Marta Sanz-Solé, Iván Torrecilla-Tarantino
We prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager, PTRF 2002, to an infinite dimensional setting.
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We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.
A Stochastic Reconstruction Theorem on Rectangular Increments with an Application to a Mixed Hyperbolic SPDE
该研究将随机重构定理推广至矩形增量情形，证明了结合Walsh随机积分与Young乘积的新型混合双曲型SPDE的适定性。
We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space.
We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $p\in[2,\infty)$, of SPDEs in divergence form in $\bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable.
We consider divergence form uniformly parabolic SPDEs with bounded and measurable leading coefficients and possibly growing lower-order coefficients in the deterministic part of the equations. We look for solutions which are summable to the second power with respect to the usual Lebesgue measure along with their first derivatives with respect to the spatial variable.
In this paper we prove the existence and uniqueness of strong solutions for SPDE in Hilbert space with locally monotone coefficients, which is a generalization of the classical result of Krylov and Rozovskii for monotone coefficients. Our main result can be applied to different types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic $p$-Laplace equation and stochastic porous media equation with some non-monotone perturbations.
We present several results on the smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.
This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new i…
We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p\geq2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory.
Motivated by applications to a manifold of semilinear and quasilinear stochastic partial differential equations (SPDEs) we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main result of our paper by showing how it can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.
Global well-posedness for hyperbolic SPDEs with non-Lipschitz coefficients driven by space-time Lévy white noise
本文研究了由时空莱维白噪声驱动的超几何随机微分方程组在局部利普希茨和线性增长系数下的全局良好性问题。
We provide new regularity results for the solutions of the Kolmogorov equation associated to a SPDE with nonlinear diffusion coefficients and a Burgers type nonlinearity. This generalizes previous results in the simpler cases of additive or affine noise. The basic tool is a discrete version of a two sided stochastic integral which allows a new formulation for the derivatives of these solutions. We show that this can be used to generalize the weak order analysis performed in [16]. The tools we develop are very general and can be used to study many other examples of applications.
We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.
In this paper we develop an $L_2$-theory for stochastic partial differential equations driven by Lévy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of the coefficients is assumed.
Non-local quasilinear singular SPDEs
本文研究二维环面上系数为非局部泛函的拟线性广义抛物型安德森模型，在扩散系数正性假设下，利用抛物控制微分框架建立了局部时间解理论。
Renormalisation of Singular SPDEs with Correlated Coefficients
本文研究了具有相关系数的奇异随机偏微分方程的局部良定性，通过随机重整化函数和随机估计方法解决了传统重整化常数导致的方差问题。
For solutions of a certain class of SPDEs in divergence form we present some estimates of their $L_{p}$-norms and the $L_{p}$-norms of their first-order derivatives. The main novelty is that the low-order coefficients are supposed to belong to certain Morrey classes instead of $L_{p}$-spaces. Our results are new even if there are no stochastic terms in the equation.
Sample path properties of parabolic SPDEs with non constant coefficients
研究由非恒定系数抛物型算子及高斯白噪声驱动的SPDE，证明解的存在唯一性，并刻画其时空样本路径正则性。
In this article, we study stochastic partial differential equations with two reflecting walls, driven by space-time white noise with non-constant diffusion coefficients under periodic boundary conditions. The existence and uniqueness of invariant measures is established under appropriate conditions. The strong Feller property is also obtained.
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense…
In this paper we study the existence and uniqueness of the $L_ρ^{2p}(\mathbb{R}^d;\mathbb{R}^1)\times L_ρ^2(\mathbb{R}^d;\mathbb{R}^d)$ valued solution of backward doubly stochastic differential equations with polynomial growth coefficients using week convergence, equivalence of norm principle and Wiener-Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding backward doubly stochastic differential equations (BDSDEs). This probabilistic representation is the…
In this paper we establish the strong existence, pathwise uniqueness and a comparison theorem to a stochastic partial differential equation driven by Gaussian colored noise with non-Lipschitz drift, Hölder continuous diffusion coefficients and the spatial domain in finite interval, $[0,1]$, and with Dirichlet, Neumann or mixed nonhomogeneous random conditions imposed on the endpoints. The Hölder continuity of the solution both in time and in space variables is also studied.
