Abstract
We prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.
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1 Introduction
Stochastic partial differential equations with fractional Laplacian appear in many fields such as physics, fractal medium, image analysis, risk management and other fields (see Debbi [4], Mueller [12] and Xie [15]). In this paper, we discuss the existence and uniqueness of solutions to the initial value problem for the following stochastic partial differential equation which is denoted by SPDE:
where \(-(-\Delta )^{\alpha /2}\) is the fractional Laplacian with order \(1<\alpha <2\), f is an initial value, \({\dot{W}}(t,x)\) is a space-time white noise on \([0,\infty )\times {\mathbb {R}}\), and \(\gamma \) is a parameter satisfying \(1/2<\gamma <1\).
We will explain some known results related to (1.1). Consider the case where \(\alpha =2\) and coefficients of noise are Lipschitz continuous, that is,
where \(a:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is Lipschitz continuous. Equation (1.2) was studied by Funaki [7] and Walsh [18]. They showed the unique existence of a solution to (1.2) and spatial regularity of the solution. Later Mueller–Perkins [11] and Shiga [17] studied the SPDE (1.2) without assuming the Lipschitz continuty of coefficients a. They imposed the following growth condition on a(u):
for some \(0<\theta <1\). They showed the existence of probability space \(({\bar{\Omega }},\bar{{\mathcal {F}}},{\bar{P}})\) on which there is a space-time white noise \(\dot{{\bar{W}}}(t,x)\) such that (1.2) with \({\dot{W}}(t,x)\) replaced by \(\dot{{\bar{W}}}(t,x)\) has a mild solution X(t, x). In addition, they proved that, if f is positive, then any solution X(t, x) to (1.2) is positive for all \(t\ge 0\) and \(x\in {\mathbb {R}}\). Moreover, it was shown that, if f(x) is sufficiently rapidly decreasing as \(x\rightarrow \infty \), then any solution X(t, x) to (1.2) is also sufficiently rapidly decreasing. Mytnik [13] proved the weak uniqueness of solutions to the following equation which is a special one of (1.2):
with \(1/2<\gamma <1\). The idea of his proof is based on a duality argument developed by Ethier–Kurtz [5]. Mytnik [14] studied the dual process Y described by the following SPDE:
where \({\dot{L}}\) is a stable noise on \({\mathbb {R}}\times {\mathbb {R}}_+\) with nonnegative jumps. He coordinated a probability space \(({\tilde{\Omega }},\tilde{{\mathcal {F}}},{\tilde{P}})\) on which there exists a stable noise \({\dot{L}}\) and random field Y(t, x), where Y(t, x) is a mild solution of (1.5).
Recently SPDEs with fractional Laplacian have been discussed by several authors (see; e.g. Chen [1], Debbi [4], Niu–Xie [15]). Consider (1.2) with \(\Delta \) replaced by \(-(-\Delta )^{\alpha /2}\) for \(1<\alpha <2\) and Lipschitz continuous coefficients of noise, that is,
where \(a:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is Lipschitz continuous. Chen [1], Debbi [4] and Niu–Xie [15] have shown the unique existence of mild solution to (1.6) and its regularity.
However, it is still an open problem to show the existence and uniqueness of solution to (1.6) without assuming the Lipschitz continuty of a(u). The purpose of this paper is to establish the existence and weak uniqueness of a mild solution to (1.1) which is a special case of (1.6) where \(a(u)=|u|^\gamma \) with \(\frac{1}{2}<\gamma <1\). We intend to use similar aguments to Mueller–Perkins [11], Shiga [17], and Mytnik [13]. The large difference between the fractional Laplacian and the usual Laplacian can be found in the decay properties of fundamental solutions as \(x\rightarrow \infty \). Since the fundamental solution of the usual Laplacian has an exponential decay property, the solution X(t, x) to (1.2) has the exponential decay property if the condition (1.3) holds. Similarly the fundamental solution of fractional Laplacian decays polynomially, we can show that the solution to (1.1) has the polinomial decay property if the condition (1.3) holds. To prove the uniqueness of solution to (1.4) Mytnik [13] used the exponential decay property. It is enough to use the polynomial decay property in order to prove the uniqueness of solution to (1.1).
