega \in \Omega \), the reaction \(f(\cdot ,\cdot ,\omega )\) has deterministic strong exclusive front speeds in all directions \(e\in {\mathbb {S}}^{d-1}\). We will first define these, following Definitions 1.3, 1.6, and Remark 3 after Hypothesis H’ in [16], and then prove their existence.
Definition 6.1
Let f satisfy (H1) and let \(e\in {\mathbb {S}}^{d-1}\). If there is \(c^*(e)\in {\mathbb {R}}\) and \(\Omega _e\subseteq \Omega \) with \({\mathbb {P}}(\Omega _e)=1\) such that for each \(\omega \in \Omega _e\) and compact \(K\subseteq {{\mathcal {H}}}^{+}_e=\{x\in {\mathbb {R}}^d\,|\, x\cdot e>0\}\),
holds for the solution to (5.1) with \(l=0\) and some \(u_{0,{{\mathcal {H}}}^{-}_e}\) satisfying (1.9) and (2.3) with \(S={{\mathcal {H}}}^{-}_e\), then we say that \(c^*(e)\) is a deterministic front speed in direction e for (1.1).
This speed is strong if for each such \(\omega \) and K, and each \(\Lambda \geqq 0\), we have
And if, in addition, for each such \(\omega \) and K there is \(\lambda _{K,\omega ,e}:(0,1]\rightarrow (0,1]\) satisfying \(\lim _{a\rightarrow 0}\lambda _{K,\omega ,e}(a)=0\) such that for each \(\Lambda >0\) and \(a\in (0,1]\) we have
where \(w_{e,a}(\cdot ,\cdot , \omega )\) solves (1.1) with initial data
then \(c^*(e)\) is a deterministic strong exclusive front speed in direction e for (1.1).
Remark
1. Lemma 2.1 and the comparison principle show that all these definitions are independent of the choice of \(u_{0,{{\mathcal {H}}}^{-}_e}\) satisfying (1.9) and (2.3). We could equivalently choose \(u_{0,{{\mathcal {H}}}^{-}_e}:=(1-\theta ^*)\chi _{{{\mathcal {H}}}^{-}_e}\) here, but having solutions with \(u_t\geqq 0\) will be more convenient.
2. We will show that in Theorem 1.7, \(\lambda _{K,\omega ,e}(a)=a\) for all \((K,\omega ,e)\) as above and all small enough \(a>0\) (depending on \( M, \theta _1,m_1,\alpha _1, \mu _*,K\)).
Let us first show that the reactions we consider here have deterministic strong front speeds, and then we will show that all these speeds are also exclusive.
Proposition 6.2
Assume that f either satisfies (H2’) and has a finite range of dependence, or satisfies (H3) and (H4’). For each \(e\in {\mathbb {S}}^{d-1}\), let \(\bar{T}(e)\) be from Proposition 5.1. Then \(c^*(e):=\bar{T}(e)^{-1}\in [c_0,c_1]\) is the deterministic strong front speed in direction e for (1.1).
Proof
Fix any \(e\in {\mathbb {S}}^{d-1}\), \(\Lambda \geqq 0\), and compact \(K'\subseteq {{\mathcal {H}}}_e^+\). Let \(K\subseteq {{\mathcal {H}}}_e^+\) be compact and such that \(K'\subseteq K^0\), let \({d_K}:=d(K,{{\mathcal {H}}}_e^-)>0\), and let \(A_K:=1+{{\,\mathrm{diam}\,}}(K)^d<\infty \). Let us also use the notation (5.2), and for any \(\eta \in (0,\theta ^*]\) and \(t\geqq 0\), let
Assume that \(u(t,x,\Upsilon _y\omega )\geqq 1-\theta ^*\) for some \((t,x,y,\omega )\in [0,\infty )\times {\mathbb {R}}^{2d}\times \Omega \). Since for all \(y'\in {\mathbb {R}}^d\) we have \(u(\cdot ,\cdot ,\Upsilon _{y'}\omega )=u(\cdot ,\cdot +y',\omega ;{{\mathcal {H}}}_e^-+(y'\cdot e)e)\), Lemma 2.1 and comparison principle yield for all \(y'\in B_{\sqrt{d}}(y)\),
with \(\tau _0:=\kappa _0+2c_0^{-1}(R_0+\sqrt{d})\). Applying Lemma 2.1 again, we obtain for all \(y'\in B_{\sqrt{d}}(y)\),
Then Lemma 2.8 shows that if \(\eta \in (0,\theta ^*]\) and \(\tau _\eta :=2\tau _0+4c_0^{-1}\sqrt{d}+D_1\eta ^{1-{m_1}}\), then
for all \(y'\in B_{\sqrt{d}}(y)\). This shows that we can only have \(\omega \in I_t^\eta (K,\Lambda )\) for some \(t\geqq \tau _\eta \) if
where \( Z_{K,\Lambda ,t}:=((c^*(e)e-K)t\cap {\mathbb {Z}}^d)\times (B_{\Lambda (t+1)}(0)\cap {\mathbb {Z}}^d). \) From this we obtain
If \(t\geqq 2(\tau _\eta +\kappa _0+\frac{2}{c_0})\), then from Lemma 2.1 we have \(T(x,\Upsilon _y\omega )\leqq t-\tau _\eta \) whenever \(x\cdot e\leqq 1 + \frac{c_0t}{4}\). Hence, with \(Z_{K,\Lambda ,t}':=\{(x,y)\in Z_{K,\Lambda ,t}\,|\, x\cdot e> 1 + \frac{c_0t}{4} \}\) we obtain
Note also that there is \(C>0\) such that this sum has at most \(C(1+\Lambda ^d)A_K t^{2d}\) terms.
