1 Introduction

We are interested in the statistical properties of solutions to the Cauchy problem for the scalar conservation law

(1.1)

where \(\xi = \xi (x)\) is a stochastic process. In the special case

$$\begin{aligned} H(p) = - \frac{p^2}{2} \end{aligned}$$
(1.2)

(1.1) is the well-known inviscid Burgers’ equation, which has often been considered with random initial data. Burgers himself, in his investigation of turbulence [8], was already moving in this direction, though it would be some time before problems in this area were addressed rigorously.

Several papers, including [4, 5, 9–11, 27], have developed a theory showing a kind of integrability for the evolution of the law of \(\rho (\cdot ,t)\) for certain initial data. This article is concerned with pushing stochastic integrability results beyond the Burgers setting to general scalar conservation laws, an effort begun in earnest by Menon and Srinivasan [22, 25]. Before stating our main result, we first survey the major developments in this subject.

This work is adapted from [19].

1.1 Background

The Burgers case \(H(p) = - p^2/2\) has seen extensive interest. Recall that if

$$\begin{aligned} u(x,t) = \int _{-\infty }^x \rho (x', t) \, dx' \end{aligned}$$
(1.3)

then u(x, t) formally satisfies

$$\begin{aligned} u_t + \frac{1}{2}(u_x)^2 = 0 \end{aligned}$$
(1.4)

and is determined by the Hopf–Lax formula [17]:

$$\begin{aligned} u(x,t) = \inf _y \left\{ \int _{-\infty }^y \xi (x') \, dx' + \frac{(x-y)^2}{2t}\right\} . \end{aligned}$$
(1.5)

The backward Lagrangian y(x, t) is the rightmost minimizer in (1.5), and it is standard [13] that this determines both the viscosity solution u(x, t) to (1.4) and the entropy solution to \(\rho _t + \rho \rho _x = 0\),

$$\begin{aligned} \rho (x,t) = \frac{x - y(x,t)}{t}. \end{aligned}$$
(1.6)

Groeneboom, in a paper [15] concerned not with PDE but concave majorantsFootnote 1 and isotonic estimators, determined the statistics of y(x, t) for the case where \(\xi (x)\) is white noise, which we understand to mean that the initial condition for u(x, t) in (1.4) is a standard Brownian motion B(x). Among the results of [15] is a density for

$$\begin{aligned} \sup \{x \in \mathbb {R}: B(x) - cx^2 \text { is maximal}\} \end{aligned}$$
(1.7)

expressed in terms of the Airy function. It has been observed (see e.g. [21]) that the Airy function seems to arise in a surprising number of seemingly unrelated stochastic problems, and it is notable that Burgers’ equation falls in this class.

Several key papers appeared in the 1990s; we rely on the introduction of Chabanol and Duchon’s 2004 paper [11], which recounts some of this history, and discuss those portions most relevant for our present purposes. In 1992, Sinai [27] explicitly connected Burgers’ equation with white noise initial data to convex minorants of Brownian motion. Three years later Avellaneda and Weinan [3] showed for the same initial data that the solution \(\rho (x,t)\) is a Markov process in x for each fixed \(t > 0\).

Carraro and Duchon’s 1994 paper [9] defined a notion of statistical solution to Burgers’ equation. This is a time-indexed family of probability measures \(\nu (t, \cdot )\) on a function space, and the definition of solution is stated in terms of the characteristic functional

$$\begin{aligned} \hat{\nu }(t,\phi ) = \int \exp \left( i \int \rho (x) \phi (x) \, dx \right) \nu (t,d\rho ) \end{aligned}$$
(1.8)

for test functions \(\phi \). Namely, \(\hat{\nu }(t,\phi )\) must satisfy for each \(\phi \) a differential equation obtained by formal differentiation under the assumption that \(\rho \) distributed according to \(\nu (t,d\rho )\) solves Burgers’ equation. This statistical solution approach was further developed in 1998 by the same authors [10] and by Chabanol and Duchon [11]. It does have a drawback: given a (random) entropy solution \(\rho (x,t)\) to the inviscid Burgers’ equation, the law of \(\rho (\cdot , t)\) is a statistical solution, but it is not clear that a statistical solution yields a entropy solution, and at least one example is known [4] when these notions differ. Nonetheless, [9, 10] realized that it was natural to consider Lévy process initial data, which set the stage for the next development.

In 1998, Bertoin [4] proved a remarkable closure theorem for Lévy initial data. We quote this, with adjustments to match our notation.

Theorem 1.1

([4, Theorem 2]) Consider Burgers’ equation with initial data \(\xi (x)\) which is a Lévy process without positive jumps for \(x \ge 0\), and \(\xi (x) = 0\) for \(x < 0\). Assume that the expected value of \(\xi (1)\) is positive, \(\mathbb {E}\xi (1) \ge 0\). Then, for each fixed \(t > 0\), the backward Lagrangian y(x, t) has the property that

$$\begin{aligned} y(x,t) - y(0,t) \end{aligned}$$
(1.9)

is independent of y(0, t) and is in the parameter x a subordinator, i.e. a nondecreasing Lévy process. Its distribution is the same as that of the first passage process

$$\begin{aligned} x \mapsto \inf \{z \ge 0 : t \xi (z) + z > x\}. \end{aligned}$$
(1.10)

Further, denoting by \(\psi (s)\) and \(\varTheta (t,s)\) \((s \ge 0)\) the Laplace exponents of \(\xi (x)\) and \(y(x,t) - y(0,t)\),

$$\begin{aligned} \mathbb {E}\exp (s \xi (x))&= \exp (x \psi (s)) \end{aligned}$$
(1.11)
$$\begin{aligned} \mathbb {E}\exp [s(y(x,t) - y(0,t))]&= \exp (x \varTheta (t,s)), \end{aligned}$$
(1.12)

we have the functional identity

$$\begin{aligned} \psi (t\varTheta (t,s)) + \varTheta (t,s) = s. \end{aligned}$$
(1.13)

Remark 1.1

The requirement \(\mathbb {E}\xi (1) \ge 0\) can be relaxed, with minor modifications to the theorem, in light of the following elementary fact. Suppose that \(\xi ^1(x)\) and \(\xi ^2(x)\) are two different initial conditions for Burgers’ equation, which are related by \(\xi ^2(x) = \xi ^1(x) + cx\). It is easily checked using (1.5) that the corresponding solutions \(\rho ^1(x,t)\) and \(\rho ^2(x,t)\) are related for \(t > 0\) by

$$\begin{aligned} \rho ^2(x,t) = \frac{1}{1+ct} \left[ \rho ^1\left( \frac{x}{1+ct},\frac{t}{1+ct}\right) + cx\right] . \end{aligned}$$
(1.14)

This observation is found in a paper of Menon and Pego [24], but (as this is elementary) it may have been known previously. Using (1.14) we can adjust a statistical description for a case where \(\mathbb {E}\xi (1) \ge 0\) to cover the case of a Lévy process with general mean drift.

We find Theorem 1.1 remarkable for several reasons. First, in light of (1.6), it follows immediately that the solution \(\rho (x,t) - \rho (0,t)\) is for each fixed t a Lévy process in the parameter x, and we have an example of an infinite-dimensional, nonlinear dynamical system (the PDE, Burgers’ equation) which preserves the independence and homogeneity properties of its random initial configuration. Second, the distributional characterization of y(x, t) is that of a first passage process, where the definition of y(x, t) following (1.5) is that of a last passage process. Third, (1.13) can be used to show [23] that if \(\psi (t,q)\) is the Laplace exponent of \(\rho (x,t) - \rho (0,t)\), then

$$\begin{aligned} \psi _t + \psi \psi _q = 0 \end{aligned}$$
(1.15)

for \(t > 0\) and \(q \in \mathbb {C}\) with nonnegative real part. This shows for entropy solutions what had previously been observed by Carraro and Duchon for statistical solutions [10], namely that the Laplace exponent (1.12) evolves according to Burgers’ equation!

In 2007 Menon and Pego [24] used the Lévy–Khintchine representation for the Laplace exponent (1.12) and observed that the evolution according to Burgers’ equation in (1.15) corresponds to a Smoluchowski coagulation equation [2, 28], with additive collision kernel, for the jump measure of the Lévy process \(y(\cdot , t)\). The jumps of \(y(\cdot , t)\) correspond to shocks in the solution \(\rho (\cdot , t)\). The relative velocity of successive shocks can be written as a sum of two functions, one depending on the positions of the shocks and the other proportional to the sum of the sizes of the jumps in \(y(\cdot , t)\). Regarding the sizes of the jumps as the usual masses in the Smoluchowski equation, it is plausible that Smoluchowski equation with additive kernel should be relevant, and [24] provides the details that verify this.

It is natural to wonder whether this evolution through Markov processes with simple statistical descriptions is a miracle [23] confined to the Burgers–Lévy case, or an instance of a more general phenomenon. However, extending the results of Bertoin [4] beyond the Burgers case \(H(p) = -\frac{1}{2} p^2\) remains a challenge. A different particular case, corresponding to \(L(q) = (-H)^*(q) = |q|\), is a problem of determining Lipschitz minorants, and has been investigated by Abramson and Evans [1]. From the PDE perspective this is not as natural, since \((-H)^*(q) = |q|\) corresponds to

$$\begin{aligned} -H(p) = +\infty \, \mathbf {1}(|p| > 1), \end{aligned}$$
(1.16)

i.e. \(-H(p)\) takes the value 0 on \([-1,+1]\) and is equal to \(+\infty \) elsewhere. So [1], while very interesting from a stochastic processes perspective, has a specialized structure which is rather different from those cases we will consider.

