Zhen-Qing Chen, Kyeong-Hun Kim
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove a Stroock-Varadhan's type support theorem for a stochastic partial differential equation (SPDE) on the real line with a noise term driven by a cylindrical Wiener process on $L_2 (\mathbb{R})$. The main ingredients of the proof are V. Mackevicius's approach to support theorem for diffusion processes and N.V. Krylov's $L_p$-theory of SPDEs.
This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical approximation processes.
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.
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.
An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.
A General Framework for Solving Singular SPDEs with Applications to Fluid Models Driven by Pseudo-differential Noise
针对漂移与噪声系数均含奇异性、Gelfand 三元组方法失效的非线性 SPDE，提出 proper regularization 一般框架，并统一证明多类伪微分噪声驱动流体模型解的局部与全局存在性。
In this paper we study a class of stochastic partial differential equations in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1$, driven by a Gaussian noise white in time and correlated in space. The differential operator is a fractional derivative operator. We show the existence, uniqueness and Hölder's regularity of the solution. Then by means of Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure.
We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.
In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by Lé{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs under consideration are random functions depending on time and space variables.
Nonlinear SPDE driven by Levy noise: Well-posedness, optimal control and invariant measure
针对乘性Lévy噪声驱动的非线性SPDE，论文结合紧性、Skorokhod表示与压缩法证明强解唯一存在，并通过变分法获得最优控制，同时证明无控制方程的不变测度存在。
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。
By a coupling method, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include $α$-stable process with $α\in (0,2)$
Fokker-Planck equation for McKean-Vlasov SPDEs driven by time-space Brownian sheet
本文研究由时空布朗单驱动的McKean-Vlasov型SPDE，证明混沌传播、非线性方程解的存在唯一性，并建立其解律的Fokker-Planck方程。
Blow-up estimates for a system of semilinear SPDEs driven by mixed fractional Brownian motions
本文针对混合布朗运动驱动的半线性随机偏微分方程组，证明了解的存在性与有限时间爆破，给出了爆破解的上下界、全局弱解存在的充分条件，并利用Malliavin微积分估计了爆破概率的下界。
Ergodicity for SPDEs driven by divergence-free transport noise
本文研究了由分歧自由传输噪声驱动的SPDEs的遍历性，证明强混合噪声可确保唯一不变测度，即使确定性部分存在多个稳态。
In this paper, the strong averaging principle is researched for a class of Hölder continuous drift slow-fast SPDEs with $α$-stable process by the Zvonkin's transformation and the classical Khasminkii's time discretization method. As applications, an example is also provided to explain our result.
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $α$-stable noise.
We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.
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.
We mainly investigate the log-Harnack inequality for the reflected stochastic partial differential equation driven by multiplicative noises based on the gradient estimate of the associated Markov semigroup. To do it, the penalization method and the comparison principle for stochastic partial differential equations are adopted. As its applications, we also establish the entropy-cost inequality and study some important estimates of the transition density relative to its invariant measure.
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective d…
In this paper, we first study the well-posedness of a class of McKean-Vlasov stochastic partial differential equations driven by cylindrical $α$-stable process, where $α\in(1,2)$. Then by the method of the Khasminskii's time discretization, we prove the averaging principle of a class of multiscale McKean-Vlasov stochastic partial differential equations driven by cylindrical $α$-stable processes. Meanwhile, we obtain a specific strong convergence rate.
We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup arguments then yields the existence of a continuous strong solution.
In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying first the nonlocal SPDEs and showing then the convergence of the family of these equations and the limit gives the solution to the SPDE.
Yamada-Watanabe uniqueness results for SPDEs driven by Wiener and pure jump processes
本文在变分框架下利用 Kurtz 结果，将 Yamada-Watanabe 理论推广到由 Wiener 过程与 Poisson 随机测度共同驱动的非线性 SPDE，证明鞅解存在加轨道唯一性可推出唯一强解存在。
We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space-time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space-time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition…
Some refinements of existence results for SPDEs driven by Wiener processes and Poisson random measures
本文改进了由Wiener过程和Poisson随机测度驱动的SPDEs存在性结果，通过移动框架方法将SPDE问题转化为SDE问题。
In this paper, we study a class of slow-fast stochastic partial differential equations with multiplicative Wiener noise. Under some appropriate conditions, we prove the slow component converges to the solution of the corresponding averaged equation with optimal orders 1/2 and 1 in the strong and weak sense respectively. The main technique is based on the Poisson equation.
We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.
We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a small ball probability of $Z_t$, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs.
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 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…
Stochastic Calculus in Infinite Dimensions and SPDEs
本文严格构造了柱形布朗运动驱动的Hilbert空间值过程的随机积分，并扩展至无限维随机微分方程（特别是Stratonovich方程和无界噪声算子）及非线性随机偏微分方程的存在性理论。
The (Ξ, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the Ξ-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.
We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. We prove that up to an error term bounded by a power of $\varepsilon$ depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so called amplitude equation, and in second order by a fast infinite dimensional Ornstein-Uh…
We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.