Consider any \((x,y)\in Z_{K,\Lambda ,t}'\), where \(t\geqq 2(\tau _\eta +\kappa _0+\frac{2}{c_0})\). With the notation from (5.3) resp. (5.4), and \(\beta ':=\frac{1+\beta }{2}\in (0,1)\), Proposition 5.1 and stationarity of f yield
Since \(x\cdot e\leqq (c^*(e)-{d_K})t\) and \(c^*(e)={\bar{T}(e)}^{-1}\leqq c_1\), we obtain
whenever
Hence for such t, (5.5) yields \(C>0\) (defining \({\bar{C}}_\rho \) via (5.3) resp. (5.4)) such that
when \((x,y)\in Z_{K,\Lambda ,t}'\), where we also used that \(\frac{c_0t}{4}\leqq x\cdot e\leqq c_1t\) and \(\tau _\eta \leqq \frac{d_K}{4c_1}t\) (recall that C can change from line to line). This and (6.4) show that for all large enough t we have
Then \(\sum _{n\geqq 1}{\mathbb {P}}[I_n^\eta (K,\Lambda )]<\infty \) since \(\beta _3> \frac{2d+1}{2d+1+m_4'}\), so the Borel-Cantelli Lemma shows that for a.e. \(\omega \in \Omega \), there is \(N_\omega \) such that \(\omega \notin \bigcup _{n\geqq N_\omega } I_n^\eta (K,\Lambda )\). But since \(K'\subseteq K^0\) means there is \(\tau \) such that
for all \(t\geqq \tau \), from \(u_t\geqq 0\) and the definition of \(I_t^\eta (K,\Lambda )\) we obtain
for all such \(\omega \) and all \(t\ge \max \{N_\omega ,\tau \}\). Applying this argument with \(\eta =\frac{1}{n}\), \(\Lambda =n\), and \(K'=\{x\in {\mathbb {R}}^d\,|\, x\cdot e\in [\frac{1}{n},n]\text { and }|x-(x\cdot e)e|\leqq n\}\) for each \(n\in {\mathbb {N}}\) yields \(\Omega _1\subseteq \Omega \) with \({\mathbb {P}}(\Omega _1)=1\) for which the first statement in (6.1) holds.
It remains to prove the the second statement for some \(\Omega _2\) with \({\mathbb {P}}(\Omega _2)=1\), as we can then take \(\Omega _e:=\Omega _1\cup \Omega _2\). The proof is similar to that of the first statement. With the setup from the start of its proof, we now let
Assume that \(u(t,x,\Upsilon _y\omega )> \eta \) for some \((t,x,y,\omega )\in [0,\infty )\times {\mathbb {R}}^{2d}\times \Omega \). Then (2.6) shows that with \(L_{t,\eta }:= \mu _{*}^{-1}(1+t^{\alpha _2} )\eta ^{-{m_2}}\), there is \(x'\in B_{L_{t,\eta }}(x)\) such that
Lemma 2.1 now shows that
In the same way as we obtained (6.3) (but using Lemma 2.1 instead of Lemma 2.8), we now get for all \(y'\in B_{\sqrt{d}}(y)\) and with \(\tau _{t,\eta }:=3\kappa _0+\frac{2}{c_0}(L_{t,\eta }+R_0+2\sqrt{d})\),
So similarly to (6.4), with \(Z_{K,\Lambda ,t}:=((c^*(e)e+K)t\cap {\mathbb {Z}}^d )\times (B_{\Lambda t}(0)\cap {\mathbb {Z}}^d)\) we get for all \(t\geqq 0\),
And again, there is \(C>0\) such that this sum has at most \(C(1+\Lambda ^d)A_K t^{2d}\) terms.
Let \(d_K':=d_H(K,{{\mathcal {H}}}_e^-)\) and consider any \(t\geqq \frac{1}{c_0}\) and \((x,y)\in Z_{K,\Lambda ,t}\). Then \(c^*(e)t\geqq 1\) and
Thus Proposition 5.1 and \(c^*(e)=\bar{T}(e)^{-1}\) imply, as above (with \(\beta '=\frac{1+{\beta }}{2}\)),
whenever
where \(C_{K,\eta }\) is such that \(\tau _{t,\eta }\leqq \frac{d_K t}{4c_1}\) for all \(t\geqq C_{K,\eta }\) (this exists because \(\alpha _2<1\), and will be used next). Hence for such t, (5.5) yields \(C>0\) (defining \({\bar{C}}_\rho \) via (5.3) resp. (5.4)) such that
when \((x,y)\in Z_{K,\Lambda ,t}\), where we also used (6.6) and \(\tau _{t,\eta }\leqq \frac{d_K t}{4c_1}\) in the last inequality. This and (6.5) show that for all large enough t we have
We can now conclude the proof of the second statement in (6.1) as we did the proof of the first statement, this time using that
for all large enough t. \(\quad \square \)
Remark
This proof shows that (6.1) holds with \(\Lambda t\) replaced by \(\exp (t^\gamma )\) for any \(\gamma <2-2\beta \).
Proposition 6.3
Under the hypotheses of Proposition 6.2, for each \(e\in {\mathbb {S}}^{d-1}\), the speed \(c^*(e)\) is also a deterministic strong exclusive front speed in direction e for (1.1).
Proof
Having Proposition 6.2, this proof is now similar to the one of [16, Theorem 1.7(i)].