The biggest step toward understanding the problem for a wide class of H is found in a 2010 paper of Menon and Srinivasan [25]. Here it is shown that when the initial condition \(\xi \) is a spectrally negative Footnote 2 strong Markov process, the backward Lagrangian process \(y(\cdot , t)\) and the solution \(\rho (\cdot , t)\) remain Markov for fixed \(t > 0\), the latter again being spectrally negative. The argument is adapted from that of [4] and both [4, 25] use the notion of splitting times (due to Getoor [14]) to verify the Markov property according to its bare definition. In the Burgers–Lévy case, the independence and homogeneity of the increments can be shown to survive, from which additional regularity is immediate using standard results about Lévy processes [18]. As [25] points out, without these properties it is not clear whether a Feller process initial condition leads to a Feller process in x at later times. Nonetheless, [25] presents a very interesting conjecture for the evolution of the generator of \(\rho (\cdot , t)\), which has a remarkably nice form and follows from multiple (nonrigorous, but persuasive) calculations.

We now give a partial statement of this conjecture. The generator \(\mathscr {A}\) of a stationary, spectrally negative Feller process acts on test functions \(J \in C^\infty _c(\mathbb {R})\) by

$$\begin{aligned} (\mathscr {A}J)(y) = b(y) J'(y) + \int _{-\infty }^y (J(z) - J(y)) \, f(y, dz), \end{aligned}$$
(1.17)

where b(y) characterizes the drift and \(f(y, \cdot )\) describes the law of the jumps. If we allow b and f to depend on t, we have a family of generators. The conjecture of [25] is that the evolution of the generator \(\mathscr {A}\) for \(\rho (\cdot , t)\) is given by the Lax equation

$$\begin{aligned} \dot{\mathscr {A}} = [\mathscr {A}, \mathscr {B}] = \mathscr {A}\mathscr {B}- \mathscr {B}\mathscr {A}\end{aligned}$$
(1.18)

for \(\mathscr {B}\) which acts on test functions J by

$$\begin{aligned} (\mathscr {B}J)(y) = - H'(y) b(t,y) J'(y) - \int _{-\infty }^y \frac{H(y) - H(z)}{y-z} (J(z) - J(y)) \, f(t, y, dz). \end{aligned}$$
(1.19)

An equivalent form of the conjecture (1.18) involves a kinetic equation for f. The key result in the present article verifies that this kinetic equation holds in the special case we will consider. Before we state this, let us establish our working notation.

1.2 Notation

Here we collect some of the various notation used later in the article.

Write \(H[p_1,\ldots ,p_n]\) for the nth divided difference of H through \(p_1,\ldots ,p_n\). For each n this function is symmetric in its arguments, and given by the following formulas in the cases \(n = 2\) and \(n = 3\) where \(p_1,p_2,p_3\) are distinct:

$$\begin{aligned} H[p_1,p_2] = \frac{H(p_2) - H(p_1)}{p_2 - p_1} \quad \text {and} \quad H[p_1,p_2,p_3] = \frac{H[p_2,p_3] - H[p_1,p_2]}{p_3 - p_1}. \end{aligned}$$
(1.20)

The definition for general n and standard properties can be found in several numerical analysis texts, including [16].

We write \(\delta _x\) for the usual point mass assigning unit mass to x and zero elsewhere.

For \(C > 0\) and n a positive integer, write

$$\begin{aligned} \varDelta ^C_n = \{(a_1,\ldots ,a_n) \in \mathbb {R}^n : 0 < a_1 < \cdots < a_n < C\} \end{aligned}$$
(1.21)

and \(\overline{\varDelta ^C_n}\) for the closure of this set in \(\mathbb {R}^n\). We write \(\partial \varDelta ^C_n\) for the boundary of the simplex, and \(\partial _i \varDelta ^C_n\) for its various faces:

$$\begin{aligned} \begin{aligned} \partial _0 \varDelta ^C_n&= \{(a_1,\ldots ,a_n) \in \partial \varDelta ^C_n : a_1 = 0\} \\ \partial _i \varDelta ^C_n&= \{(a_1,\ldots ,a_n) \in \partial \varDelta ^C_n : a_i = a_{i+1}\}, \qquad i = 1,\ldots ,n-1 \\ \partial _n \varDelta ^C_n&= \{(a_1,\ldots ,a_n) \in \partial \varDelta ^C_n : a_n = C\}. \end{aligned} \end{aligned}$$
(1.22)

Write \(\mathscr {M}\) for the set of finite, regular (signed) measures on [0, P], which is a Banach space when equipped with the total variation norm \(\Vert \cdot \Vert _{\textsc {tv}}\). Call its nonnegative subset \(\mathscr {M}_+\).

Let \(\mathscr {K}\) denote the set of bounded signed kernels from [0, P] to [0, P], i.e. the set of \(k : [0,P] \mapsto \mathscr {M}\) which are measurable when [0, P] is endowed with its Borel \(\sigma \)-algebra and \(\mathscr {M}\) is endowed with the \(\sigma \)-algebra generated by evaluation on Borel subsets of [0, P], and which satisfy

$$\begin{aligned} \Vert k\Vert = \sup \{ \Vert k(\rho _-, \cdot )\Vert _{\textsc {tv}} : \rho _- \in [0,P]\} < \infty . \end{aligned}$$
(1.23)

Observe that \(\mathscr {K}\) is a Banach space; completeness holds since a Cauchy sequence \(k_n\) has \(k_n(\rho _-,\cdot )\) Cauchy in total variation for each \(\rho _-\), and we obtain a pointwise limit \(k(\rho _-,\cdot )\). Measurability of \(k(\cdot , A)\) for each Borel \(A \subseteq [0,P]\) then holds since this is a real-valued pointwise limit of \(k_n(\cdot , A)\). Write \(\mathscr {K}_+\) for the subset of \(\mathscr {K}\) with range contained in \(\mathscr {M}_+\).

1.3 Main result

In this section we provide a statistical description of solutions to the scalar conservation law when the initial condition is an increasing, pure-jump Markov process given by a rate kernel g. For this we require some assumptions on the rate kernel g and the Hamiltonian H.

Assumption 1.1

The initial condition \(\xi = \xi (x)\) is a pure-jump Markov process starting at \(\xi (0) = 0\) and evolving for \(x > 0\) according to a rate kernel \(g(\rho _-,d\rho _+)\). We assume that for some constant \(P > 0\) the kernel g is supported on

$$\begin{aligned} \{(\rho _-,\rho _+) : 0 \le \rho _- \le \rho _+ \le P\} \end{aligned}$$
(1.24)

and has total rate which is constant in \(\rho _-\):

$$\begin{aligned} \lambda = \int g(\rho _-, d\rho _+) \end{aligned}$$
(1.25)

for all \(0 \le \rho _- \le P\). In particular, \(g \in \mathscr {K}_+\).

Assumption 1.2

The Hamiltonian function \(H : [0,P] \rightarrow \mathbb {R}\) is smooth, convex, has nonnegative right-derivative at \(p = 0\) and noninfinite left-derivative at \(p = P\).

We provide for \(x \ge 0\) a statistical description consisting of a one-dimensional marginal at \(x = 0\) and a rate kernel generating the rest of the path. The evolution of the rate kernel is given by the following kinetic equation, and the evolution of the marginal will be described in terms of the solution to the kinetic equation.

Definition 1.3

We say that a continuous mapping \(f : [0,\infty ) \rightarrow \mathscr {K}_+\) is a solution of the kinetic equation

$$\begin{aligned} \left\{ \begin{aligned} f_t&= \mathscr {L}^\kappa f \\ f(0,\rho _-,d\rho _+)&= g(\rho _-,d\rho _+), \end{aligned} \right. \end{aligned}$$
(1.26)

where

$$\begin{aligned} \begin{aligned}&\mathscr {L}^{\kappa } f(t,\rho _-,d\rho _+) \\&\quad = \int (H[\rho _*,\rho _+] - H[\rho _-,\rho _*]) f(t,\rho _-,d\rho _*) f(t,\rho _*,d\rho _+) \\&\qquad - \left[ \int H[\rho _+,\rho _*] f(t,\rho _+,d\rho _*) - \int H[\rho _-,\rho _*] f(t,\rho _-,d\rho _*)\right] f(t,\rho _-,d\rho _+), \end{aligned} \end{aligned}$$
(1.27)

provided that \(\theta \mapsto \mathscr {L}^{\kappa } f(\theta , \cdot , \cdot ) \in \mathscr {K}\) is Bochner-integrable and

$$\begin{aligned} f(t,\cdot ,\cdot ) - f(s,\cdot ,\cdot ) = \int _s^t \mathscr {L}^\kappa f(\theta , \cdot , \cdot ) \, d\theta \end{aligned}$$
(1.28)

for all \(0 \le s \le t < \infty \).