We give general sufficient conditions which imply upper and lower bounds for the probability that a multiparameter process hits a given set E in terms of a capacity of E related to the process. This extends a result of Khoshnevisan and Shi [Ann. Probab. 27 (1999) 1135-1159], where estimates for the hitting probabilities of the (N,d) Brownian sheet in terms of the (d-2N) Newtonian capacity are obtained, and readily applies to a wide class of Gaussian processes. Using Malliavin calculus and, in particular, a result of Kohatsu-Higa [Probab. Theory Related Fields 126 (2003) 421-457], we apply thes…
Renormalised Models for Variable Coefficient Singular SPDEs
本文在正则性结构框架下，将重正规化模型的收敛性结果推广至变系数奇异SPDE，引入局部重正规化函数并证明模型对系数场的连续依赖性。
In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the $L^1$-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.
Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application
本文针对具有完全局部单调系数的随机偏微分方程，通过Faedo-Galerkin方法等建立Wong-Zakai逼近，并应用于确定解的拓扑支集，涵盖多种流体模型。
Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
本文针对加性噪声驱动的半线性随机波动方程，结合标准有限元空间离散与指数函数有理逼近时间离散，利用 Malliavin 分析证明弱收敛速率约为强速率的两倍，并通过数值模拟验证了理论结果。
A central limit theorem for nonlinear conservative SPDEs
本文证明了适用于非线性退化扩散及非正则噪声系数的保守型随机偏微分方程的中心极限定理，并揭示其涨落与相互作用粒子系统流体力学极限涨落的一阶一致性。
Milstein-type Schemes for Hyperbolic SPDEs
研究乘性高斯噪声驱动的双曲型半线性随机发展方程的时间逼近，通过Milstein型格式获得路径一致强误差的最优收敛阶，将结果从抛物型推广到双曲型SPDE。
We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singular terminal condition. For this degenerate BSPDE, we prove existence and uniqueness of a nonnegative solution. Our existence result requires a novel gradient estimate for degenerate BSPDEs.
Large deviation principles for the stationary solutions and invariant measures of a class of SPDE with locally monotone coefficients
本文研究了具有局部单调系数的SPDE的平稳解的存在性，并证明了这些解的大偏差原理，通过收缩原理推导出不变测度的大偏差原理，避免了构造准势能和验证Dembo-Zeitouni均匀大偏差原理的需要。
We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure $L^2$ condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function $a(t,x,u)=Cu^γ$, $1/2\leγ\le1$ and $C\ne0$. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.
Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is an Hilbert-Schmidt perturbation of a fixed bounded operator. In this paper we investigate a class of semi-linear SPDEs with strongly multiplicative noise whose coefficient is even allowed to be unbounded which is thus no way to be Hilbert-Schmidt. Gradient estimates, log-Harnack inequality and applications are derived. Applications to stochastic reaction-diffusion equations driven by space-time white noise are pres…
We establish n-th order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial d…
Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs
本文证明驯化指数欧拉格式对超线性双曲型随机偏微分方程具有1/2阶强收敛率，将结果从全局推广至局部Lipschitz非线性，并应用于多种非线性方程。
In this paper we develop the $l_p$-theory of space-time stochastic difference equations which can be considered as a discrete counterpart of N.V. Krylov's $L_p$-theory of stochastic partial differential equations. We also prove a Calderon-Zygmund type estimate for deterministic parabolic finite difference schemes with variable coefficients under relaxed assumptions on the coefficients, the initial data and the forcing term.
McKean-Vlasov SPDEs with coefficients exhibiting locally weak monotonicity: existence, uniqueness, ergodicity, exponential mixing and limit theorems
本文针对具有局部弱单调性系数的McKean-Vlasov SPDE，利用Galerkin投影和广义耦合方法证明了解的存在唯一性，进而建立指数遍历性、指数混合性以及强大数定律和中心极限定理。
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution t…
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of t…
In the first part of the paper we develop the sensitivity analysis for the nonlinear McKean-Vlasov diffusions stressing precise estimates of growth of solutions and their derivatives with respect to the initial data, under rather general assumptions on the coefficients. The exact estimates become particularly important when treating the extension of these equations having random coefficient, since the noise is usually assumed to be unbounded. The second part contains our main results dealing with the sensitivity of stochastic McKean-Vlasov diffusions. By using the method of stochastic characte…
Pathwise regularization by noise for semilinear SPDEs driven by a multiplicative cylindrical Brownian motion
本文研究乘性圆柱布朗运动驱动的半线性SPDE，通过无穷维推广、极大正则性分析和Volterra-sewing技术，证明了奇异扩散系数下的噪声正则化现象。
Nonzero-Sum Stochastic Differential Games for Controlled Convection-Diffusion SPDEs
本文研究了受控对流-扩散SPDE主导的双玩家非零和随机微分博弈，证明了正向和反向SPDE在Hilbert空间中的解存在性，利用哈密顿方法推导最大原理，并通过复合材料示例展示界面传输条件。
In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique $L^{2}$-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the $L^{2}$-bounded solution possesses the same recurrent properties (e.g. periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as…
Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness
针对时空分数阶SPDE，在局部Lipschitz系数满足线性增长等条件下，证明其存在唯一有限矩解，将Foondun等人2025年的工作推广到时空分数阶情形。
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving path properties of the solution to a parabolic Cauchy problem in evolution form.