For any \((e,a,\omega )\in {\mathbb {S}}^{d-1}\times (0,1)\times \Omega \), let \(u(\cdot ,\cdot ,\omega ;{{\mathcal {H}}}_e^-)\) and \( w_{e,a}(\cdot ,\cdot ,\omega )\) be from Definition 6.1, and let \(\tau _a:=1+D_1a^{1-m_1}\). Lemma 2.9 (see Lemma 3.7) shows that if \(a\in (0,\frac{1}{2}\min \{\theta ^*,M_*^{-1}\}]\), then
is a supersolution to (1.1) on \((\kappa _*,\infty )\times {\mathbb {R}}^d\). Moreover, \(u_t\geqq 0\) and Lemma 2.8 show that
The comparison principle now yields for all \(t\geqq 0\),
It now follows from Proposition 6.2 that (6.2) holds with \(\lambda _{K,\omega ,e}(a)=a\) for all \(\Lambda \geqq 0\) and all compact \(K\subseteq {{\mathcal {H}}}_e^++M_*a\,c^*(e)e\). This is true for all \(a\in (0,\frac{1}{2}\min \{\theta ^*,M_*^{-1}\}]\), so the result follows after letting \(\lambda _{K,\omega ,e}(a):=1\) for all \(a\in (\frac{1}{2}\min \{\theta ^*,M_*^{-1}\},1]\). \(\quad \square \)
6.1 Proof of Theorem 1.7
For any \(e\in {\mathbb {S}}^{d-1}\), let \(\Omega _e\subseteq \Omega \) with \({\mathbb {P}}(\Omega _e)=1\) be the set from Definition 6.1 and let \(c^*(e)\) be the corresponding deterministic strong exclusive front speed for (1.1) from Proposition 6.3. Let \(A\subseteq {\mathbb {S}}^{d-1}\) be a dense countable set and let \(\Omega _0:=\bigcap _{e\in A} \Omega _e\). Then \({\mathbb {P}}(\Omega _0)=1\), and for each \(\omega \in \Omega _0\), (1.1) with this fixed \(\omega \) has a strong exclusive front speed \(c^*(e)\) in each direction \(e\in A\) (i.e., (6.1) and (6.2) hold for this fixed \(\omega \) and each \(e\in A\), \(\Lambda \geqq 0\), and compact \(K\subseteq {{\mathcal {H}}}_e^+\)).
Then [16, Theorem 4.4(i)] shows that (1.1) with this fixed \(\omega \) has a strong exclusive front speed \(c^*_\omega (e)\) in each direction \(e\in {\mathbb {S}}^{d-1}\), and \(c^*_\omega \) is Lipschitz with Lipschitz constant only depending on M. But then \(c^*_\omega (e)\) must be independent of \(\omega \in \Omega _0\) for each \(e\in {\mathbb {S}}^{d-1}\) (instead of just all \(e\in A\)), and hence equals \(c^*(e)\) from Proposition 6.3 because \({\mathbb {P}}(\Omega _0)=1\).
Theorem 1.7 now follows directly from [16, Theorem 5.4] applied separately to each \(\omega \in \Omega _0\) (see also the remarks after Hypothesis H’ in [16]).
7 Proof of Theorem 1.8
In this section we will show how to extend the above analysis to the cases considered in Theorem 1.8. We can obviously assume \(\alpha _2'> 0\) without loss, and all constants with C in them may depend on (2.10) as well as on \(\alpha _2'\).
Since we now replace (H2’) by (H2”), the estimates (2.6) instead become
for all \(a\in [0,a_2]\) and either all \(u\in {{\mathcal {U}}}_{f,a}\) (when assuming (H2”)) or all \(u\in \bigcup _{n\geqq n_4} {{\mathcal {U}}}_{f_n,a}\) (when assuming (H3) and (H4”)), again with some \(\mu _*,\kappa _*>0\).
We will first assume without loss that \(a_2=0\). Then of course also \(a=0\) and \({{\mathcal {U}}}_{f,a}={{\mathcal {U}}}_f\) above, so (7.1) is just (2.6) with the extra factor of \(t^{\alpha _2'}\) in the second estimate. We will now show how the results in Sects. 2–6 and their proofs change due to this.
Of the results in Sect. 2, clearly only Lemmas 2.9 and 2.10 are affected by this change. They will instead become the following two results.
Lemma 7.1
Let \(f_1\) satisfy (H2”) and \(f_2\) satisfy (H1’), and let \(M_*:=2^{\alpha _2'}\frac{1+M}{\mu _*}\), with \(\mu _*,\kappa _*\) from (7.1) for all \(u\in {{\mathcal {U}}}_{f_1}\). Fix some \(\omega \in \Omega \) and let \(u_1,u_2:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solve (1.1) with \(f_1,f_2\) in place of f, respectively. If \(u_1\in {{\mathcal {U}}}_{f_1}\), \(t_0\geqq 0\), \(T\geqq 2\kappa _*\), and for some \(\eta \in [0, \frac{1}{2}\min \{\theta ^*,M_*^{-1}(T+t_0)^{-\alpha _2'}\} ]\) we have
then
is a supersolution to (1.1) with \(f_2\) in place of f on \((\kappa _*,T)\times {\mathbb {R}}^d\), and
is a subsolution to (1.1) with \(f_2\) in place of f on \( (2\kappa _*,T)\times \{ x\in {\mathbb {R}}^d \,|\,u_1(t_0,x)<1-\eta \}. \)
Moreover, there is \(D_2=D_2(M,\theta _1,m_1,\alpha _1)\geqq 1\) such that if also \({T}_{u_2}(y)\leqq T\), and
for some \(y\in {\mathbb {R}}^d\) and \(R\geqq D_2(1+{T}_{u_2}(y)),\) then
Proof
The proof is the same as that of Lemma 2.9, replacing (2.6) by (7.1) and using \(\tau _\pm (t):=(1\pm M_*(T+t_0)^{\alpha _2'}\eta )t+t_0\). In particular, we use in it that for \(t\leqq T\) we have
We also have \(D_2:=2\sqrt{Md}\ln \frac{4d}{\theta ^*}\) as before. \(\quad \square \)
Lemma 7.2
Let \(f_1\) satisfy (H2”) and \(f_2\) satisfy (H1’), with at least one satisfying (H3) with \(\alpha _3\leqq 1\), and let \(M_*,D_2\) be from Lemma 7.1. Fix some \(\omega \in \Omega \) and let \(u_1,u_2:[0,\infty )\times {\mathbb {R}}^d\rightarrow [0,1]\) solve (1.1) with \(f_1,f_2\) in place of f, respectively. If \(u_1\in {{\mathcal {U}}}_{f_1}\), for some \(y\in {\mathbb {R}}^d\), \(R\geqq D_2(1+{T}_{u_2}(y))\), and \( \eta \in [0, \frac{1}{2}\min \{\theta ^*,M_*^{-1}(\max \{{T}_{u_2}(y),2\kappa _*\}+t_0)^{-\alpha _2'}\} ] \) we have
and \(u_2(0,\cdot )\leqq u_1(t_0,\cdot )\) for some \(t_0\geqq 0\) and all \(x\in B_R(y)\), then
Proof
The proof is the same as that of Lemma 2.10, replacing (2.6) and Lemma 2.9 by (7.1) and Lemma 7.1 with \(T:=\max \{{T}_{u_2}(y),2\kappa _*\}\), and using \(\tau _+(t):=(1+ M_* (T+t_0)^{\alpha _2'}\eta )t+t_0\). \(\quad \square \)
We can now extend all of Sects. 3–5 to (H2”) in place of (H2’), and obtain the following analog of Proposition 6.2.