This paper is organized as follows: in Sect. 2 we prepare some tools and lemma to prove our main results. In Sect. 3, we show the existence and uniqueness of solution by applying the Banach fixed point theorem with Lipschitz continuous coefficients a(X). In addition, we follow the argument of Mueller–Perkins [11] and Shiga [17] to prove positivity and polynomial decay properties of solutions to (1.6). From Sect. 4 we begin with consideration (1.1). We prove the existence of solution to (1.1) by tightness arguments. In fact, we can prove a uniqueness of solution in the distributional sense by applying to duality method. Since the proof is almost same as the paper [5], we omit the detail.
2 Preliminaries
2.1 Fractional differential operator
For \(1<\alpha \le 2\), let \(-(-\Delta )^{\alpha /2}\) be a fractional differential operator defined by the Fourier transform \({\mathcal {F}}\):
where
Let G(t, x) be a fundamental solution to the Cauchy problem
where \(\delta _0\) denotes the Dirac measure. Then G(t, x) can be expressed using the Fourier transform:
and has the following properties (cf. Debbi–Dozzi [4] and Kotelenez [9]):
Lemma 2.1
Let G(t, x) be the fundamental solution of (2.1) and define
Then there exists a constant \(C_\alpha >0\) such that for all \(0\le s \le t\le T,~x\in {\mathbb {R}}\), and \(0<\rho \le (\alpha +1)/2\), the following properties hold:
Hereafter, we sometimes write \(G(t,x-y)=G(t,x,y)\). Note that there exists a constant \(C_1,C_2>0\) such that \(\lambda ^{-\rho }(y)\le C_1\lambda ^{\rho }(x-y)\lambda ^{-\rho }(x)\) and from Lemma 2.1 (ii) we obtain the estimate
2.2 Definition of the solution
Let \((\Omega ,{\mathcal {F}},{\mathcal {F}}_t,P)\) be a complete probability space with filtration and \({\dot{W}}(t,x)\) be an \(\{{\mathcal {F}}_t\}\)-space-time Gaussian white noise with covariance given by
for \(t,t'\ge 0\) and \(x,x'\in {\mathbb {R}}\). For an \(\{{\mathcal {F}}_t\}\)-predictable functional \(\phi (t,x,\omega ):[0,\infty )\times {\mathbb {R}}\times \Omega \rightarrow {\mathbb {R}}\) satisfying
we can define stochastic integral (cf. Walsh [18])
with quadratic variational process
The Eq. (1.1) makes sense if we integrate the equation in time and space and use the initial condition.
Definition 2.1
An \(({\mathcal {F}}_t)\)-adapted random field \(\{X(t,x),t\ge 0,x\in {\mathbb {R}}\}\) is said to be a solution in the sense of generalized functions of (1.1) if for any \(\phi \in C_0^\infty ({\mathbb {R}})\), the following equality holds:
Using the Green function, we can describe a solution of (1.1) in a mild form.
Definition 2.2
An \(({\mathcal {F}}_t)\)-adapted random field \(\{X(t,x),t\ge 0,x\in {\mathbb {R}}\}\) is said to be a mild solution of (1.1) with initial function f if the following stochastic integral equation holds:
where G(t, x, y) denotes the Green function of (2.1).
We introduce a martingale problem induced by (1.1).
Definition 2.3
Let S be a Banach space. A solution to martingale problem for (1.1) we mean a measurable process X with values in S defined on some probability space \((\Omega ,{\mathcal {F}},P,\{{\mathcal {F}}_t\})\) with a filtration satisfyng for all \(\phi \in {\mathcal {D}}(-\Delta )^{\alpha /2}\)
is an \({\mathcal {F}}_t^X\) square integrable martingale with the quadratic variation given by
Definition 2.4
Let S be a Banach space and \(X_1\) and \(X_2\) be S-valued mild solutions to the SPDE (1.1) with the same initial value. We say that the SPDE (1.1) has pathwise uniqueness if
holds.
To this end, it is required that all the terms in (2.4) and (2.3) are well defined. Here, a relationship between a solution in the sense of generalized functions and a mild solution is well known (cf. [18]).
Proposition 2.1
A solution in the sense of generalized functions of (1.1) is equivalent to the mild solution.
