Anna Karczewska, Carlos Lizama
In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of $α$-times resolvent families.
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On the Analysis of a Singular Stochastic Volterra Differential Equation driven by a Wiener Noise
本文研究了由奇异漂移向量场和Wiener噪声驱动的随机Volterra微分方程的强解唯一性及初始值对强解的Sobolev可微性。
This paper is devoted to a construction of the stochastic Itô integral with respect to infinite dimensional cylindrical Wiener process. The construction given is an alternative one to that introduced by DaPrato and Zabczyk [3]. The connection of the introduced integral with the integral defined by Walsh [9] is provided as well.
Following the ideas of F. Russo and P. Vallois we use the notion of forward integral to introduce a new stochastic integral respect to the cylindrical Winer process. This integral is an extension of the classical integral. As an application, we prove existence of solution of a parabolic stochastic differential partial equation with anticipating stochastic initial date.
In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach spa…
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.
We consider a class of one-dimensional nonlinear stochastic parabolic problems associated with Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and …
In the paper stochastic Volterra equations with noise terms driven by series of independent scalar Wiener processes are considered. In our study we use the resolvent approach to the equations under consideration. We give sufficient condition for the existence of strong solution to the class of stochastic Volterra equations of convolution type. We provide regularity of stochastic convolution, as well.
The aim of this note is to provide some results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolution of the form $W^Ψ(t):=\int_0^t S(t-τ)Ψ(τ)dW(τ)$, $t\geq 0$, where $S(t), t\geq 0$, is so-called {\em resolvent} for Volterra equation considered,$Ψ$ is an appropriate process and $W$ is a cylindrical Wiener process.
In this article we illustrate the relation between the existence of Wiener integrals with respect to a Levy process in a separable Banach space and radonifying operators. For this purpose, we introduce the class of theta-radonifying operators, i.e. operators which map a cylindrical measure theta to a genuine Radon measure. We study this class of operators for various examples of infinitely divisible cylindrical measures theta and highlight the differences from the Gaussian case.
We study Volterra processes $X_t = \int_0^t K(t,s) dW_s$, where $W$ is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \int_s^t b(u) c(u-s) du$. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$. We establish smoothness properties of $X$, including continuity and Holder property. It happens that its Holder smoothness is close to well-known Holder smoothness of fBm but is a bit worse. We give a comparison with fBm for any smoothness theorem. Then we investigate the problem of inverse representation of $W$ via $X$ in the…
Local times of self-intersection and sample path properties of Volterra Gaussian processes
本文研究由连续核与 Wiener 过程定义的 Volterra 高斯过程，建立一维重对数律、局部时存在性及其对核的连续依赖，并证明平面情形 Rosen 重整化自相交局部时存在。
Space-time regularity of linear stochastic partial differential equations is studied. The solution is defined in the mild sense in the state space $L^p$. The corresponding regularity is obtained by showing that the stochastic convolution integrals are Hölder continuous in a suitable function space. In particular cases, this allows to show space-time Hölder continuity of the solution. The main tool used is a hypercontractivity result on Banach-space valued random variables in a finite Wiener chaos.
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。
We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric $α$-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance operator. In particular, in the equation with diagonal pure $α$-stable cylindrical noise introd…
We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in…
In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Lévy processes in separable Banach spaces. Here, a cylindrical Lévy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornst…
Limits of stochastic Volterra equations driven by Gaussian noise
研究高斯噪声驱动的随机Volterra方程，推导温和形式，证明解的存在唯一性、稳定性及极限分布，并刻画其与初始状态无关的条件。
We consider the Navier-Stokes equation on the 2D torus, with a stochastic forcing term which is a cylindrical fractional Wiener noise of Hurst parameter $H$. Following [3,8] which dealt with the case $1/2$, we prove a local existence and uniqueness result when $7/16< H< 1/ 2$ and a global existence and uniqueness result when $ 1/2<H<1$.
Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times.…
This paper focuses on the time-changed Q-Wiener process, a Hilbert space-valued sub-diffusion. It is a martingale with respect to an appropriate filtration, hence a stochastic integral with respect to it is definable. For the resulting integral, two change of variables formulas are derived. Via a duality theorem for integrals, existence and uniqueness theorems for stochastic differential equations (SDEs) driven by the time-changed Q-Wiener process are discussed. Associated fractional Fokker-Planck-Kolmogorov equations are derived using either a time-changed Itô formula or duality. Connections …
We investigate the concept of cylindrical Wiener process subordinated to a strictly $α$-stable Lévy process, with $α\in\left(0,1\right)$, in an infinite dimensional, separable Hilbert space, and consider the related stochastic convolution. We then introduce the corresponding Ornstein-Uhlenbeck process, focusing on the regularizing properties of the Markov transition semigroup defined by it. In particular, we provide an explicit, original formula -- which is not of Bismut-Elworthy-Li's type -- for the Gateaux derivatives of the functions generated by the operators of the semigroup, as well as a…
A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges…
We study a class of stochastic evolution equations in a Banach space $E$ driven by cylindrical Wiener process. Three different concept of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay equation with additive noise. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic McKendrick equation with delay.
A class of stochastic delay equations in Banach space $E$ driven by cylindrical Wiener process is studied. We investigate two concepts of solutions: weak and generalised strong, and give conditions under which they are equivalent. We present an evolution equation approach in a Banach space $\Ep{p}:=E\times L^p(-1,0;E)$ proving that the solutions can be reformulated as $\Ep{p}$-valued Markov processes. Based on the Markovian representation we prove the existence and continuity of the solutions. The results are applied to stochastic delay partial differential equations with an application to neu…
Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction
研究具有相互作用的随机流中Volterra高斯过程的自交局部时，通过建立变元公式将过程x的自交局部时表示为过程u的加权自交局部时，并证明其存在性及大时间渐近性。
We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.
We study stochastic Korteweg - de Vries equation driven by Lévy noise consisting of the compensated time homogeneous Poisson random measure and a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. In proof of the existence theorem we use the Galerkin approximation and several auxiliary results suitable for the problem considered.
This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.
A Schauder-Tychonoff fixed-point approach for nonlinear Lévy driven reaction-diffusion systems
本文提出了一种基于Schauder-Tychonoff不动点定理的随机方法，用于解决由莱维过程驱动的非线性反应扩散系统的问题。
We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musiela's stochastic PDE.
We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder regularity $C^α$, $α>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having Hölder regularity $C^α$, $α>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.
We introduce the local martingale problem associated to semilinear stochastic evolution equations driven by a cylindrical Wiener process and establish a one-to-one correspondence between solutions of the martingale problem and (analytically) weak solutions of the stochastic equation. We also prove that the solutions of well-posed equations are strong Markov processes. We apply our results to semilinear stochastic equations with additive noise where the semilinear term is merely measurable and to stochastic reaction-diffusion equations with Hölder continuous multiplicative noise.
Martingale-driven integrals and singular SPDEs
本文针对逼近柱Wiener过程的càdlàg鞅建立多重随机积分的混沌展开与矩界，并结合正则结构离散化框架证明粒子系统向奇异SPDE收敛。
This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. …
We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $Σ$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on $Σ$.
In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity $C^α,α>1/2$ in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager's conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying Hölder regularity $C^α,α>1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.
Global Strong Well-Posedness of the stochastic bidomain equations with FitzHugh-Nagumo transport
研究受电流噪声影响的FitzHugh-Nagumo输运双域方程，证明其在临界空间中存在唯一全局强路径解，方法结合随机与确定性极大正则性及外推空间技术。
We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the form \begin{align*} \text{d}X_t=b(t,X)\text{d}t+σ(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*} and show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here $W$ is a cylindrical Wiener process in a separable Hilbert space $U$ and the equation is considered in a Gelfand triple $V \subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional) Hilbert space. This genera…
We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.