Proposition 7.3
Assume that f either satisfies (H2”) and has a finite range of dependence, or satisfies (H3) and (H4”). Then for each \(e\in {\mathbb {S}}^{d-1}\), (1.1) has a deterministic strong front speed \(c^*(e)\in [c_0,c_1]\) in direction e.
Proof
In the whole proof, we assume without loss that \(a_2=0\) (and so \(a=0\) as well). When extending results from Sect. 3, we assume that f satisfies (H2”) and has range of dependence at most \(\rho \in [1,\infty )\); in Sect. 4 we instead assume (H3) and (H4”); and in Sects. 5 and 6 we assume either of these two cases, as before.
Most of Sect. 3 is unchanged, with (3.3) replaced by
which is still in (0, 1) thanks to \(\alpha _2'<\min \{ \frac{1}{m_1-1}, \frac{1-\alpha _2}{m_2} \}\). The first adjustment is required in Lemma 3.4, where we used the second claim in (2.6) to obtain (3.8). We have here \(u(\tau (x,\omega ),y,\omega )\ge \theta ^*\) for some \(y\in \overline{B_1(x)}\), and can instead use Lemma 2.1 to get
Since the first claim in (7.1) (with \(a=0\) and \(\eta =\theta ^*\)) yields \(C>0\) such that
(recall that \(u\in {{\mathcal {U}}}_f\)) and Lemma 2.1 also implies \(\tau (x,\omega )\leqq C(1+t)\) whenever \(\omega \in F_{t,x}\), we have \(T(x,\cdot )\leqq \tau (x,\cdot )+C(1+t^{\alpha _2})\) on \(F_{t,x}\). Hence the first claim of Lemma 3.4 becomes
while the second holds only for \(s\in [0,t-C(1+t^{\alpha _2})]\).
As for Proposition 3.5, instead of (3.13) we let
where \(C_1>0\) will be chosen shortly and
Note that then, similarly to (3.14), we have
(in particular, \(\gamma >\alpha _2'\)), and (3.15) and (3.16) continue to hold. Now we pick \(C_1\) so that with with \(T:=\max \{T(x_0,\omega ), {T}_i(x_0,\omega ),2\kappa _*\}\) we have
which is possible due to \(\max \{T(x_0,\omega ),{T}_i(x_0,\omega ),t_0\}\leqq C(1+d(x_0,S))\) and (3.16). Then we can use Lemma 7.1 with this T and \(\eta \) (instead of Lemma 2.9) to see that (3.17) becomes
because \(\beta _1\geqq 1+\alpha _2'-\gamma \). This ends the first half of the proof. Using again Lemma 7.1 instead of Lemma 2.9 in the second half of it, with \(\eta \) as above, shows that
is a subsolution to (1.1) on \( (2\kappa _*,T)\times ({\mathbb {R}}^d\backslash \Gamma _{u,1-\eta }(t_0,\omega ))\). The rest of the proof does not use the second claim in (2.6) and is unchanged (using this \(u_-\) and also (7.4)), with (3.23) becoming
This finishes the proof.
The proof of Lemma 3.6 remains the same. In the proof of Proposition 3.1, the change in Lemma 3.4 turns the C in (3.25) and (3.30) into \(C(1+t^{\alpha _2})\) (recall that \(s\leqq t\) in the argument), which is then added to the right-hand sides of (3.26) and (3.29). Then (3.30) shows that there is \(C>0\) such that \(X_t=T(x,\cdot )\) for all \(t\geqq C(1+d(x,S))\), and we now pick N to be the smallest integer with \(N\tau \geqq C(1+d(x,S))\). The estimate \(N\leqq C_1d(x,S)(\rho +d(x,S)^{\beta _1})^{-1}\) now still holds (recall that \(d(x,S)>\rho \geqq 1\) here), and (3.31) remains unchanged because the term added to (3.29) is estimated by \(C(1+(N\tau )^{\alpha _2})\leqq C(1+d(x,S)^{\beta _1})\) because \(\alpha _2<\beta _1\). The rest of the proof of Proposition 3.1 remains the same.
Lemmas 3.7 and 4.1 are unchanged except for replacement of (H2’), (H4’), \({{\mathcal {U}}}_f\), \({{\mathcal {U}}}_f'\), Lemma 2.9, and Lemma 2.10 in their statements and proofs by (H2”), (H4”), \({{\mathcal {U}}}_{f,a}\), \({{\mathcal {U}}}_{f,a}'\), Lemma 7.1, and Lemma 7.2, respectively (here we can even allow any fixed \(a_2\in [0,\frac{1}{2}\theta ^*]\) in (H2”), although \(a_2=0\) is sufficient). The proof of Proposition 3.8 also remains the same, using Lemma 7.1 (with \(T=\infty \) because \(\eta =0\)) instead of Lemma 2.9.