A solution to the martigale problem and a mild solution in weak sense are equivalent (cf. [7]).
Proposition 2.2
The following (1) and (2) are equivalent.
-
(1)
X(t, x) is a solution to the martigale problem for (1.1).
-
(2)
There exists an \(\{{\mathcal {F}}_t\}\)-space-time Gaussian white noise \({\dot{W}}(t,x)\) and stochastic process X(t, x) such that X(t, x) is a mild solution of the SPDE (1.1) on a suitable probability space with filtration \((\Omega ,{\mathcal {F}},P,\{{\mathcal {F}}_t\}).\)
3 Some properties of solutions in Lipschitz case
3.1 Existence and uniqueness of mild solutions
In order to prove the existence and uniqueness to SPDE (1.1), we first consider the case where coefficients are Lipschitz continuous:
where \(a:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is Lipschitz continuous. Using the Green function, we can write the solution of Eq. (3.1) in a mild form:
For any \(0<\rho <(\alpha +1)/2\), define a weighted \(L^2\)-norm defined by
Theorem 3.1
Assume that \(f\in L^2_\rho ({\mathbb {R}})\) and \(a:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Lipschitz function satisfying linear growth condition, that is, there exists \(C>0\) such that
Then (3.2) admits the pathwise unique mild solution \(X(t,x)_{(t,x)\in [0,T]\times {\mathbb {R}}}\) such that for each \(T>0\),
For every \(0<\rho <(\alpha +1)/2\), define a function space
The regularity of solution to (3.1) is well known (cf. [4], [15]).
Theorem 3.2
Under the conditions of Theorem 3.1 with the initial condition \(f\in C_\rho ({\mathbb {R}})\), the solution X(t, x) of (3.1) has the continuous modification on \([0,\infty )\times {\mathbb {R}}\). In addition, the solution X(t, x) to (3.1) has the \(\beta \)-Hölder continuous modification for \(t\in [0,\infty )\) and \(\eta \)-Hölder continuous modification for \(x\in {\mathbb {R}}\) with \(\beta \in (0,\frac{\alpha -1}{2\alpha })\) and \(\eta \in (0,\frac{\alpha -1}{2})\).
3.2 Positivity of solution
We will follow Shiga’s arguments [17] to prove the positivity of the solution to (3.1). At first, we need to prepare a boundness of the solution to (3.1).
Lemma 3.1
Assume that for every \(T>0\) there exists \(C_T>0\) and such that
Then for \(p\ge 1\) can be written by \(p=2^m,~m\in {\mathbb {N}}\cup {\{0\}}\) and \(0<\rho <(\alpha +1)/2\), the mild solution X(t, x) to (3.2) with initial condition \(f\in C_\rho ({\mathbb {R}})\) satisfies
Proof
Assume that \(p=1\). Then for every \(0<\rho <(\alpha +1)/2\) and \(t\ge 0\), we have
We will apply an induction argument. For \(p\ge 2\), using the Burkhorder’s inequality, the Hölder’s inequality shows that
Multiply both sides by \(\lambda ^\rho (x)\) and integrate with variable x, we obtain that
Setting \(p=2\), (3.6) and Grownwall’s lemma imply that
From an induction argument, we complete the proof for \(p=2^m\) with \(m\in {\mathbb {N}}\cup \{0\}\). \(\square \)
Remark 3.1
We can get in a similarly way to the proof of Lemma 3.1 and Hölder’s inequality together,
For any \(0<\rho <(\alpha +1)/2\) define the function space \(C_\rho ^+({\mathbb {R}})\) by
Theorem 3.3
Let X(t, x) be the mild solution of (3.2) with the initial function \(f\in C_\rho ^+({\mathbb {R}})\). Assume that a(u) satisfies the condition (3.4). In particular, \(a(0)=0\). Then, we have
Proof