Definition 1.4

We say that a continuous mapping \(\ell : [0,\infty ) \rightarrow \mathscr {M}_+\) is a solution of the marginal equation

$$\begin{aligned} \left\{ \begin{aligned} \ell _t&= \mathscr {L}^0 \ell \\ \ell (0,d\rho _0)&= \delta _0(d\rho _0), \end{aligned} \right. \end{aligned}$$
(1.29)

where

$$\begin{aligned} \mathscr {L}^0 \ell (t,d\rho _0)= & {} \int H[\rho _*,\rho _0] \ell (t,d\rho _*) f(t,\rho _*,d\rho _0) \nonumber \\&- \left[ \int H[\rho _0,\rho _*] f(t, \rho _0, d\rho _*)\right] \ell (t,d\rho _0), \end{aligned}$$
(1.30)

provided that \(\theta \mapsto \mathscr {L}^0 \ell (\theta , \cdot ) \in \mathscr {M}\) is Bochner-integrable and

$$\begin{aligned} \ell (t,\cdot ) - \ell (s,\cdot ) = \int _s^t \mathscr {L}^0 \ell (\theta , \cdot ) \, d\theta \end{aligned}$$
(1.31)

for all \(0 \le s \le t < \infty \).

Theorem 1.2

Suppose Assumptions 1.1 and 1.2 hold. Then the kinetic and marginal equations have unique solutions, in the sense of Definitions 1.3 and 1.4. Furthermore, the total integrals are conserved:

$$\begin{aligned} \lambda&= \int f(t,\rho _-,d\rho _+) \end{aligned}$$
(1.32)
$$\begin{aligned} 1&= \int \ell (t,d\rho _0) \end{aligned}$$
(1.33)

for all \(t \ge 0\) and all \(0 \le \rho _- \le P\). Finally, if

$$\begin{aligned} g(\rho _-, [0,\rho _-] \cup \{P\}) = 0 \end{aligned}$$
(1.34)

for all \(0 \le \rho _- < P\), then \(f(t,\cdot ,\cdot )\) has the same property for \(t > 0\).

The kernels described by the Theorem 1.2 are precisely what we need to describe the statistics of the solution \(\rho \), which brings us to our main result:

Theorem 1.3

When Assumptions 1.1 and 1.2 hold, the solution \(\rho \) to

(1.35)

for each fixed \(t > 0\) has \(x = 0\) marginal given by \(\ell (t,d\rho _0)\) and for \(0 < x < \infty \) evolves according to rate kernel \(f(t,\rho _-,d\rho _+)\).

Remark 1.2

Theorems 1.2 and 1.3 establish rigorously the conjectured [25] evolution according to the Lax pair (1.18), within the present hypotheses. See [25, Section 2.7] for a calculation showing the equivalence of the kinetic and Lax pair formulations, which simplifies considerably in the present case due to the absence of drift terms. The Lax pair and integrable systems approach (in the case of finitely many states, where the generator is a triangular matrix) have been further explored by Menon [22] and in a forthcoming work by Li [20].

1.4 Organization

The remainder of this article is organized as follows. In Sect. 2 we show that Theorem 1.3 will follow from a similar statistical characterization for the solution to the PDE over \(x \in [0,L]\) with a time-dependent random boundary condition at \(x = L\). The latter we can study using a sticky particle system whose dimension is random and unbounded, but almost surely finite. Elementary arguments are used to check that our candidate for the law matches the evolution of the random initial condition according to the dynamics. Next in Sect. 3 we show existence and uniqueness of the solutions to marginal and kinetic equations of Theorem 1.2, which are needed to construct the candidate law. The concluding Sect. 4 indicates some desired extensions and similar questions for future work. To keep the main development concise, proofs of lemmas have been deferred to Appendix.

2 A random particle system

As functions of x, the solutions \(\rho (x,t)\) we consider all have the form depicted in Fig. 1. From a PDE perspective this situation is standard—a concatenation of Riemann problems for the scalar conservation law—and we can describe the solution completely in terms of a particle system. Each shock consists of some number of particles stuck together, and the particles move at constant velocities according to the Rankine–Hugoniot condition except when they collide. The collisions are totally inelastic.

Fig. 1
figure 1

For each \(t > 0\), the solution \(\rho (x,t)\) is a nondecreasing, pure-jump process in x. We will see that for any fixed \(L > 0\), we have a.s. finitely many jumps for \(x \in [0,L]\) and that \(\rho (\cdot ,t)\) on this interval can be described by two (finite) nondecreasing sequences \((x_1,\ldots ,x_N; \rho _0,\ldots ,\rho _N)\)

Full size image

The utility of this particle description is lessened, however, by the fact that the dynamics are quite infinite dimensional for our pure-jump initial condition \(\xi (x)\), \(x \in [0,+\infty )\). To have a simple description as motion at constant velocities, punctuated by occasional collisions, we might argue that on each fixed bounded interval of space we have finitely many collisions in a bounded interval of time, and piece together whatever statistical descriptions we might obtain for the solution on these various intervals. Inspired by those situations in statistical mechanics where boundary conditions become irrelevant in an infinite-volume limit, we pursue a different approach. We construct solutions to a problem on a bounded space interval [0, L], and choose a random boundary condition at \(x = L\) to obtain an exact match with the kinetic equations. The involved analysis will all pertain to the following result.

Theorem 2.1

Suppose Assumptions 1.1 and 1.2 hold. For any fixed \(L > 0\), consider the scalar conservation law

(2.1)

with initial condition \(\xi \) (restricted to [0, L]), open boundaryFootnote 3 at \(x = 0\), and random boundary \(\zeta \) at \(x = L\). Suppose the process \(\zeta \) has \(\zeta (0) = \xi (L)\) and evolves according to the time-dependent rate kernel \(H[\rho _,\rho _+] f(t,\rho _,d\rho _+)\) independently of \(\xi \) given \(\xi (L)\). Then for all \(t > 0\) the law of \(\rho (\cdot , t)\) is as follows:

  1. (i)

    the \(x = 0\) marginal is \(\ell (t,d\rho _0)\), and

  2. (ii)

    the rest of the path is a pure-jump process with rate kernel \(f(t,\rho _-,d\rho _+)\).

To prove our main result we can send \(L \rightarrow \infty \), applying Theorem 2.1 on each [0, L], and use bounded speed of propagation to limit the respective influences of far away particles (unbounded system) or truncation with random boundary (bounded system). The argument is quite short.

Proof

(Theorem 1.3) Fix any \(t > 0\). We will write \(\hat{\rho }(x,t)\) for the solution to (1.35) over the semi-infinite x-interval \([0,+\infty )\) with initial data \(\hat{\xi }(x)\). We take the right-continuous version of the solution. For L to be specified shortly, write \(\rho (x,t)\) for the solution and \(\xi (x)\) for the initial data corresponding to (2.1).

Fix any \(x_1,\ldots ,x_k \in [0,+\infty )\), and let \(X = \max \{x_1,\ldots ,x_k\}\). Choose \(L > X + t H'(P)\). We couple the bounded and unbounded systems by requiring

$$\begin{aligned} \hat{\xi }|_{[0,L]} = \xi , \end{aligned}$$
(2.2)

allowing the random boundary \(\zeta \) to evolve independently of \(\hat{\xi }\) given \(\hat{\xi }(L)\)

Recall [13] that the scalar conservation law has finite speed of propagation. Our solutions are bounded in [0, P], so the speed is bounded by \(H'(P)\). Since \(\hat{\rho }(x,0)\) and \(\rho (x,0)\) are a.s. equal on [0, L], we have also \(\hat{\rho }(x,t) = \rho (x,t)\) a.s. for \(x \in [0,L-tH'(P)] \supset [0,X]\). From this we deduce the distributional equality

$$\begin{aligned} (\hat{\rho }(x_1,t),\ldots ,\hat{\rho }(x_k,t)) \mathop {=}\limits ^{d} (\rho (x_1,t),\ldots ,\rho (x_k,t)). \end{aligned}$$
(2.3)

By Theorem 2.1, the latter distribution is exactly that of a process started according to \(\ell (t,d\rho _0)\), evolving according to rate kernel \(f(t,\rho _-,d\rho _+)\). This process is terminated deterministically at \(x = L\), which does not alter finite-dimensional distributions prior to \(x = L\). Since \(\rho (x,t)\) is right-continuous and has the correct finite-dimensional distributions, the result follows. \(\square \)

2.1 The dynamics

Our work in the remainder of this section is to prove Theorem 2.1. We begin by describing precisely those particle dynamics which determine the solution to the PDE.

Figure 1 illustrates a parametrization of a nondecreasing pure-jump process on \([0,+\infty )\) with heights \(\rho _0,\rho _1,\rho _2,\ldots \) and jump locations \(x_1,x_2,\ldots \). Our sign restriction on the jumps excludes rarefaction waves, and we have only constant values separated by shocks. Going forward in time, the shocks move according to the Rankine–Hugoniot condition

$$\begin{aligned} \dot{x}_i = -H[\rho _{i-1},\rho _i] \end{aligned}$$
(2.4)

until they collide. We say that each shock consists initially of one particle moving at the velocity indicated above. Then result of a collision can be characterized in two equivalent ways:

  1. (i)

    At the first instant when \(x_i = x_{i+1}\), the ith particle is annihilated, and the velocity of the \((i+1)\)th particle changes from \(-H[\rho _i,\rho _{i+1}]\) to

    $$\begin{aligned} \dot{x}_{i+1} = - H[\rho _{i-1},\rho _{i+1}]. \end{aligned}$$
    (2.5)

    In the caseFootnote 4 where several consecutive particles collide with each other at the same instant, all but the rightmost particle is annihilated. Since we only seek a statistical description for \(x \ge 0\), we annihilate the first particle when \(x_1 = 0\), replace \(\rho _0\) with \(\rho _1\), and relabel the other particles accordingly.