In Proposition 4.2, we replace (4.1) by
which is still in (0, 1) thanks to \(\alpha _2'<\frac{m_4}{m_3}\). Then when we use Lemma 2.10 in the proof, we replace it by Lemma 7.2 with the same \(\eta :=(\frac{\alpha _4}{\alpha _3}n^{-m_4})^{1/m_3}\), but now we need to pick n so that
with \(T:=\max \{T(x,\omega ;S),T_n(x,\omega ;S),2\kappa _*\}\). Since \(T \leqq C(1+d(x,S))\) due to Lemma 2.1, and \(\beta _3\frac{m_4}{m_3}>\alpha _2'\) due to \(\alpha _2'<\frac{m_4}{m_3}\) and (7.5), there is again \(C_1>0\) such that it suffices to let n be the smallest integer for which (4.5) holds. Then a double application of Lemma 7.2 replaces (4.4) by
Since there is again \(C_2>0\) such that
because \( 1+\alpha _2' - \beta _3\frac{m_4}{m_3} \leqq \frac{1+\beta _3}{2}\), the rest of the proof of Proposition 4.2 is unchanged.
Most of Sect. 5 is also unchanged, with the only two adjustments needed in the proof of Proposition 5.5. We used Lemma 2.9 when proving (5.8), and we can just replace it by Lemma 7.1 without any other change because there we had \(\eta =0\). We also used Lemma 2.9 when proving (5.13), and the change to Lemma 7.1 now requires us to replace (5.9) by
with \(\gamma \) from (7.3) (recall that (7.4) shows \(\gamma \geqq 1+\alpha _2'-\beta _1\geqq 1-\beta _1\), so \(\eta \leqq l^{\beta _1-1}\) as well; in fact, we could have chosen this \(\eta \) in (5.9) as well) and \(C_1\geqq 2\) such that with \(T=\max \{\kappa _0+\frac{2(l+m)}{c_0},2\kappa _*\}\) we have \(\eta \leqq \frac{1}{2M_*}(T+\tau _0^l)^{-\alpha _2'}\). This is possible because of (5.12) and \(\gamma \geqq \alpha _2'\). Then replacing Lemma 2.9 by Lemma 7.1 with this T yields
instead of (5.13). But the addition of \((T+\tau _0^l)^{\alpha _2'}\) here does not require further changes because
by (7.4) (recall that we assume here \(l\geqq m\geqq \frac{4c_1}{c_0}l_0\)).
The proofs of Propositions 5.1 and 6.2 then remain unchanged, finishing the proof. \(\quad \square \)
We are only able to obtain an (H2”)-version of Proposition 6.3 when \(a_2>0\), and we do so below. But even without that, we can already prove Theorem 1.8(ii).
Proof of Theorem 1.8(ii)
This is identical to the proof of Theorem 1.7 above, with the word “exclusive” and (6.2) dropped, and using Proposition 7.3 and [16, Theorem 1.4(iii)] instead of Proposition 6.3 and [16, Theorem 5.4], respectively. Note that f is also stationary ergodic in [16, Theorem 1.4(iii)], but this is only used in the first paragraph of its proof to show existence of deterministic strong front speeds for all \(e\in {\mathbb {S}}^{d-1}\) (which we proved in Proposition 7.3), so that result extends to the case in hand. \(\quad \square \)
To prove Theorem 1.8(i), we need to show that the deterministic strong front speeds from Proposition 7.3 are exclusive. For this, we will need some uniform estimates on the reactions
for \((x,u,\omega )\in {\mathbb {R}}^d\times [0,1]\times \Omega \). Note that the transformation \(u\mapsto \frac{u-a}{1-a}\) turns solutions u to (1.1) for which \(a\leqq u\leqq 1\) into solutions to (1.1) with \(f_a\) in place of f for which \(0\leqq u\leqq 1\).
Proposition 7.4
Assume that f either satisfies (H2”) with \(a_2>0\) and has a finite range of dependence, or satisfies (H3) and (H4”) with \(a_2>0\). Then for each \((a,e)\in (0,a_2]\times {\mathbb {S}}^{d-1}\), (1.1) with \(f_a\) in place of f has a deterministic strong front speed \(c^*_a(e)\in [c_0,c_1]\) in direction e, and \(\lim _{a\rightarrow 0} c^*_a(e)=c^*(e)\) holds uniformly in \(e\in {\mathbb {S}}^{d-1}\) (with \(c^*(e)\) from Proposition 7.3).
Proof of Theorem 1.8(ii)
Clearly \(f_a\) satisfies (H1) with the same M and \(m_1\), and \(\theta _1\) and \(\alpha _1\) replaced by \(\frac{1}{2}\theta _1\) and \(\alpha _1(1-\frac{1}{8} {\theta _1})^{m_1-1}\), respectively (recall that \(a\leqq a_2\leqq \frac{1}{8}{\theta _1}\)).
If we now assume (H2”) and finite range of dependence of f, and let \(u^a:=\frac{u-a}{1-a}\) for some \(u\in {{\mathcal {U}}}_{f,a}\) with initial datum \(u_{0,k,a}\), then \(u^a_t=\Delta u^a+f_a(x,u^a,\omega )\) on \((0,\infty )\times {\mathbb {R}}^d\) (with the same \(\omega \)) and its initial datum \(\frac{1}{1-a}(u_{0,k,a}-a)\) satisfies (1.8) and (1.9) (with F now defined via \(f_a\)). Moreover, (1.12) and \(a_2\leqq \frac{1}{2}\) show that we also have
Hence for each such \(f_a\) we have (H2”) with \(a_2=0\), \(\theta ^*\) replaced by \(2\theta ^*\), the above constants in (H1), and \({{\mathcal {U}}}_{f_a}:=\{u^a\,|\,u\in {{\mathcal {U}}}_{f,a}\}\) (and the same \(m_2,\alpha _2,\alpha _2'\)). Moreover, there are \(\mu _*,\kappa _*>0\) such that (7.1) holds for all \(a\in [0,a_2]\) and \(u\in {{\mathcal {U}}}_{f_a}\), with \(\theta ^*\) replaced by \(2\theta ^*\). This and the remark after (2.1) (which shows that replacing \(\theta ^*\) by \(2\theta ^*\) in (7.1) does not change any of the above proofs) now show that Proposition 7.3 holds for all the \(f_a\), and all constants in its proof are uniform in \(a\in [0,a_2]\). In particular, (7.1) holds with the same \(\mu _*,\kappa _*>0\) (and \(\theta ^*\) replaced by \(2\theta ^*\)) for all \(a\in [0,a_2]\) and \(u\in {{\mathcal {U}}}_{f_a}'\) (see Lemma 3.7), and the first claim in Proposition 5.1 holds for \(f_a\) with \(\beta _1\) from (7.2) and \(C_\delta \) uniform in \(a\in [0,a_2]\).