For every \(\varepsilon >0\), choose non-negative, symmetric and smooth function \(\rho _\varepsilon (x)\) such that
Define
Notice that, for every \(x\in {\mathbb {R}}\), \(W_x^\varepsilon (t)\) is a one-dimensional Brownian motion. Set \(\Delta _\varepsilon =(G(\varepsilon )-I)/\varepsilon \) for \(\varepsilon >0\) where \(G(\varepsilon )f(x)=\left( G(\varepsilon ,\cdot )*f\right) (x).\) We consider the following equation
Since a(u) is Lipschitz continuous and \(\Delta _\varepsilon \) is a bounded operator on \(L^2({\mathbb {R}})\), the above equation has the unique strong solution and continuous version(cf. [2]). From now on, we claim that
Let \(a_n=-2(n^2+n+2)^{-1}\) be a non-increasing sequence. Immediately, we obtain that \(a_n\rightarrow 0\) as \(n\rightarrow \infty \) and \(\int _{a_{n-1}}^{a_n}x^{-2}dx=n\). Let \(\psi _n(x)\) be a nonnegative continuous function such that
Define
Then we can get \(\Phi _n(x)\in C^2({\mathbb {R}})\) with \(\Phi ^{\prime \prime }_n(x)=\psi _n(x)~\), \(-1\le \Phi _n^{\prime }(x)=\int _0^x\psi _n(z)dz\le 0\) for \(x<0\) and \(\Phi _n(x)=0\) for \(x\ge 0\). Note that, for \(x<0\) there exists \(n_0\in {\mathbb {N}}\) such that for all \(n\ge n_0\) we have \(a_n> x\). About \(\Phi _n\) we can get the following properties as \(n\rightarrow \infty \),
By \(\mathrm{It{\hat{o}}^{\prime }s}\) formula
From the Lipschitz condition and \(a(0)=0\),
Hence
Taking the limit as \(n\rightarrow \infty \) and by monotone convergence theorem
For the second term, we notice that
Then, since \(|x|1(x<0)=\phi (x)\), we get
Therefore, by Grownwall’s lemma for \(\sup _{x\in {\mathbb {R}}}E(\phi (X_\varepsilon (t,x)))\), we obtain
for every \(t>0\) and \(x\in {\mathbb {R}}\), which yields (3.8). Let
We will prove that
To prove this fact, we need the following lemma (cf. Appendix of [1]).
Lemma 3.2
- \(\mathrm{(i)}\) :
-
$$\begin{aligned} \int _{\mathbb {R}}|R_\varepsilon (t,x,y)-G(t,x,y)|dy\le e^{-t/\varepsilon }+C(\varepsilon /t)^{1/2}~~~{\forall }t>0,\varepsilon >0. \end{aligned}$$
- \(\mathrm{(ii)}\) :
-
For some \(\alpha >0\) and \(\beta >0\)
$$\begin{aligned} \int _{\mathbb {R}}R_\varepsilon (t,x,y)^2dy\le Ct^{-1/\alpha }~~~{\forall }>0. \end{aligned}$$ - \(\mathrm{(iii)}\) :
-
$$\begin{aligned} \lim _{\varepsilon \rightarrow \infty }\int _0^t\int _{\mathbb {R}}(R_\varepsilon (t,x,y)-G(t,x,y))^2=0~~~{\forall }t>0,x\in {\mathbb {R}}. \end{aligned}$$
Notice that, \(X_\varepsilon (t,x)\) can be written in the following mild form:
where the last term equals to
Since f is bounded, it follows that for every \(T>0\) (see Remark 3.1),
Then we have
By using Lemma 3.2 and boundness of f(x)
As for \(I_2(t,x,\varepsilon )\)
This inequality and (3.9) imply that
By the Hölder’s inequality, the Lipschitz continuty of function a(x) and Lemm 3.2, we have
and similarly
According to the inequality
the definition of \(\rho _\varepsilon \) gives that
By Hölder’s inequality and (3.10),
and applying Lemma 3.2 we obtain
In a same way, we have
Applying the fact
we can get
Now, we set
Then there exists some constant \(C>0\) such that
where
Therefore, Grownwall’s inequality implies that
and thus completes the proof of Theorem 3.3. \(\square \)
3.3 Polynomial decay
In this section, we show that the solution of (3.1) has modification in the class \(C_\rho ({\mathbb {R}})\). The following lemma is a variant of a Kolmogorov’s continuity criterion theorem.