  2. (ii)

    When \(x_{i-1} < x_i = x_{i+1} = \cdots = x_{j-1} = x_j < x_{j+1}\), the particles i through j all move with common velocity \(-H[\rho _{i-1},\rho _j]\). Following Brenier et al [6, 7] we call these a sticky particle dynamics. We can additionally take the position \(x = 0\) to be absorbing: \(\dot{x_i} = 0\) whenever \(x_i = 0\).

We adopt the first viewpoint for convenience, as this is compatible with [12] (see the text following Definition 2.2 below), but a suitable argument could be given for the second alternative as well.

Definition 2.1

For L as in Theorem 2.1, the configuration space Q for the sticky particle dynamics is

$$\begin{aligned} Q = \bigsqcup _{n=0}^\infty Q_n, \qquad Q_n = \varDelta ^L_n \times \overline{\varDelta ^P_{n+1}}. \end{aligned}$$
(2.6)

A typical configuration is \(q = (x_1,\ldots ,x_n;\rho _0,\ldots ,\rho _n) \in Q_n\) when \(n>0\), or \(q = (\rho _0) \in Q_0=\{\rho _0: \ 0\le \rho _0\le P\}\) when \(n=0\).

Definition 2.2

Our notation for the particle dynamics is as follows:

  1. (i)

    For \(0 \le s \le t\) and \(q \in Q\), write

    $$\begin{aligned} \phi _s^t q = \phi _0^{t-s} q \end{aligned}$$
    (2.7)

    for the deterministic evolution from time s to t of the configuration q according to the annihilating particle dynamics for the PDE, without random entry dynamics at \(x = L\).

  2. (ii)

    Given a configuration \(q = (x_1,\ldots ,x_n,\rho _0,\ldots ,\rho _n)\) and \(\rho _+ > \rho _n\), write \(\epsilon _{\rho _+} q\) for the configuration \((x_1,\ldots ,x_n,L,\rho _0,\ldots ,\rho _n,\rho _+)\).

  3. (iii)

    Write \(\varPhi _s^t q\) for the random evolution of the configuration according to deterministic particle dynamics interrupted with random entries at \(x = L\) according to the boundary process \(\zeta \) of (2.1), where the latter has been started at time s with value \(\rho _n\). In particular, if the jumps of \(\zeta \) between times s and t occur at times \(s < \tau _1 < \cdots < \tau _k < t\) with values \(\rho _{n+1} < \cdots < \rho _{n+k}\), then

    $$\begin{aligned} \varPhi _s^t q = \phi _{\tau _k}^t \epsilon _{\rho _{n+k}} \phi _{\tau _{k-1}}^{\tau _k} \epsilon _{\rho _{n+k-1}} \cdots \phi _{\tau _1}^{\tau _2} \epsilon _{\rho _{n+1}} \phi _s^{\tau _1} q. \end{aligned}$$
    (2.8)

Proposition 2.1

For any s, q, the process \(\varPhi _s^t q\) is strong Markov.

This assertion follows after recognizing \(\varPhi _s^t q\) as a piecewise-deterministic Markov process described in some generality by Davis [12]. Namely, we augment the configuration space Q to add the time parameter to each component \(Q_n\), and then have a deterministic flow according to the vector field

$$\begin{aligned} (1; -H[\rho _0,\rho _1], \ldots , -H[\rho _{n-1},\rho _n]; 0, \ldots , 0). \end{aligned}$$
(2.9)

With rate \(\int H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+)\) we jump to the indicated point in \(Q_{n+1}\), and upon hitting a boundary we transition to \(Q_k\) for suitable \(k < n\), annihilating particles in the manner described above.

2.2 Checking the candidate measure

Our goal is to take q distributed according to the initial condition and exactly describe the law of \(\varPhi _0^T q\) for each \(T > 0\). Using the kinetic (1.27) and marginal equations (1.30), we construct for each time \(t \ge 0\) a candidate law \(\mu (t,dq)\) on Q as follows. Take N to be Poisson with rate \(\lambda L\), \(x_1,\ldots ,x_N\) uniform on \(\varDelta ^L_N\), and \(\rho _0,\ldots ,\rho _N\) distributed on \(\overline{\varDelta ^P_{N+1}}\) according to the marginal \(\ell \) and transitions f independently of the \(x_i\):

$$\begin{aligned} e^{-\lambda L} \sum _{n=0}^\infty \delta _n(dN)\mu _n(t,dq), \end{aligned}$$
(2.10)

where \(\mu _0(t,dq) = \ell (t,d\rho _0)\) and

$$\begin{aligned} \mu _n(t,dq) = \mathbf {1}_{\varDelta ^L_n}(x_1,\ldots ,x_n) \, dx_1 \cdots dx_n \, \ell (t,d\rho _0) \prod _{j=1}^n f(t,\rho _{j-1},d\rho _j), \end{aligned}$$
(2.11)

for \(n>0\). When we have verified (1.32), it will be immediate that the total mass of this is one.

We decompose the mapping from configurations q to solutions (as functions of x over [0, L]) into the map from q to the measure

$$\begin{aligned} \pi (q, dx) = \rho _0 \delta _0 + \sum _{i=1}^n (\rho _i - \rho _{i-1}) \delta _{x_i} \qquad ( q \in Q_n) \end{aligned}$$
(2.12)

and integration over [0, x]. We claim that when q is distributed according to \(\mu (0,dq)\), the law of the measure \(\pi (\varPhi _0^t q, \cdot )\) is identical to that of \(\pi (q', \cdot )\) where \(q'\) is distributed according to \(\mu (t,dq')\).

We now describe the structure of the proof of Theorem 2.1. Fix some time \(T > 0\) and consider \(F(t,q) = \mathbb {E}G(\varPhi _t^T q)\) where G takes the form of a Laplace functional:

$$\begin{aligned} G(q) = \exp \left( -\int J(x) \, \pi (q,dx) \right) = \exp \left( -\rho _0 J(0) - \sum _{i=1}^n (\rho _i - \rho _{i-1}) J(x_i)\right) \end{aligned}$$
(2.13)

for \(J \ge 0\) a continuous function on [0, L]. We aim to show that

$$\begin{aligned} \frac{d}{dt} \int \mathbb {E}G(\varPhi _t^T q) \mu (t,dq) = 0 \end{aligned}$$
(2.14)

for \(0 < t < T\), from which it will follow that

$$\begin{aligned} \int \mathbb {E}G(\varPhi _0^T q) \, \mu (0,dq) = \int G(q) \, \mu (T,dq) \end{aligned}$$
(2.15)

for all G of the form in (2.13). Using the standard fact that Laplace functionals completely determine the law of any random measure [18], this will suffice to show that law of \(\pi (\varPhi _0^T q, dx)\) for q distributed as \(\mu (0, dq)\) is precisely the pushforward through \(\pi \) of \(\mu (T, dq)\), and obtain the result.

We might verify (2.14) by establishing regularity for \(F(t,q) = \mathbb {E}G(\varPhi _t^T q)\) in t and the x-components of q. We speculate that it should be possible to do so if J is smooth with \(J'(0) = 0\) and we content ourselves to divide the configuration space into finitely many regions, each of which corresponds to a definite order of deterministic collisions in \(\phi \). On the other hand, the measures \(\mu (t, dq)\) enjoy considerable regularity (uniformity, in fact) in x, and we pursue instead an argument along these lines. Here continuity of F(t, q) in t will suffice.

Lemma 2.1

Let G take the form of (2.13). Then \(F(t,q) = \mathbb {E}G(\varPhi _t^T q)\) is a bounded function which is uniformly continuous in t uniformly in q:

$$\begin{aligned} w(\delta ) = \sup \{ |F(t,q) - F(s,q)| \, : \, s,t \in [0,T], \, |t-s| < \delta , \, q \in Q\} \rightarrow 0 \end{aligned}$$
(2.16)

as \(\delta \rightarrow 0+\).

To differentiate in (2.14) we will need to compare \(\mathbb {E}G(\varPhi _s^T q)\) and \(\mathbb {E}G(\varPhi _t^T q)\) for \(0 < s < t < T\). Our next observation is that when the \(t-s\) is small, we can separate the deterministic and stochastic portions of the dynamics over the time interval [s, t]. The idea is that in an interval of time [s, t], the probability of multiple particle entries is \(O((t-s)^2)\), and a single particle entry has probability \(O(t-s)\) which permits additional o(1) errors.