We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) & = [-μV_{t}'(x) + \frac{1}{2} (σ_{M}^{2} + σ_{I}^{2})V_{t}"(x)]dt - σ_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian motion, has a unique solution in the class of finite-measure valued processes. The solution has a smooth density process which has a probabilistic representation and shows degeneracy near the absorbing boundary. In the language of weighted Sobolev spaces, we describe the precise order of integrability of …
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques …
The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with fully locally monotone coefficients covering a large variety of physical systems, a wide class of quasilinear SPDEs and a good number of fluid dynamic models is carried out in this work. The aim of this work is to develop the large deviation theory for small Gaussian as well as Poisson noise perturbations of the above class of SPDEs. We establish a Wentzell-Freidlin type large deviation principle for the strong solutions to such SPDEs perturbed by Lévy noise in a suitable Polish space using a variation…
We determine sufficient conditions on the volatility coefficient of Musiela's stochastic partial differential equation driven by an infinite dimensional L{é}vy process so that it admits a unique local mild solution in spaces of functions whose first derivative is square integrable with respect to a weight.
We prove a general theorem that the $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the $L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes L_ρ^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipsch…
We consider stochastic partial differential equations on $\mathbb{R}^{d}, d\geq 1$, driven by a Gaussian noise white in time and colored in space, for which the pathwise uniqueness holds. By using the Skorokhod representation theorem we establish various strong stability results. Then, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and uniqueness of solutions as well as the continu…
The purpose of this paper is twofold. Firstly, we prove transportation inequalities ${\bf T_2}(C)$ on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the ${\bf T_1}(C)$ property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic $2$-D Navier-Stokes equation.
This paper, is an attempt to extend the notion of stochastic viscosity solution to reflected semi-linear stochastic partial differential equations (RSPDEs, in short) with non-Lipschitz condition on the coefficients. Our method is fully probabilistic and use the recently developed theory on reflected backward doubly stochastic differential equations (RBDSDEs, in short). Among other, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula to reflected SPDEs, like one appear in \cite{2}. Indeed, in their recent work, Aman and Mrhardy \cite…
McKean-Vlasov SDE and SPDE with Locally Monotone Coefficients
本文在状态变量和分布变量均满足局部单调条件的弱假设下，证明了McKean-Vlasov随机（偏）微分方程的强、弱适定性，并推导了大偏差原理，从而可放宽颗粒介质、等离子体、Navier-Stokes等多类模型中的典型假设。
Optimal parameter estimation for linear SPDEs from multiple measurements
本文研究基于多次局部化测量的二阶抛物型线性随机偏微分方程参数估计，通过分析再生核Hilbert空间提出Gaussian下界方案，证明了收敛速度的minimax最优性及一致估计的充要条件。
On the well-posedness of SPDEs with locally Lipschitz coefficients
本文研究具有局部Lipschitz系数的随机偏微分方程的适定性，通过截断论证结合矩界和尾估计方法，在初始条件有界可测、系数至多线性增长假设下证明适定性，并推广至时间依赖系数和随机波动方程。
In Rajeev (2013), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one to one correspondence between solutions of a class of finite dimensional SDEs and solutions of a class of SPDEs in $\mathcal{S}'$, the space of tempered distributions, driven by the same Brownian motion. There the coefficients $\barσ, \bar{b}$ of the finite dimensional SDEs were related to the coefficients of the SPDEs in $\mathcal{S}'$ in a special way, viz. through convolution with the initial value $y$ of the SPDEs. In this paper, we consider the situation where the …
By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and semi-linear SPDEs.
We prove a modification to the classical maximal inequality for stochastic convolutions in 2-smooth Banach spaces using the factorization method. This permits to study semilinear stochastic partial differential equations with unbounded diffusion operators driven by cylindrical Brownian motion via the mild solution approach. In the case of finite dimensional driving noise, provided sufficient regularity on the coefficients, we establish existence and uniqueness of strong solutions.
Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x) +\bigl|u(t,x)\bigr|^γ\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<γ<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently …
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