Lemma 3.3
-
(i)
Suppose that for every \(0<\rho <(\alpha +1)/2\) there exist \(p>0,\gamma >2\) and \(C_\rho >0\) such that
$$\begin{aligned} E[|X(t,x)-X(t',x')|^p]\le C_\rho (|t-t'|^\gamma +|x-x'|^\gamma )\lambda ^{-\rho }(x), \end{aligned}$$for \(0\le t,t'\le 1\) and \(x,x'\in {\mathbb {R}}\) with \(|x-x'|\le 1\). Then \(X(t,\cdot )\) has \(C_{\rho }\)-valued continuous version P-a.s.
-
(ii)
Let \(\{X_n(t,\cdot );~t\ge 0,n\in {\mathbb {N}}\}\) be a sequence of continuous \(C_\rho \)-valued processes. Suppose that for every \(0<\rho <(\alpha +1)/2\) and \(T>0\) there exists \(p>0,\gamma >2\) and \(C_\rho >0\) such that
$$\begin{aligned} E[|X_n(t,x)-X_n(t',x')|^p]\le C_\rho (|t-t'|^\gamma +|x-x'|^\gamma )\lambda ^{-\rho }(x), \end{aligned}$$for \(t,t'\in [0,T]\) and \(x,x'\in {\mathbb {R}}\) with \(|x-x'|\le 1\). Then the sequence of probability distributions on \(C([0,\infty );C_\rho ({\mathbb {R}}))\) induced by \(X_n(\cdot )\) is tight.
Theorem 3.4
Under the conditions of Lemma 3.1, \(X(t,\cdot )\) has \(C_\rho ({\mathbb {R}})\) valued continuous version P-a.s.
Proof
Since \(f\in C_\rho ({\mathbb {R}})\), it is enough to prove that for all \(0<\rho <(\alpha +1)/2\) there exist \(p>0\), \(\gamma >2\) and \(C_\rho >0\) such that
for \(X(t,x)=\int _0^t\int _{\mathbb {R}}G(t-s,x,y)a(X(s,y))W(dy,ds)\), \(0\le t,t'\le 1\) and \(x,x'\in {\mathbb {R}}\) with \(|x-x'|\le 1\). We first show (3.11) with \(t=t'\). From the Burkholder’s inequality and the Hölder’s inequality, we have for every \(p=2^m,~m\in {\mathbb {N}}\),
where \(\rho < \frac{\alpha +1}{p-2}\). Lemma 3.1 implies that
From Lemma 2.1
where \(x(\theta )=x+(1-\theta )x'\) for \(0<\theta <1\). Therefore, for every \(\kappa <\alpha -1\)
Note that,
so we have
Choosing \(p=2^m\) satisfying that \(\frac{p-2}{2}\kappa >2\), we can get (3.11) with \(t=t'\). Next, we prove (3.11) with \(x=x'\). In the same way as above, for \(0\le t\le t'\le T\), we can show that
for \(\rho <\frac{\alpha +1}{p-2}\). Immediately, we can get from Lemma 2.1
Hence combining with (3.5)
From now on, we claim that
The change of variable \(s=\theta v\) with \(\theta =t'-t\),
Note that,
Again, by the change of variable \(z=\theta ^{-\frac{1}{\alpha }}(x-y)\), we have
Therefore, to prove (3.11) we need to show that
Let us write
The first integral is finite, since
For the second term one, we use Lemma 2.1 and the change of variable \(z=(1+v)^{1/\alpha }z'\)
Further,
and by Lemma 2.1
Notice that,
Thus we can estimate
Since \(0<\rho <(\alpha +1)/2\), we have
By the mean value theorem, we can get
Hence
Since \(-2(\alpha +1)/\alpha +1/\alpha <-1\), the last integral is finite and thus completes the proof of Theorem 3.4. \(\square \)
4 Existence in non-Lipschitz case
In this section, we consider SPDE (1.1). We prove the existence of solutions by using the results in the previous section.