Lemma 2.2

Let \(0 < s < t \le T\) and \(q \in Q_n\). There exist a random variable \(\tau \in (s,t)\) a.s., with law depending on s, t,  and q only through \(\rho _n\), and a constant \(C_{\lambda ,H'(P)}\) independent of q, s, t so that

$$\begin{aligned}&\left| F(s,q) - F(t,\phi _s^t q) - (t-s) \int [\mathbb {E}F(t, \epsilon _{\rho _+} \phi _s^\tau q) - F(t,q)] H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+)\right| \nonumber \\&\quad \le C_{\lambda ,H'(P)} [(t-s)^2 + (t-s) w(t-s)]. \end{aligned}$$
(2.17)

We proceed to an analysis of the deterministic portion of the flow, \(\phi _s^t q\), where q is distributed according to \(\mu (t,dq)\). For this we introduce some notation. Given \(q \in Q_n\), we write

$$\begin{aligned} \sigma (q) = \min \left\{ r \ge s \, : \phi _s^r q \in (\partial \varDelta ^L_n) \times \overline{\varDelta ^P_{n+1}}\right\} . \end{aligned}$$
(2.18)

For each fixed n, we consider several subsets of the set of n-particle configurations \(Q_n\). In particular we need to separate those configurations which experience an exit at \(x = 0\) or a collision in a time interval shorter than \(t-s\). For \(i = 0,\ldots ,n-1\), write:

$$\begin{aligned} \begin{aligned} A_i&= \left\{ q \in Q_n : \sigma (q) \le t \text { and } \phi _s^{\sigma (q)} q \in (\partial _i \varDelta ^L_n) \times \overline{\varDelta ^P_{n+1}}\right\} \\ U&= Q_n \setminus \bigcup _{i=0}^{n-1} A_i \\ B&= \left\{ q \in Q_n : x_n \ge L - (t-s) H[\rho _{n-1},\rho _n]\right\} \\ V&= Q_n \setminus B. \end{aligned} \end{aligned}$$
(2.19)

Figure 2 illustrates these sets in the case \(n = 2\). We observe in particular that \(\phi _s^t U\) and V differ by a \(\mu _n(t,\cdot )\)-null set, and that the terms associated with the boundary faces are related to configurations with one fewer particle.

Lemma 2.3

For each positive integer n and with errors bounded by

$$\begin{aligned} C_{L,n,\lambda ,H'(P)} [(t-s) + w(t-s)] \end{aligned}$$
(2.20)

we have the following approximations:

$$\begin{aligned}&(t-s)^{-1} \int F(t,\phi _s^t q) \mathbf {1}_{A_0}(q) \mu _n(t,dq) \nonumber \\&\quad \approx \int F(t,q) \frac{H[\rho _*,\rho _0] \ell (t,d\rho _*) f(t,\rho _*,d\rho _0)}{\ell (t,d\rho _0)} \mu _{n-1}(t,dq), \end{aligned}$$
(2.21)
$$\begin{aligned}&(t-s)^{-1} \int F(t,\phi _s^t q) \mathbf {1}_{A_i}(q) \mu _n(t,dq) \nonumber \\&\quad \approx \int F(t,q) \frac{(H[\rho _*,\rho _i] - H[\rho _{i-1},\rho _*]) f(t,\rho _{i-1},d\rho _*) f(t,\rho _*,d\rho _i)}{f(t,\rho _{i-1},d\rho _i)} \mu _{n-1}(t,dq) \end{aligned}$$
(2.22)

for \(i = 1, \ldots , n-1\), and

$$\begin{aligned}&(t-s)^{-1} \int F(t,\phi _s^t q) \mathbf {1}_B(q) \mu _n(t,dq) \nonumber \\&\quad \approx \int F(t,\epsilon _{\rho _+} q) H[\rho _{n-1},\rho _+] f(t,\rho _{n-1},d\rho _+) \mu _{n-1}(t,dq). \end{aligned}$$
(2.23)

Remark 2.1

It is essential that the integral over B can be approximated by an integral over \((x_1,\ldots ,x_{n-1},L;\rho _0,\ldots ,\rho _{n-1},\rho _+)\). The measure \(\mathscr {L}^* \mu \) below does not have any singular factors like \(\delta (x_n = L)\). In particular, the result of integrating over B will partially cancel with the term arising from random particle entries.

Fig. 2
figure 2

The deterministic flow \(\phi \) on the x-simplex is translation at constant velocity unless this translation crosses a boundary of \(\varDelta ^L_n\). In the case \(n = 2\) pictured above, points in \(A_0\) and \(A_1\) hit the boundary faces \(x_1 = 0\) and \(x_1 = x_2\), respectively. The remaining portion of the simplex is mapped to the simplex minus the set B

Full size image

The final ingredient for the proof of Theorem 2.1 is the time derivative of \(\mu (t,dq)\), which we now record.

Lemma 2.4

For any \(n \ge 0\) and any \(0 \le s < t\) we have

$$\begin{aligned} \Vert \mu _n(t, \cdot ) - \mu _n(s, \cdot ) - (t-s) (\mathscr {L}^* \mu _n)(t,\cdot )\Vert _{\textsc {tv}} = o(t-s) \end{aligned}$$
(2.24)

where the norm is total variation and \((\mathscr {L}^* \mu _n)(t,dq)\) is defined to be the signed kernel

$$\begin{aligned}&\left[ \frac{(\mathscr {L}^0 \ell )(t, d\rho _0)}{\ell (t, d\rho _0)} + \sum _{i=1}^n \frac{(\mathscr {L}^{\kappa } f)(t, \rho _{i-1}, d\rho _i)}{f(t, \rho _{i-1}, d\rho _i)}\right] \nonumber \\&\quad \times \, e^{-\lambda L} \mathbf {1}_{\varDelta ^L_n}(x_1,\ldots ,x_n) \, dx_1 \cdots dx_n \, \ell (t,d\rho _0) \prod _{j=1}^n f(t, \rho _{j-1}, d\rho _j).\quad \quad \end{aligned}$$
(2.25)

The expression for the measure above is to be understood formally; the correct interpretation involves replacement, not division. All of the “divisors” above are present as factors of \(\ell (t, d\rho _0) \prod _{j=1}^n f(t,\rho _{j-1}, d\rho _j)\), and the fractions indicate that the appearance of the denominator in this portion is to be replaced with the indicated numerator.

So that we know what to expect, before proceeding we note that when we sum over i in (2.25), some of the terms arising from \(\mathscr {L}^0\) and \(\mathscr {L}^{\kappa }\) cancel. Namely, the bracketed portion of (2.25) expands as

$$\begin{aligned}&\frac{\int H[\rho _*,\rho _0] \ell (t, d\rho _*) f(t,\rho _*, d\rho _0)}{\ell (t, d\rho _0)} - \int H[\rho _0,\rho _*] f(t,\rho _0, d\rho _*) \nonumber \\&\quad + \sum _{i=1}^n \left[ \frac{\int (H[\rho _*,\rho _i] - H[\rho _{i-1},\rho _*]) f(t,\rho _{i-1}, d\rho _*) f(t,\rho _*, d\rho _i)}{f(t,\rho _{i-1}, d\rho _i)} \right. \nonumber \\&\qquad \quad \qquad \left. - \int (H[\rho _i,\rho _*] f(t,\rho _i, d\rho _*) - H[\rho _{i-1},\rho _*] f(t,\rho _{i-1}, d\rho _*)) \right] \quad \quad \quad \quad \end{aligned}$$
(2.26)

The gain terms associated with the kinetic equations we leave as they are, but note that the “loss” terms telescope, and the above may be shortened to

$$\begin{aligned}&\frac{\int H[\rho _*,\rho _0] \ell (t, d\rho _*) f(t,\rho _*, d\rho _0)}{\ell (t, d\rho _0)} - \int H[\rho _n,\rho _*] f(t,\rho _n,d\rho _*) \nonumber \\&\quad + \sum _{i=1}^n \frac{\int (H[\rho _*,\rho _i] - H[\rho _{i-1},\rho _*]) f(t,\rho _{i-1},d\rho _*) f(t,\rho _*,d\rho _i)}{f(t,\rho _{i-1},d\rho _i)}.\quad \quad \end{aligned}$$
(2.27)

We are ready to prove our statistical characterization of the bounded system.

Proof

(Theorem 2.1) For times s and t with \(0 \le s < t \le T\), consider the difference \(\int F(t,q) \, \mu (t,dq) - \int F(s,q) \, \mu (s,dq)\):

$$\begin{aligned}&\int [F(t,q) - F(s,q)] \, \mu (t, dq) + \int F(t,q) [\mu (t, dq) - \mu (s, dq)] \nonumber \\&\quad - \int [F(t,q) - F(s,q)] [\mu (t,dq) - \mu (s,dq)] = (\hbox {I}) + (\hbox {II}) + (\hbox {III})\quad \quad \quad \end{aligned}$$
(2.28)

Since F(t, q) is uniformly continuous in t uniformly in q by Lemma 2.1 and \(\mu \) is a probability measure, (I) \(\rightarrow 0\) as \(t-s \rightarrow 0\). Using Lemma 2.4 and the fact that \(|F| \le 1\), (II) \(\rightarrow 0\) as \(t-s \rightarrow 0\), and in fact

$$\begin{aligned} \int F(t,q) \frac{\mu (t,dq) - \mu (s,dq)}{t-s} \rightarrow \int F(t,q)(\mathscr {L}^* \mu )(t,dq) \end{aligned}$$
(2.29)

as \(s \rightarrow t-\) with t fixed. Using both Lemmas 2.1 and 2.4, we see (III) is \(o(t-s)\). Thus \(\int F(t,q) \, \mu (t,dq)\) is continuous in t; we will show additionally that it is differentiable from below in t with one-sided derivative equal to 0 for all \(0 < t < T\). In light of (2.29), our task is to show that \(-\int F(t,q) \, (\mathscr {L}^* \mu )(t,dq)\) approximates (I) up to an \(o(t-s)\) error.