The same argument applies when we assume (H3)+(H4”), where the passage to \(f_a\) and \(f_{n,a}\) also replaces \(\alpha _3 \) by \(\alpha _3(1-\frac{\theta _1}{8})^{m_3-1}\) in (H3) and \(\alpha _4\) by \(\alpha _4 (1-\frac{\theta _1}{8})^{-1}\) in (H4”). Again we obtain Proposition 7.3 for all the \(f_a\), as well as that (7.1) holds with some \(\mu _*,\kappa _*>0\) (and with \(\theta ^*\) replaced by \(2\theta ^*\)) for all \(a\in [0,a_2]\) and \(u\in {{\mathcal {U}}}_{f_a}'\), and the second claim in Proposition 5.1 holds for \(f_a\) with \(\beta _3\) from (7.5) and \(C_\delta \) uniform in \(a\in [0,a_2]\).
It therefore remains to prove the last claim, with the above deterministic strong front speeds denoted \(c^*_a(e)\) (where clearly \(c^*_0(e)=c^*(e)\)). To achieve this, we will use uniformity of the estimates in Proposition 5.1 in \(a\in [0,a_2]\). We therefore denote by \(v_{e,a}(\cdot ,\cdot ,\omega )\) the solution to (5.1) with f replaced by \(f_a\), some \((e,\omega )\in {\mathbb {S}}^{d-1}\times \Omega \), and \(l=0\) (then of course \(v_{e,a}(\cdot ,\cdot ,\omega )\in {{\mathcal {U}}}_{f_a}'\)). We also let
with \({{\mathcal {U}}}_{f,a}'\) obtained from \({{\mathcal {U}}}_{f,a}\) as in Lemma 3.7, and for any \(x\in {\mathbb {R}}^d\),
These definitions, Lemma 2.1, and \(a_2\leqq \frac{1}{2}\theta ^*\) show that there is C such that
We will treat both cases (H2”)+finite range and (H3)+(H4”) at once, using the notation from either (5.3) in the first case or (5.4) in the second. Then the claims from Proposition 5.1, with \(\delta :=\frac{1-\beta }{2}\) and \({\bar{C}}'_\rho :=C_{(1-\beta )/2} \bar{C}_\rho \) independent of \((e,a,\omega )\) become
for all \((e,a,\omega )\in {\mathbb {S}}^{d-1}\times [0,a_2]\times \Omega \) and all \(l\geqq 1\).
Lemma 2.1 shows that \(u_{e,a}(\tau _0,\cdot ,\cdot )\geqq u_{e,0}(0,\cdot ,\cdot )\) with \(\tau _0:=\kappa _0+\frac{2R_0}{c_0}\), hence the comparison principle yields \(u_{e,a}(\tau _0+t,\cdot ,\cdot )\geqq u_{e,0}(t,\cdot ,\cdot ) \) for all \(t\geqq 0\). This and (7.6) immediately imply
so \(c_a^*(e)\geqq c^*(e)\) for all \((e,a)\in {\mathbb {S}}^{d-1}\times [0,a_2]\) by (7.7) (this also shows that \(c_a^*(e)\geqq c_0\)).
Since \(T_{e,a}'(le,\cdot )\leqq Cl\) for all \(l\geqq 1\) by Lemma 2.1, Lemma 7.1 with \(f_1=f_2=f\) and
yields
as long as \(l \in [1,(C'a)^{-1/\alpha _2'}]\) (for some \(C,C'>0\)). It follows by (7.7) that for such l we have
Picking \(l:=a^{-1/2\alpha _2'}\) and using (7.8) now yield for all small enough a (depending only on (2.10) and \(\alpha _2\)),
Since \(\beta <1\), \(\alpha _2'>0\), and \(c_a^*(e)\geqq c^*(e)\), the uniform convergence claim follows. \(\quad \square \)
We can now extend Proposition 6.3 to the case \(a_2>0\).
Proposition 7.5
Under the hypotheses of Proposition 7.4, for each \(e\in {\mathbb {S}}^{d-1}\), the speed \(c^*(e)\) is a deterministic strong exclusive front speed in direction e for (1.1).
Proof of Theorem 1.8(ii)
Fix any \(e\in {\mathbb {S}}^{d-1}\), and let \(u_{e,a}\) be from (7.6) and \(c_a^*(e)\) from Proposition 7.4. From that proposition and \(f(\cdot ,a,\cdot )\equiv 0\) for all \(a\in [0,a_2]\) we know that for almost all \(\omega \in \Omega \) we have
for each \(a\in [0,a_2]\cap {\mathbb {Q}}\), \(\Lambda >0\), and compact \(K\subseteq {{\mathcal {H}}}^{+}_e=\{x\in {\mathbb {R}}^d\,|\, x\cdot e>0\}\). Fix any such \(\omega \), and then any compact \(K\subseteq {{\mathcal {H}}}^{+}_e\). Let \(K'\subseteq {{\mathcal {H}}}_e^+\) be compact and such that \(K\subseteq (K')^0\). Proposition 7.4 then yields \(\delta \in (0,a_2]\) such that for all \(a\in [0, \delta ]\) we have
Since \(\alpha _2'<\min \{ \frac{1}{m_1-1}, \frac{1-\alpha _2}{m_2} \}\), there exists \(T_0\geqq \kappa _*\) such that for all \(T\geqq T_0\) we have
where \(\eta _T:=T^{-\gamma }\) with \(\gamma :=\frac{1}{2}(\alpha _2'+\frac{1}{m_1-1})\), and \(\tau _T:=1+D_1\eta _T^{1-m_1}\) with \(D_1\) from Lemma 2.8.