Stochastic Tamed Navier--Stokes Equations with Wiener and Jump Noise on $\mathbb R^3$. I. Maximal Local $L^p$-Well-Posedness
研究$\mathbb{R}^3$上由乘性Wiener噪声和补偿Lévy跳跃噪声驱动的随机驯化Navier-Stokes方程，证明了局部和全局强解的存在性。
In the paper, we consider a stochastic hybrid Korteweg - de Vries - Burgers type equation with multiplicative noise in the form of cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied. The proof of the existence of the solution is based on two approximations of the considered problem and the compactness method. First, we introduce an auxiliary problem corresponding to the equation studied. Then, we prove the existence of a martingale solution to this problem. Finally, we show that the solution of the auxiliary problem converges, in some sense, to …
Uniqueness results of a nonlinear stochastic diffusion-convection equation with reflection
本文研究带反射的非线性伪单调随机扩散演化问题解的唯一性，指出因缺乏一般 Itô 公式无法得到一般唯一性结果，但在解或反射具有更高正则性时可证明比较原理。
The primitive equations with stochastic wind driven boundary conditions
本文研究随机风驱动边界条件下地球物理流原始方程，提出结合随机与确定性方法的严格处理，证明在各向异性临界空间中存在唯一局部路径wise解。
The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$
本文研究了具有非局部算子的时间分数随机偏微分方程的适定性和正则性，结合调和分析与随机分析技术，解决了低正则性情况下方程的局部解问题。
We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distr…
We present an existence theory for martingale and strong solutions to doubly nonlinear evolution equations in a separable Hilbert space in the form $$d(Au) + Bu\,dt \ni F(u)\,dt + G(u)\,dW$$ where both $A$ and $B$ are maximal monotone operators, possibly multivalued, $F$ and $G$ are Lipschitz-continuous, and $W$ is a cylindrical Wiener process. Via regularization and passage-to-the-limit we show the existence of martingale solutions. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô's formula. If either $A$ or $B$ is linear and s…
We continue with the study of the mollified stochastic heat equation in $d\geq 3$ given by $d u_{ε,t}=\frac 12Δu_{ε,t}+ βε^{(d-2)/2} \,u_{ε,t} \,d B_{ε,t}$ with spatially smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac solution describes the partition function of the continuous directed polymer. In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending on the value of $β>0$ in the limiting object of the smoothened solution $u_ε$ as the smoothing parameter $ε\to 0$ This partition function naturally defines a quenched polymer path measure and we pr…
Strong order of convergence of a fully discrete approximation of a linear stochastic Volterra type evolution equation
针对带正型记忆项和加性高斯噪声的Hilbert空间值线性随机发展方程，采用隐式Euler时间离散、Laplace变换卷积求积和连续有限元空间离散，证明了全离散解在均方意义下关于时间和空间步长的收敛阶。
We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an applica…
Maximum principle for recursive optimal control problem of stochastic delay evolution equations
本文研究由柱面 $Q$-Wiener 过程驱动的随机时滞发展方程递归最优控制问题，通过引入预期倒向随机发展方程作为伴随方程并运用对偶分析，推导并证明了 Pontryagin 最大值原理及其充分性。
Maximum Principles for Partially Observed Controls of Forward SPDEs and Backward SDEs with Jumps
本文提出两种最大原理，用于部分观测控制的耦合前向和后向随机偏微分方程与随机微分方程，解决了具有跳跃的SPDE的适定性问题。
In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$α$ Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by $\sqrtα$. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as $α$ goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term o…
An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes
本文提出了一种包含跳跃扩散噪声的随机Allen-Cahn方程，用于建模生物损伤与修复过程，并分析了其数学性质与数值模拟方法。
We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these …
Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces
本文在Hilbert空间中研究一类随机微分方程，在比已有文献更弱的条件下建立了其转移半群的对数Harnack不等式，并给出了若干应用。
Large deviations for long-time occupation measures of stochastic evolution equations with small, asymptotically rough noise
研究可分希尔伯特空间中一类耗散随机演化方程长时间小噪声行为，建立占据测度大偏差原理，涵盖多种无穷维方程。通过解析半群等技术，结合弱收敛方法，给出速率函数显式公式并构建近最优控制。
We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers' equations and for three-dimensional stochastic Cahn-Hilliard type equations …
We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also pr…
In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ing…
Weak uniqueness for stochastic partial differential equations in Hilbert spaces
本文研究希尔伯特空间中随机微分方程的弱唯一性，在无 Hölder 连续性假设下，证明了任意维数热方程和阻尼方程的弱唯一性。
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