Using Lemma 2.2 we have the following approximation of the portion of (I) involving \(\mu _n\), with error bounded by \(C_{\lambda ,H'(P)}[(t-s)^2 + (t-s) w(t-s)]\):

$$\begin{aligned}&\int \left[ F(t,q) (\mathbf {1}_B(q) + \mathbf {1}_V(q)) - F(t,\phi _s^t q) \left( \mathbf {1}_U(q) + \sum _{i=0}^{n-1} \mathbf {1}_{A_i}(q)\right) \right. \nonumber \\&\quad + \,(t-s) F(t, q) \int H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \nonumber \\&\quad \left. - \,(t-s) \int \mathbb {E}F(t, \epsilon _{\rho _+} \phi _s^\tau q) H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \right] \mu _n(t,dq).\quad \quad \end{aligned}$$
(2.30)

We have \(\int [F(t,q) \mathbf {1}_V(q) - F(t,\phi _s^t q) \mathbf {1}_U(q)] \, \mu _n(t,dq) = 0\) because the deterministic flow \(\phi _s^t\) maps U to V (modulo lower-dimensional sets) and preserve the Lebesgue measure in spatial coordinates. Making replacements using Lemma 2.3 and reordering the terms, we find with an error bounded by \(C_{L,n,\lambda ,H'(P)}[(t-s) + w(t-s)]\) that

$$\begin{aligned}&\int \frac{F(t,q) - F(s,q)}{t-s} \mu _n(t,dq) \nonumber \\&\quad \approx \int F(t,\epsilon _{\rho _+} q) H[\rho _{n-1},\rho _+] f(t,\rho _{n-1},d\rho _+) \mu _{n-1}(t,dq) \nonumber \\&\quad \quad - \int \mathbb {E}F(t, \epsilon _{\rho _+} \phi _s^\tau q) H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \mu _n(t,dq) \nonumber \\&\quad \quad - \left[ \int F(t,q) \left( \sum _{i=1}^{n-1} \frac{(H[\rho _*,\rho _i] - H[\rho _{i-1},\rho _*]) f(t,\rho _{i-1},d\rho _*) f(t,\rho _*,d\rho _i)}{f(t,\rho _{i-1},d\rho _i)}\right. \right. \nonumber \\&\left. \qquad \quad \quad \qquad \qquad \qquad + \frac{H[\rho _*,\rho _0] \ell (t,d\rho _*) f(t,\rho _*,d\rho _0)}{\ell (t,d\rho _0)} \right) \mu _{n-1}(t,dq) \nonumber \\&\left. \qquad \qquad - \int F(t,q) \int H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \mu _n(t,dq)\right] \end{aligned}$$
(2.31)

for positive integers n. In the case \(n = 0\), we have \(\phi _s^t q = q\), and the approximation is

$$\begin{aligned} \int \frac{F(t,q) - F(s,q)}{t-s} \mu _0(t,dq) \approx \int [F(t,\epsilon _{\rho _*} q) - F(t,q)] H[\rho _0,\rho _*] \mu _0(t,dq). \end{aligned}$$
(2.32)

We observe that the bracketed portion of (2.31) nearly matches (2.27), except that part involves \(\mu _{n-1}\) and part involves \(\mu _n\). Furthermore, we have

$$\begin{aligned} \begin{aligned}&\left| \int \mathbb {E}F(t, \epsilon _{\rho _+} \phi _s^\tau q) H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \mu _n(t,dq) \right. \\&\qquad \left. - \int F(t, \epsilon _{\rho _+} q) H[\rho _n,\rho _+] f(t,\rho _n,d\rho _+) \mu _n(t,dq) \right| \le C_{H'(P),n,L} (t-s). \end{aligned}\nonumber \\ \end{aligned}$$
(2.33)

This follows since \(\tau \) is conditionally independent of q given \(\rho _n\), and \(\tau \in (s,t)\) a.s. so that all but \(O(t-s)\) volume of the x-simplex is simply translated by \(\phi _s^\tau \) to a region of identical volume. We make the replacement indicated by (2.33) in (2.31) without changing the form of the error.

For any positive integer N we define

$$\begin{aligned} \varGamma _N(t) = \int F(t,q) \sum _{n=0}^N \mu _n(t,dq). \end{aligned}$$
(2.34)

Summing (2.31), (2.32), and our approximation for \(\mu _n(t,\cdot ) - \mu _n(s,\cdot )\) from (2.27) gives

$$\begin{aligned}&\frac{\varGamma _N(t) - \varGamma _N(s)}{t-s} \nonumber \\&\quad \approx \int F(t,q) \left( \sum _{i=1}^N \frac{(H[\rho _*,\rho _i] - H[\rho _{i-1},\rho _*]) f(t,\rho _{i-1},d\rho _*) f(t,\rho _*,d\rho _i)}{f(t,\rho _{i-1},d\rho _i)}\right. \nonumber \\&\left. \quad \qquad \qquad \qquad \qquad + \frac{H[\rho _*,\rho _0] \ell (t,d\rho _*) f(t,\rho _*,d\rho _0)}{\ell (t,d\rho _0)} \right) \mu _N(t,dq) \nonumber \\&\quad \quad - \int F(t,\epsilon _{\rho _+} q) H[\rho _N,\rho _+] f(t,\rho _N,d\rho _+) \mu _N(t,dq) \end{aligned}$$
(2.35)

with an \(o(t-s)\) error. Call the right side \(\gamma _N(t)\), so that the left derivative of \(\varGamma _N\) is \(\gamma _N\). Since \(\varGamma _N\) is a Lipschitz function, we deduce

$$\begin{aligned} \varGamma _N(T) - \varGamma _N(0) = \int _0^T \gamma _N(t) \, dt. \end{aligned}$$
(2.36)

We have \(\gamma _N(t)\) bounded uniformly in t by

$$\begin{aligned} |\gamma _N(t)| \le \frac{3H'(P)L^N \lambda ^{N+1}}{(N-1)!}, \end{aligned}$$
(2.37)

and thus \(\gamma _N \rightarrow 0\) uniformly in t. It follows that

$$\begin{aligned} \varGamma _N(T) - \varGamma _N(0) \rightarrow 0 \end{aligned}$$
(2.38)

as \(N \rightarrow \infty \). Recognizing the limits of \(\varGamma _N(T)\) and \(\varGamma _N(0)\) as

$$\begin{aligned} \int G(q) \mu (T,dq) \quad \text {and} \quad \int \mathbb {E}G(\varPhi _0^T q) \mu (0,dq), \end{aligned}$$
(2.39)

respectively, we have verified (2.14) and completed the proof. \(\square \)

Having shown the candidate measure \(\mu (t,dq)\) constructed using the solutions \(\ell (t,d\rho _0)\) and \(f(t,\rho _-,d\rho _+)\) given by Theorem 1.2, our next task is to verify the latter, which we undertake in the next section.

3 The kinetic and marginal equation

The primary goal of the present section is to prove Theorem 1.2 concerning the kernel \(f(t,\rho _-,d\rho _+)\) and marginal \(\ell (t,d\rho _0)\), so that the candidate measure \(\mu (t,dq)\) in the previous section is well-defined and has the properties required for the argument there.

While we introduce our notation, we also discuss the intuitive meaning of the terms of our kinetic equation, comparing with that of Menon and Srinivasan [25]. Let us write \(\mathscr {L}^{\kappa }\) for the operator on \(\mathscr {K}\) given by the right-hand side of (1.27),

$$\begin{aligned}&(\mathscr {L}^{\kappa } k)(\rho _-,d\rho _+) \nonumber \\&\quad = \int (H[\rho _*,\rho _+] - H[\rho _-,\rho _*]) k(\rho _-,d\rho _*) k(\rho _*,d\rho _+) \nonumber \\&\quad \quad - \left[ \int H[\rho _+,\rho _*] k(\rho _+,d\rho _*) - \int H[\rho _-,\rho _*] k(\rho _-,d\rho _*)\right] k(\rho _-,d\rho _+),\quad \quad \quad \end{aligned}$$
(3.1)

so that the kinetic equation is \(f_t = \mathscr {L}^{\kappa } f\).

The first term, which we call the gain term, corresponds to the production of a shock connecting states \(\rho _-\) and \(\rho _+\) by means of collision of shocks connecting states \(\rho _-,\rho _*\) and \(\rho _*,\rho _+\). Such shocks have relative velocity given by

$$\begin{aligned} H[\rho _*,\rho _+] - H[\rho _-,\rho _*] = H[\rho _-,\rho _*,\rho _+] (\rho _+ - \rho _-). \end{aligned}$$
(3.2)

In the Burgers case, the second divided difference \(H[\cdot ,\cdot ,\cdot ]\) is constant, and the above is proportional to the sum of the increment \(\rho _+ - \rho _-\), analogous to mass in the Smoluchowski equation.

The second line of (3.1) we call the “loss” term, though this need not be of definite sign. To better understand this, note that when we have proven Theorem 1.2, we will know that \(f(t,\rho _-,d\rho _+)\) has total mass which is constant in \((t,\rho _-)\). In this case the loss term may be rewritten equivalently as

$$\begin{aligned}&\left[ \int \big (H[\rho _+,\rho _*] - H[\rho _-,\rho _+]\big ) f(t,\rho _+,d\rho _*) \right. \nonumber \\&\quad \left. - \int (H[\rho _-,\rho _*] - H[\rho _-,\rho _+]) f(t,\rho _-,d\rho _*) \right] f(t,\rho _-,d\rho _+), \end{aligned}$$
(3.3)

which corresponds precisely with the kinetic equation of [25]. The meaning of the first line of (3.3) is clear: we lose a shock connecting states \(\rho _-,\rho _+\) when a shock connecting \(\rho _+,\rho _*\) collides with this, and the relative velocity is precisely \(H[\rho _+,\rho _*] - H[\rho _-,\rho _+]\).