We develop a solution theory in Hölder spaces for a quasilinear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE Lemmas which establish a priori Hölder bounds for an equation with irregular right hand side written in divergence form. We apply these deterministic bounds to the case of a noise term which is white in time and trace class in space to obtain stretched exponential bounds for the Hölder semi-norms of the solution for the stochastic equation.
In this paper, the averaging principle is studied for a class of multiscale stochastic partial differential equations driven by $α$-stable process, where $α\in(1,2)$. Using the technique of Poisson equation, the orders of strong and weak convergence are given $1-1/α$ and $1-r$ for any $r\in (0,1)$ respectively. The main results extend Wiener noise considered by Bréhier in [6] and Ge et al. in [17] to $α$-stable process, and the finite dimensional case considered by Sun et al. in [39] to the infinite dimensional case.
Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions
本文扩展了奇异随机偏微分方程的能量解理论，通过引入格尔福三元组和希尔伯特空间中的无限维分析，消除了傅里叶级数和混沌展开的需要，拓宽了能量解的应用范围。
In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the results obtained for nonlinear PDE of McKean-Vlasov type generated by stable-like processes in the previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.
In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of si…
We consider a general class of $L^2$-valued stochastic processes that arise primarily as solutions of parabolic SPDEs on p.c.f. fractals. Using a Kolmogorov-type continuity theorem, conditions are found under which these processes admit versions which are function-valued and jointly continuous in space and time, and the associated Hölder exponents are computed. We apply this theorem to the solutions of SPDEs in the theories of both da Prato--Zabczyk and Walsh. We conclude by discussing a version of the parabolic Anderson model on these fractals and demonstrate a weak form of intermittency.
We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control probl…
In this paper we solve a Lévy driven linear stochastic first order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita (1984, 1997).
In this article, we established a large deviation principle for invariant measures of solutions of stochastic partial differential equations with two reflecting walls driven by space-time white noise.
Effective dynamics of interfaces for nonlinear SPDEs driven by multiplicative white noise
本文研究乘性白噪声驱动的无穷维SPDE界面动力学，利用有限维SDE推导慢流形上的有效方程，验证SPDE与耦合系统在小随机扰动下的等价性，并在四类随机方程中展示应用。
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we…
Concentration estimates for SPDEs driven by fractional Brownian motion
本文针对柱形分数布朗运动驱动的慢时变SPDE，避开鞅方法，利用高斯过程上确界尾概率最优估计，得到Hurst指数H>1/4时解的样本路径浓度估计，还附带得到任意H∈(0,1)下一维分数SDE的浓度估计。
Irreducibility and ergodicity of SPDEs driven by pure jump noise
本文提出一种新方法，在温和且近乎必要充分的条件下，证明乘性纯跳跃噪声驱动的SPDEs/SDEs的不可约性，并应用于消除限制条件，进而获得多类奇异随机演化方程的遍历性。
Markov property and path regularity for the solutions to SPDEs driven by cylindrical-martingale valued measures
研究由柱形正交鞅值测度驱动的随机偏微分方程的马尔可夫性，证明在时间依赖系数下解具有马尔可夫性，在时间独立系数下解具有Feller性，并在C0-半群拟压缩时解存在càdlàg版本。
Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise
本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。
In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: $du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum_{k = 1}^m g^k (u(t, x)) dw^k (t).$ We prove the convergence of a Wong-Zakai type approximation scheme of the above equation in the space $ C^{θ} ([0, T], H^γ_p (\mathbb{R}^d)) $ in probability, for some $ θ\in (0,1/2), γ\in (1, 2)$, and $p > 2$. We also prove a Stroock-Varadhan's type support theorem. To prove the results we combine V. Mackevicius ideas from his papers on Wong-Zakai theorem and the support theorem for…
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's le…
This is a review article which presents part of the contribution of Sergio Albeverio to the study of existence and uniqueness of solutions of SPDEs driven by jump processes and their stability properties. The results on stability properties obtained in Albeverio et al. [1] are presented in a slightly simplified and different way.
We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itô's backward integral. For this we use the technique developed by R.Buckdahn to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce …
Consider a random process s solution of the stochastic partial differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w so that the rescaled versions of s converges in law to a self-similar process of order H at coarse scales and at fine scales. The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor indices associated to the Lévy white noise w. Finally, we apply ou…
In this paper, we address the question of the discretization of Stochastic Partial Differential Equations (SPDE's) for excitable media. Working with SPDE's driven by colored noise, we consider a numerical scheme based on finite differences in time (Euler-Maruyama) and finite elements in space. Motivated by biological considerations, we study numerically the emergence of reentrant patterns in excitable systems such as the Barkley or Mitchell-Schaeffer models.
We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal mono…
We give a short proof of Itô's formula for stochastic Hilbert-space valued processes in the setting $V\subset H\subset V^{*}$ based on the possibility to lift the stochastic differentials, which are originally in $V^{*}$, into $H$. Using this result we also prove the maximum principle for second-order SPDEs in arbitrary domains.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。