Zhen-Qing Chen, Kyeong-Hun Kim
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.
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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 prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift in divergence form involving maximal monotone graphs defined everywhere, on which neither growth nor continuity assumptions are imposed.
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.
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.
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.
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。
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.
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, an…
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.
We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} γ(\nabla \cdot)$, where $γ$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $γ$ is single-valued, a solution with better integrability and regularity prop…
We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein--Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the inte…
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。
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 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.
The asymptotic behavior of a class of stochastic reaction-diffusion-advection equations in the plane is studied. We show that as the divergence-free advection term becomes larger and larger, the solutions of such equations converge to the solution of a suitable stochastic PDE defined on the graph associated with the Hamiltonian. Firstly, we deal with the case that the stochastic perturbation is given by a singular spatially homogeneous Wiener process taking values in the space of Schwartz distributions. As in previous works, we assume here that the derivative of the period of the motion on the…
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.
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 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.
This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical pr…
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.
Our first result is a stochastic sewing lemma with quantitative estimates for mild incremental processes, with which we study SPDEs driven by fractional Brownian motions in a random environment. We obtain uniform $L^p$-bounds. Our second result is a fractional averaging principle admitting non-stationary fast environments. As an application, we prove a fractional averaging principle for SPDEs.
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.
We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is Hölder-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to …
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。
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.
We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model finds a connection between all known definitions of CARMA random fields, and especially for dimension 1 we obtain the classical CARMA process.
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 …
SPDE for stochastic SIR epidemic models with infection-age dependent infectivity
本研究为具有感染年龄依赖性传染力的随机SIR模型建立了FLLN和FCLT，分别得到PDE和SPDE极限，并显式刻画解及在更强假设下证明SPDE解的唯一性。
Early-Warning Signs for SPDEs with Continuous Spectrum
本文研究连续谱随机偏微分方程的早期预警信号，通过方差标度律刻画分岔阈值逼近，在一维域获精确发散速率，在高维域证得上界，并经数值验证。
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.
SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula
本文研究希尔伯特空间中标准对称$\alpha$-稳定柱形噪声驱动的随机偏微分方程，证明温和解存在性、给出解的长期正则性与Itô公式，针对柱形噪声特性发展了新的论证方法。
Strong averaging principle for nonautonomous slow-fast SPDEs driven by $α$-stable processes
本文针对由$\alpha$-稳定过程驱动的非自治慢快随机偏微分方程，通过演化测度系统建立平均原理，证明慢分量在$L^p$意义下强收敛，并处理了缺乏有限二阶矩的技术难题。
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 prove Itô's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
In this paper we further study the stochastic partial differential equation first proposed by Xiong (2013). Under localized conditions on the coefficients we show that the solution is in fact distribution-function-valued and we establish the pathwise uniqueness of the solution. As applications we obtain the well-posedness of the martingale problems for two classes of measure-valued diffusions: interacting super-Brownian motions and interacting Fleming-Viot processes. Properties of the two superprocesses such as the existence of density fields and the extinction behaviors are also studied.
We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.
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.
A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The li…
In this article, we give a new proof of the Itô formula for some integral processes related to the space-time Lévy white noise introduced in Balan (2015) as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the $p$-th moment of the stochastic integral, and the Itô representation theorem leading to a chaos expansion similar to the Gaussian case.
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。
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。
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…
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.
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, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
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…
In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^α_tu=L(ω,t,x)u+f(u)+\partial^β_t \sum_{k=1}^{\infty}\int^t_0 ( Λ^k(ω,t,x)u+g^k(u))dw^k_t, $$ where $α\in (0,2)$, $β<α+\frac{1}{2}$, and $\partial^α_t$ and $\partial^β_t$ denote the Caputo derivative of order $α$ and $β$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $Ω$, $L$ is a second order operator of either divergence or non-div…
A pathwise approach to semilinear SPDEs with Lévy drivers
本文提出一种路径方法，将鲁棒粘性解推广到 Lévy 驱动的半线性粗糙偏微分方程，建立适定性与流性质，并给出基于粗糙 BSDE 的随机表示，为相应 SPDE 提供路径解释。
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.
In this paper, we extend Walsh's stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang's one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equati…
In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment estimates, and thus allows for a simple treatment of the Malliavin operators. The main part of the theory is developed for general Poisson random measures, defined on a $σ$-finite measure space, with minimal conditions. The theory is shown to apply to a space-time setting, suitable for studying stochastic partial differential equations. As an application, we analyze the weak order of convergence of spac…
This work considers weak approximations of stochastic partial differential equations (SPDEs) driven by Lévy noise. The SPDEs at hand are parabolic with additive noise processes. A weak-convergence rate for the corresponding Galerkin approximation is derived. The convergence result is derived by use of the Malliavin derivative rather then the common approach via the Kolmogorov backward equation.
We study linear stochastic partial differential equations of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution either over time and probabilistic space for forward SPDEs and over time for backward SPDEs. Well-posedness, existence, uniqueness, and a regularity of the solution for this new problem are obtained. In particular, this can be considered as a possibility to recover a solution of a forward SPDE in a setting where its values at the initial time are unknown, and where the average of the solution over time…
Linear filtering problem for infinite-dimensional Gaussian processes is studied, the observation process being finite-dimensional. Integral equations for the filter and for covariance of the error are derived. General results are applied to linear SPDEs driven by Gauss-Volterra process observed at finitely many points of the domain.
We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Lévy processes in this context.
Higher order approximation of nonlinear SPDEs with additive space-time white noise
本文研究了由加性时空白噪声驱动的1+1维随机偏微分方程的强近似，证明了基于随机卷积采样而非基础Wiener过程增量的方案能实现接近1的时序收敛率，显著优于传统1/4的最优速率。
We present an Itô formula for the $L_p$-norm of jump processes having stochastic differentials in $L_p$-spaces. The main results extend well-known theorems of Krylov to the case of processes with jumps, and which can be used to prove existence and uniqueness theorems in $L_p$-spaces for SPDEs driven by Lévy processes.
Brownian-time Change of measure
本文证明了布朗时间布朗运动及其相关过程的测度变换定理，为分析由布朗时间噪声驱动的SDEs和SPDEs行为提供了关键基础。
Nonlinear Filtering and Spatial Asymptotic Consistency for SPDEs Observed via Spatio-Temporal Point Processes
本文发展了用于生物物理应用的滤波框架，其中数据来自共聚焦激光扫描显微镜记录的细胞内生物物理量时空动态。信号由随机偏微分方程描述，观测可建模为标记点过程的功能，其强度依赖于底层信号。研究推导了未归一化和归一化滤波方程，展示了渐近一致性和有限维观测方案的近似。
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