The second line of (3.3) is less easily understood. One would expect to find here a loss related to a shock connecting \(\rho _-,\rho _+\) colliding with \(\rho _*,\rho _-\) for \(\rho _* < \rho _-\), particularly if we were viewing f as a jump density as [25] views its corresponding n(t, y, dz). Viewed as a rate kernel, it is not clear that f should suffer such a loss: in some sense we must condition on being in state \(\rho _-\) for \(f(t,\rho _-,\cdot )\) to be relevant at all. At this point we cannot offer an intuitive kinetic reason for the second line of (3.3), but in light of the rigorous results of this article we can be assured that this is correct for our model.

Remark 3.1

The form (3.3) seems preferable in the more generic setting, since—as pointed out by the referee—the resulting equation \(f_t = \mathscr {L}^\kappa f\) has the property that

$$\begin{aligned} \lambda (t,\rho _-) = \int f(t,\rho _-,d\rho _+) \end{aligned}$$
(3.4)

is (formally) conserved in time for each \(\rho _-\), without assuming that \(\lambda (0,\rho _-)\) is constant. This observation will be important in attempts to generalize the results of the present paper.

We return to the task at hand, showing existence and uniqueness of \(f(t,\rho _-,d\rho _+)\). The argument proceeds through an approximation scheme for \(\exp (\lambda H'(P) t) f(t,\cdot ,\cdot )\), where we can easily maintain positivity. To avoid endowing \(\mathscr {K}\) with a weak topology, we show directly the approximations are Cauchy, rather than appealing to Arzela–Ascoli.

Proof

(Theorem 1.2, part I) Write \(c = \lambda H'(P)\) and consider for each positive integer n the continuous paths \(h^n : [0,\infty ) \rightarrow \mathscr {K}\) defined by \(h^n(0) = g\) for all n and

$$\begin{aligned} \dot{h}^n(t) = e^{-cj/n} (\mathscr {L}^{\kappa } h^n)(j/n) + c h^n(j/n), \qquad t \in (j/n, (j+1)/n). \end{aligned}$$
(3.5)

We claim the following properties of \(h^n\):

  1. (a)

    \(h^n(t) \ge 0\) for all \(t \ge 0\),

  2. (b)

    for each \(t \ge 0\) the total integral \(\int h^n(t,\rho _-,d\rho _+)\) is constant in \(\rho _- \in [0,P)\),

  3. (c)

    for all \(t \ge 0\)

    $$\begin{aligned} \Vert h^n(t)\Vert \le \lambda (1+c/n)^{\lceil nt \rceil } \le \lambda e^{c(t+1)} =: M(t), \end{aligned}$$
    (3.6)

    and

  4. (d)

    \(h^n(t,\rho _-,[0,\rho _-] \cup \{P\}) = 0\) for all t and all \(\rho _- \in [0,P)\).

Since \(h^n\) is piecewise-linear in t and the properties above are preserved by convex combinations, it suffices to verify this at \(t = j/n\) for all integers \(j \ge 0\).

We proceed by induction. Abbreviate \(h^n_j = h^n( j/n)\). The \(j = 0\) case holds by the hypotheses on g. Assume now that the claim holds for j, and consider \(j+1\). We have

$$\begin{aligned} \begin{aligned} n (h^n_{j+1} - h^n_j)&= e^{-cj/n} \left[ \int (H[\rho _*,\rho _+] - H[\rho _-,\rho _*]) h^n_j(\rho _-,d\rho _*) h^n_j(\rho _*,d\rho _+) \right. \\&\qquad \qquad \qquad + \left( \int H[\rho _-,\rho _*] h^n_j(\rho _-,d\rho _*)\right) h^n_j(\rho _-,d\rho _+) \\&\qquad \qquad \qquad \left. - \left( \int H[\rho _+,\rho _*] h^n_j(\rho _+,d\rho _*)\right) h^n_j(\rho _-,d\rho _+)\right] \\&\qquad + c h^n_j(\rho _-,d\rho _+). \end{aligned} \end{aligned}$$
(3.7)

In particular, for any \(\rho _+\) we have

$$\begin{aligned} c - e^{-cj/n} \int H[\rho _+,\rho _*] h^n_j(\rho _+,d\rho _*)\ge & {} c - e^{-cj/n} H'(P) \Vert h^n_j\Vert \nonumber \\\ge & {} c - e^{-cj/n} H'(P) \lambda (1+c/n)^j\nonumber \\= & {} c\left[ 1 - e^{-cj/n}(1+c/n)^j\right] \ge 0,\quad \quad \end{aligned}$$
(3.8)

using property (c) for case j; thus \(n(h^n_{j+1} - h^n_j) \ge 0\). This and (a) \(h^n_j \ge 0\) give \(h^n_{j+1} \ge 0\), property (a) for case \(j+1\). Furthermore, the total integral of the right-hand side of (3.7) is exactly

$$\begin{aligned} c \int h^n_j(\rho _-,d\rho _+) = c \Vert h^n_j\Vert , \end{aligned}$$
(3.9)

since (b) implies the bracketed portion integrates to 0. Also using (b) for case j, the change in the total integral from \(h^n_j\) to \(h^n_{j+1}\) is independent of \(\rho _-\), verifying (b) for case \(j+1\). Using (c) for case j we find

$$\begin{aligned} \Vert h^n_{j+1}\Vert \le \Vert h^n_j\Vert + c n^{-1}\Vert h^n_j\Vert = (1 + c/n) \Vert h^n_j\Vert \le \lambda (1 + c/n)^j, \end{aligned}$$
(3.10)

so that (c) holds for case \(j+1\). Finally, we observe that property (d) for \(h^n_j\) implies

$$\begin{aligned} \mathscr {L}^\kappa h^n_j(\rho _-,[0,\rho _-] \cup \{P\}) = 0, \end{aligned}$$
(3.11)

and thus (d) holds for case \(j+1\).

We now consider the matter of convergence. Pairing \(\mathscr {L}^{\kappa }\) with any \(k_1,k_2 \in \mathscr {K}\) and measurable \(J(\rho _+)\) with \(|J| \le 1\), we find that

$$\begin{aligned} \Vert \mathscr {L}^{\kappa } k_1 - \mathscr {L}^{\kappa } k_2\Vert \le 3 H'(P) (\Vert k_1\Vert + \Vert k_2\Vert ) \Vert k_1 - k_2\Vert . \end{aligned}$$
(3.12)

In particular \(\mathscr {L}^\kappa \) is Lipschitz on bounded sets in \(\mathscr {K}\). For brevity write

$$\begin{aligned} \mathscr {L}^h k(s,\rho _-,d\rho _+) = e^{-cs} \mathscr {L}^\kappa k(\rho _-,d\rho _+) + c k(\rho _-,d\rho _+). \end{aligned}$$
(3.13)

We compute for any \(m<n\) and \(t \ge 0\)

$$\begin{aligned} \Vert h^{m}(t) - h^n(t)\Vert\le & {} \int _0^t \Vert \dot{h}^{m}(s) - \mathscr {L}^h h^{m}(s)\Vert + \Vert \mathscr {L}^h h^{m}(s) - \mathscr {L}^h h^n(s)\Vert \nonumber \\&+\, \Vert \mathscr {L}^h h^n(s) - \dot{h}^n(s)\Vert \, ds. \end{aligned}$$
(3.14)

Observe that

$$\begin{aligned} \Vert \dot{h}^n(s) - \mathscr {L}^h h^n(s)\Vert= & {} \Vert \mathscr {L}^h h^n(n^{-1} \lfloor n s \rfloor ) - \mathscr {L}^h h^n(s)\Vert \nonumber \\\le & {} \Vert e^{-cn^{-1} \lfloor n s \rfloor } \mathscr {L}^\kappa h^n(n^{-1} \lfloor n s \rfloor ) - e^{-cs} \mathscr {L}^\kappa h^n(s)\Vert \nonumber \\&+\, c\Vert h^n(n^{-1} \lfloor n s \rfloor ) - h^n(s)\Vert \end{aligned}$$
(3.15)

where

$$\begin{aligned}&\Vert e^{-cn^{-1} \lfloor n s \rfloor } \mathscr {L}^\kappa h^n(n^{-1} \lfloor n s \rfloor ) - e^{-cs} \mathscr {L}^\kappa h^n(s)\Vert \nonumber \\&\quad \le e^{-cs} \Vert \mathscr {L}^\kappa h^n(n^{-1} \lfloor n s \rfloor ) - \mathscr {L}^\kappa h^n(s)\Vert \nonumber \\&\quad \quad +\, (e^{-cn^{-1} \lfloor n s \rfloor } - e^{-cs}) \Vert \mathscr {L}^\kappa h^n(n^{-1} \lfloor n s \rfloor )\Vert \nonumber \\&\quad \le e^{-cs} 6 H'(P) M(s) \Vert h^n(n^{-1} \lfloor n s \rfloor ) - h^n(s)\Vert \nonumber \\&\qquad + \,e^{-cs} e^{c n^{-1}} n^{-1} 3 H'(P) M(s)^2 \nonumber \\&\quad \le 6 c e^c \Vert h^n(n^{-1} \lfloor n s \rfloor ) - h^n(s)\Vert + 3 c \lambda e^{cs + 3c}n^{-1}, \end{aligned}$$
(3.16)

We have also

$$\begin{aligned} \Vert h^n(n^{-1} \lfloor n s \rfloor ) - h^n(s)\Vert\le & {} n^{-1} \Vert \mathscr {L}^h h^n(n^{-1} \lfloor n s \rfloor )\Vert \nonumber \\\le & {} n^{-1} (e^{-cs} 3 H'(P) M(s)^2 + M(s)) \nonumber \\= & {} n^{-1} (3 \lambda c e^{cs + 2c} + \lambda e^{c(s+1)}). \end{aligned}$$
(3.17)

Combining these, there is a constant \(C_1\) so that

$$\begin{aligned} \Vert \dot{h}^n(s) - \mathscr {L}^h h^n(s)\Vert \le C_1 n^{-1} e^{cs} \end{aligned}$$
(3.18)

for all positive integers n and times \(s \ge 0\).