Now fix any \(\Lambda >0\). We see from (7.9) that for each \(a\in [0,a_2]\cap {\mathbb {Q}}\), there is a function \(\varphi _{a}:[0,\infty )\rightarrow [0,\infty )\) such that \(\lim _{t\rightarrow \infty }\varphi _{a}(t)=0\) and
Pick any \(T\geqq T_0\) and \(a\in [0, \delta ]\cap {\mathbb {Q}}\), and let \(w_{e,a}\) be from Definition 6.1. Then Lemma 2.8 yields \(u_{e,a}(\tau _T,\cdot ,\cdot )\geqq 1-\eta _T\) on \({{\mathcal {H}}}_e^-\), so from \(u_{e,a}\geqq a\) and \((u_{e,a})_t\geqq 0\) we see that
for all \(t\geqq 0\). Since Lemma 7.1 and (7.11) show that
is a supersolution to (1.1) on \((\kappa _*,T+\kappa _*)\times {\mathbb {R}}^d\), the comparison principle and (7.13) yield
for all \((t,x)\in [0,T]\times {\mathbb {R}}^d\). This, (7.11), (7.10), (7.12), and \((u_{e,a})_t\geqq 0\) now show that
Hence
for all \(a\in [0,\delta ]\cap {\mathbb {Q}}\) (and the previously fixed \((\omega ,K,\Lambda )\)). Since \(w_{e,a}\) is non-decreasing in a, this extends to all \(a\in [0,\delta ]\), and since \(\delta \) does not depend on \(\Lambda \), we obtain (6.2) with \(\lambda _{K,\omega ,e}(a)=a+(1-a)\chi _{(\delta ,1]}(a)\), so the result follows. \(\quad \square \)
Proof of Theorem 1.8(i)
This is now identical to the proof of Theorem 1.7 in Sect. 6, using Proposition 7.5 in place of Proposition 6.3. \(\quad \square \)
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Acknowledgements
The authors thank Scott Armstrong and Jessica Lin for illuminating discussions. AZ also acknowledges partial support by NSF Grants DMS-1652284 and DMS-1900943.
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Appendices
Appendix A. Proof of Lemma 2.2
If \(F_0\) is the function defined before Lemma 2.1, then we have
We now claim that for each \({L}\geqq 1\), there is \(R_{L}:=R_L(M,\theta _1,m_1,\alpha _1)\) and a smooth function \(u_{L}:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) such that
hold on \({\mathbb {R}}^d\), and for each \(x\in {\mathbb {R}}^d\) with \(u_L(x)< \frac{1-\theta ^*}{3}\) we have
Note that if we also had \(1-\frac{2}{3} \theta _1\leqq \frac{1-\theta ^*}{3}\) (which is not the case), then (A.1) and (A.3) would show that for such \(u_L\) (with \(L\geqq 1\)) we have
whenever \(u_L(x)\in \left[ \frac{1-\theta ^*}{3},1-\theta ^*\right] \), so this and (A.4) would yield
on \({\mathbb {R}}^d\). Hence the result would follow with \(u_{0,S}:=u_{1}\) and \(R_0:=R_{1}\) because \(F_0\leqq F\).
Let us now prove the claim. For any \(a\in (0,\frac{1}{8})\), let \(0\not \equiv \xi _a:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) be a smooth, radially symmetric, non-negative function supported in \(B_{a}(0)\), and define
where
Notice that \({\zeta }\) is sub-harmonic on \({\mathbb {R}}^d\backslash \{0\}\), and it is supported and integrable in \(B_{1/2}(0)\). Therefore it is not hard to see that
Since \(\xi _a\) is supported in \(B_a(0)\), we also have
for all \(x\in {\mathbb {R}}^d{\setminus } B_a(0)\). Thus, for any \(R\geqq 1\), the function
satisfies
on \({\mathbb {R}}^d{\setminus } B_{aR}(0)\).
Next, for some \(N\geqq 1\) (to be determined later), take
Direct computations then yield
because \(\varphi _a\) is supported in \(B_1(0)\). Hence (A.3) will hold with \(u_L:=u\) provided \(R=R(a,\delta ,{L})\) is chosen large enough. And then \(N\geqq 1\) shows that (A.2) will also hold as long as we pick \(R_L\geqq (N+1)R\) (given this R, as well as some yet to be determined a and N).
It remains to show (A.4) when \(u(x)< \frac{1-\theta ^*}{3}\). If \(d(x,S)\geqq (N+a)R\), then (A.6) yields
so (A.4) holds. If \(d(x,S) \leqq (N+a)R\), let \(z\in \overline{S}\cap \overline{B_{(N+a)R}(x)}\). Then
with \(z':=\frac{z-x}{R}\), so \(|z'|\leqq N+a\). From (A.5) and radial symmetry of \(\varphi _a\), we get
so there are universal \(a\in (0,\frac{1}{8})\) and \(N\geqq 1\) such that the last integral is at least \(\frac{1}{3}\). Then
holds when \(d(x,S)\geqq (N+a)R\), so (A.4) holds when \(u(x)< \frac{1-\theta ^*}{3}\) and the claim is proved.
Next, to prove the lemma, recall that \(1-\frac{2}{3} \theta _1\in ( \frac{1-\theta ^*}{3},1-\theta ^*)\) and take \(u_{0,S}:=\psi (u_{L})\), for some \(L\geqq 1\) and some \(\psi :[0,1-\theta ^*]\rightarrow [0,1-\theta ^*]\) satisfying the following:
-
(i)
\(\psi \) is smooth and non-decreasing on \([0,1-\theta ^*]\);
-
(ii)
\(\psi (0)=0\), \(\psi (1-\theta ^*)=1-\theta ^*\) and \(\psi (\frac{1-\theta ^*}{3} )=1-\frac{2}{3} \theta _1\),
-
(iii)
\(\psi ''= 0\) on \([0,\frac{1-\theta ^*}{3}]\).