Theorem 4.1
Let \(f\in C_\rho ^+({\mathbb {R}})\) be an initial function. Then for every \(T>0\), there exist an \(\{{\mathcal {F}}_t\}\)-space-time Gaussian white noise \({\dot{W}}(t,x)\) and \(C([0,T];C_\rho ^+({\mathbb {R}}))\) valued solutions X to (2.3) with \(X(0)=f\) on a suitable probability space with filtration \((\Omega ,{\mathcal {F}},P,\{{\mathcal {F}}_t\}).\)
Proof
Let \(a_n(u)\) be a sequence of Lipschitz functions such that
The sequence \(a_n(u)\) converge to \(|u|^\gamma \) uniformly in \(u\in {\mathbb {R}}\) as \(n\rightarrow \infty \). Then by Theorem 3.1, 3.3 and 3.4 for every \(0<\rho <(\alpha +1)/2\), there exist the unique \(C_\rho ^+({\mathbb {R}})\)- valued solution \(X_n\) to (3.1) with \(a_n(u)\) for each \(n\ge 1\).
The solution \(X_n\) holds the moment condition (3.11), and it follows from Lemma 3.3 that the family of probability distributions on \(C\left( [0,T];C_\rho ^+({\mathbb {R}})\right) \) induced by \(\left\{ X_n\right\} \) is tight. This means that there exists a subsequence \(\left\{ n_k\right\} _{k\in {\mathbb {N}}}\) and a random field \(X\in C\left( [0,T];C_\rho ^+({\mathbb {R}})\right) \) such that
By Skorohod representation theorem (cf. [5]), we can find some random fields \(Y_n,Y\in C\left( [0,T];C_\rho ^+({\mathbb {R}})\right) \) on some probability space \(({\bar{\Omega }},\bar{{\mathcal {F}}},(\bar{{\mathcal {F}}}_t)_{0\le t\le T},{\bar{P}})\), such that
and
Then we can get for every \(\phi \in {\mathcal {D}}((-\Delta )^{\alpha /2})\)
Note that, by (3.7)
Hence, \((M_\phi ^n)_{n\in {\mathbb {N}}}\) is a sequence of uniformly integrable martingales, and therefore, there exists a martingale \(M_\phi \) such that
and
It follows from (4.1) we can get as \(n\rightarrow \infty \),
and so that
This means that there corresponds a martingale measure M(dx, dt) with the quadratic measure
Take an \({\mathcal {S}}'({\mathbb {R}})\)-valued standard Wiener process \({\bar{W}}_t\) independent of \(X_s(dx)\). We set
From the definition of M and W we can show that
Since the last term equals to 0 a.s., we have
Thus we complete the proof of Theorem 4.1. \(\square \)
References
Chen, L., Kim, K.: On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. Ann. Inst. Henri Poincare Probab. Stat. 53(1), 358–388 (2017). https://doi.org/10.1214/15-AIHP719
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Dawson, D.: The critical measure diffusion process. Z. Wahr. Verw Gebiete 40, 125–145 (1977)
Debbi, L., Dozzi, M.: On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. Stoch. Process. Appl. 115(11), 1764–1781 (2005)
Ethier, S., Kurtz, T.: Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1986)
Fleischmann, K.: Critical behavior of some measure-valued processes. Math. Nachr. 135, 131–147 (1988)
Funaki, T.: Regularity properties for stochastic partial differential equations of parabolic type. Osaka J. Math. 28(3), 495–516 (1991)
Iscoe, I.: A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71, 85–116 (1986)
Kotelenez, P.: Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stoch. Stoch. Rep. 41, 177–199 (1992)
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1997)
Mueller, C., Perkins, E.: The compact support property for solutions to the heat equation with noise. Probab. Theory Relat. Fields 93, 325–358 (1992)
Mueller, C.: The heat equation with Lévy noise. Stoch. Process. Appl. 74(1), 67–82 (1998)
Mytnik, L.: Weak uniqueness for the heat equation with noise. Ann. Probab. 26, 968–984 (1998)
Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields 123(2), 157–201 (2002)
Niu, M., Xie, B.: Regularity of a fractional partial differential equation driven by space-time white noise. Proc. Am. Math. Soc. 138, 1479–1489 (2010)
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. An Evolution Equation Approach, Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge (2007)
Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math. 46(2), 415–437 (1994)
Walsh, J.: An Introduction to Stochastic Partial Differential Equations. Ecole d’Eté de probabilités de Saint-Flour, XIV-1984, Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)
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Nakajima, S. Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients. Stoch PDE: Anal Comp 10, 255–277 (2022). https://doi.org/10.1007/s40072-021-00199-6
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DOI: https://doi.org/10.1007/s40072-021-00199-6