The remaining term in the integrand of (3.14) is

$$\begin{aligned} \begin{aligned} \Vert \mathscr {L}^h h^{m}(s) - \mathscr {L}^h h^n(s)\Vert&\le e^{-cs} \Vert \mathscr {L}^\kappa h^{m}(s) - \mathscr {L}^\kappa h^n(s)\Vert + c \Vert h^{m}(s) - h^n(s)\Vert \\&\le (e^{-cs} 6H'(P) M(s) + c) \Vert h^{m}(s) - h^n(s)\Vert \\&= (6 e^c + c) \Vert h^{m}(s) - h^n(s)\Vert = C_2 \Vert h^{m}(s) - h^n(s)\Vert . \end{aligned}\nonumber \\ \end{aligned}$$
(3.19)

All together,

$$\begin{aligned} \Vert h^{m}(t) - h^n(t)\Vert \le \int _0^t C_2 \Vert h^m(s) - h^n(s)\Vert + C_1 (n^{-1} + m^{-1}) e^{cs} \, ds, \end{aligned}$$
(3.20)

and by Gronwall

$$\begin{aligned} \Vert h^{m}(t) - h^n(t)\Vert \le C_1 (n^{-1} + m^{-1}) e^{C_2 t} \frac{e^{ct} - 1}{c}. \end{aligned}$$
(3.21)

From this we see that \(h^n\) is Cauchy, hence convergent to a continuous \(h : [0,\infty ) \rightarrow \mathscr {K}\) satisfying

$$\begin{aligned} h(t) = g + \int _0^t [e^{-cs} (\mathscr {L}^\kappa h)(s) + ch(s)] \, ds \end{aligned}$$
(3.22)

for all \(t \ge 0\), having properties (a,b,d) above and \(\Vert h(t)\Vert \le M(t)\).

We define \(f(t) = e^{-ct} h(t)\), finding that

  • the mapping \(t \mapsto f(t) \in \mathscr {K}_+\) is continuous;

  • the mapping \(t \mapsto \mathscr {L}^\kappa f(t) \in \mathscr {K}\) is continuous (using the local Lipschitz property of \(\mathscr {L}^\kappa \)), and hence Bochner-integrable; and

  • the kernels f solve \(\dot{f} = \mathscr {L}^\kappa f\) with \(f(0) = g\) by Leibniz.

Using the local Lipschitz property of \(\mathscr {L}^\kappa \), the solution is unique. Properties (a,b,d) above hold for f, and (b) in particular gives for each fixed \(\rho _-\)

$$\begin{aligned} \frac{d}{dt} \int f(t,\rho _-,d\rho _+) = \int (\mathscr {L}^\kappa f)(t,\rho _-,d\rho _+) = 0. \end{aligned}$$
(3.23)

Thus \(\int f(t,\rho _-,d\rho _+) = \lambda \) for all \(t \ge 0\), \(0 \le \rho _- < P\). \(\square \)

We now turn to \(\ell (t,d\rho _0)\), which is intended to serve as the marginal at \(x = 0\) for our solution \(\rho (x,t)\) for fixed \(t > 0\). We define a time-dependent family of operators \(\mathscr {L}^0\) acting on measures \(\nu (d\rho _0)\) in \(\mathscr {M}\) by

$$\begin{aligned} (\mathscr {L}^0 \nu )(t, d\rho _0)= & {} \int H[\rho _*,\rho _0] \nu (d\rho _*) f(t,\rho _*,d\rho _0) \nonumber \\&- \left[ \int H[\rho _0,\rho _*] f(t,\rho _0,d\rho _*)\right] \nu (d\rho _0). \end{aligned}$$
(3.24)

Again the integration is over \(\rho _*\) only, and for each t we have \((\mathscr {L}^0 \nu )(t,\cdot ) \in \mathscr {M}\). Our evolution equation for \(\ell (t,d\rho _0)\) is the linear, time-inhomogeneous \(\ell _t = \mathscr {L}^0 \ell \).

Proof

(Theorem 1.2, part II) Note that (3.24) is exactly the forward equation for a pure-jump Markov process evolving according to a time-varying rate kernel

$$\begin{aligned} H[\rho _0,\rho _+] f(t,\rho _0,d\rho _+), \end{aligned}$$
(3.25)

so we could obtain existence and uniqueness of solutions along these lines. We outline an argument similar to that employed for f, for the sake of completeness.

Define for each n the continuous path \(h^n : [0,\infty ) \rightarrow \mathscr {M}\) with \(h^n(0) = \delta _0\) and

$$\begin{aligned} \dot{h}^n(t) = e^{-cj/n} \mathscr {L}^0 h^n(j/n) + c h^n(j/n), \qquad t \in (j/n,(j+1)/n). \end{aligned}$$
(3.26)

Proceeding as in the proof for f we verify that for each j the \(h^n_j = h^n(j/n)\) are nonnegative and, since \(\mathscr {L}^0 h^n(j/n)\) has total integral zero, \(\Vert h^n_j\Vert \le (1+c/n)^{\lceil n t \rceil }\). Observing that \(\mathscr {L}^0\) is linear and bounded uniformly over t,

$$\begin{aligned} \sup _t \Vert (\mathscr {L}^0 \nu )(t,\cdot )\Vert _{\textsc {tv}} \le 2 \lambda H'(P) \Vert \nu \Vert _{\textsc {tv}}, \end{aligned}$$
(3.27)

we easily show \(h^n\) is Cauchy on bounded time intervals, with limit \(h : [0,\infty ) \rightarrow \mathscr {M}_+\) satisfying

$$\begin{aligned} h(t) = \delta _0 + \int _0^t [e^{-cs} (\mathscr {L}^0 h)(s) + ch(s)] \, ds. \end{aligned}$$
(3.28)

Take \(\ell (t) = e^{-ct} h(t)\) to find that \(\ell (t) \ge 0 \) solves

$$\begin{aligned} \ell (t) = \delta _0 + \int _0^t (\mathscr {L}^0 \ell )(s) \, ds. \end{aligned}$$
(3.29)

Uniqueness follows using (3.27). Finally, since \(\mathscr {L}^0 \nu \) as zero total integral for any \(\nu \), we find that \(\int \ell (t,d\rho _0) = 1\) for all t. \(\square \)

We close this section with a remark concerning the kinetic equation \(f_t = \mathscr {L}^\kappa f\) without the assumption that the initial kernel g has bounded support [0, P]. When the initial condition \(\xi (x)\) is unbounded, growing for example linearly in the case of quadratic H, the solution to the scalar conservation law \(\rho _t = H(\rho _x)\) on the semi-infinite domain should blow up in finite time. We likewise expect that the solution f to the kinetic equation will blow up, but control of certain moments prior to this blow up may allow us to run the argument of Theorem 2.1 with only superficial changes. At the moment we lack the sort of estimates one obtains in the Smoluchowski case (e.g. coming from a closed equation for the second moment), but we hope to revisit this in future work.

4 Conclusion

We review what has been accomplished: using an exact propagation of chaos calculation for a bounded system with a suitably selected random boundary condition, we have derived a complete description of the law of the solution \(\rho (x,t)\) to the scalar conservation law \(\rho _t = H(\rho )_x\) for \(x,t > 0\) when \(\rho (x,0) = \xi (x)\) is a bounded, monotone, pure-jump Markov process with constant jump rate. Notably, we have recovered the Markov property of the solution and a statistical description of the shocks simultaneously. This may be regarded as both a strength of our present analysis and a shortcoming of our present understanding (a soft argument for the preservation of the Markov property in the particle system would be illuminating).

We emphasize how our approach using random dynamics on a bounded interval has made things considerably easier: by constructing a bounded system which has exactly the right law, we are relieved of the burden of determining precisely how wrong the law would be with deterministic boundary.

Our result lends additional support to the conjecture of Menon and Srinivasan [25] and we hope that the sticky particle methods described herein will provide another approach, quite different from that of Bertoin [4], which might be adapted to resolve the full conjecture. One of the authors is attempting a similar particle approach in the non-monotone setting, and hopes to report on this in future work.