From (i,ii) and (A.2) we clearly have
so it suffices to take \(R_0:=R_L\) and verify (1.9).
When \( u_{L}(x)< \frac{1-\theta ^*}{3}\), (A.4) and (i,iii) yield
When \(u_{L}(x)\geqq \frac{1-\theta ^*}{3}\), (A.1) and (ii) yield \( {F_0}(\psi (u_{L}(x)))\geqq \delta \). Hence with
we get
So (1.9) follows from (A.3) and \(F_0\leqq F\), concluding the proof.
Appendix B. Proof of Lemma 2.9
Let us drop \(\omega \) from the notation. Also recall that we extend the reactions by 0 to \(u\notin [0,1]\).
Let us start with four estimates involving the reactions where \(u_1(t,x)\notin (\theta ^*,1-\theta ^*)\). From (2.1) and \(\eta \leqq \frac{\theta ^*}{2}\) we get \({\theta ^*}+\eta \leqq \theta _1\). Hence (H1) shows that for \(u\leqq {\theta ^*}\) we have
on \({\mathbb {R}}^d\), while for \(u\geqq 1-\eta \) we have
on \({\mathbb {R}}^d\). If \(u_1(t,x)\in [1-\theta ^*,1-\eta )\) for some \((t,x)\in [t_0,\infty )\times {\mathbb {R}}^d\), then \((u_1)_t\geqq 0\) shows that \(u_1(t_0,x)<1-\eta \), so (H1) and (2.7) yield
Finally, if \(u_1(t,x)\in [1-\theta ^*,1]\) for some \((t,x)\in [t_0,\infty )\times {\mathbb {R}}^d\) and \(u_1(t_0,x)<1-\eta \), then (H1) and (2.7) again yield
Denote \(\tau _\pm (t):=(1\pm {M_*}\eta )\,t+t_0\), so that \(u_\pm (t,x)=u_1(\tau _\pm (t),x)\pm \eta \). If now \(u_1(\tau _+(t),x)\notin (\theta ^*,1-\theta ^*)\) for some \((t,x)\in (0,\infty )\times {\mathbb {R}}^d\), then (B.1), (B.2), and (B.3) yield
Similarly, if \(u_1(\tau _-(t),x)\notin (\theta ^*,1-\theta ^*)\) and \(u_1(t_0,x)<1-\eta \), then (B.1), (B.2), and (B.4) yield
Let us now consider those \((t,x)\in (\kappa _*,\infty )\times {\mathbb {R}}^d\) for which \(u_1(\tau _+(t),x)\in (\theta ^*,1-\theta ^*)\). Then \((u_1)_t(\tau _+(t),x) \geqq \mu _*\) by (2.6), so \(|f_2(x,u_+(t,x))-f_2(x,u_1(\tau _+(t),x))| \leqq {M}\eta \) yields
where we again used (2.7) due to \(u_1(t_0,x)<1-\theta ^*<1-\eta \). Similarly if \(u_1(\tau _-(t),x)\in (\theta ^*,1-\theta ^*)\) for some \((t,x)\in ( 2\kappa _*,\infty )\times {\mathbb {R}}^d\) (so \(\tau _-(t)>\kappa _*\) because \(M_*\eta \leqq \frac{1}{2}\)), we obtain
This proves the claims about \(u_+\) and \(u_-\).
If now \(u_2(0,\cdot )\leqq u_1(t_0,\cdot )+\eta \) on \(B_R(y)\), from \((u_1)_t\geqq 0\) we also obtain \(u_2(0,\cdot )\leqq u_+(\kappa _*,\cdot )\) there. Since \(u_+\) is a supersolution to (1.1) with \(f_2\) in place of f on \((\kappa _*,\infty )\times B_R(y)\), Lemma 2.5 yields
for all \(t\geqq 0\). Hence,
for all \(t\in [0,{T}_{u_2}(y)]\) as long as
which will be guaranteed by taking \(D_2:=2\sqrt{Md}\ln \frac{4d}{\theta ^*}\).
It follows from \(\eta \leqq \frac{\theta ^*}{2}\) and the definition of \(T_{u_2}(y)\) that,
By Lemma 2.1, we have
Therefore
Appendix C. Proof of Lemma 2.10
We again have (B.1) and (B.2). For \((x,u)\in B_R(y)\times [1-\theta ^*,1-\eta ]\), (H3) and (2.9) yield either
(if \(f_1\) satisfies (H3)) or
(if \(f_2\) does), replacing (B.3). Hence, as in the proof of Lemma 2.9 and with \(\tau _+,u_+\) from it, we have that if \(u_1(\tau _+(t),x)\notin (\theta ^*,1-\theta ^*)\) for some \((t,x)\in (0,\infty )\times B_R(y)\), then
Let us now consider those \((t,x)\in (\kappa _*,\infty )\times B_R(y)\) for which \(u_1(\tau _+(t),x)\in (\theta ^*,1-\theta ^*)\). Then \((u_1)_t(\tau _+(t),x) \geqq \mu _*\) by (2.6), so
yields
Hence \(u_+\) is a supersolution to (1.1) with \(f_2\) in place of f on \((\kappa _*,\infty )\times B_R(y)\). Since we also have \(u_2(0,\cdot )\leqq u_+(\kappa _*,\cdot )\) on \(B_R(y)\) due to \((u_1)_t\geqq 0\), (B.5) follows via Lemmas 2.5 and 2.1 as at the end of the proof of Lemma 2.9.
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Zhang, Y.P., Zlatoš, A. Long Time Dynamics for Combustion in Random Media. Arch Rational Mech Anal 243, 33–94 (2022). https://doi.org/10.1007/s00205-021-01723-7
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DOI: https://doi.org/10.1007/s00205-021-01723-7