On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

On processes of ornstein-uhlenbeck type in hilbert space

A process fo Ornstein-Uhlenbeck type is a mild solution of the stochastic differential system in Hilbert space dXt=AX _t dt+dZ _t, where A generates a semigroup of operators and Z _tis a process with homogeneous independent increments. The explicit integral formula for the process of O-U type is given. The main purpose is to study stationary distributions for such processes. Sufficient and necessary conditions for existence and characterization are given. The difference between finite and infinite dimensional cases is illustrated by examples     

Kolmogorov equations for measures

We consider a semigroup of operators in the Banach space C _b (H) of uniformly continuous and bounded functions on a separable Hilbert space H. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to the transition semigroup and the Kolmogorov operator corresponding to a stochastic PDE in H. For this purpose, we characterize the generator of the transition semigroup on a core.      

Continuity properties of Hilbert space valued martingales

Necessary and sufficient conditions for Hölder continuity of Hilbert space valued martingales are given in terms of the associated quadratic variation. As an application one obtains a sufficient condition for a mild solution of a stochastic evolution equation to have a continuous version if the semigroup governing this equation is analytic. Further we derive Levy's modulus of continuity for the Hilbert space valued stochastic integral with the Wiener process as integrator and obtain a generalization of the loglog law for that integral.     

Stochastic bilinear spectral systems †

A class of bilinear stochastic partial differential equations is investigated using a semigroup approach. Existence of a mild solution is obtained by proving a maximal inequality for stochastic convolution integrals with a stochastic evolution operator U(t,s) as integrand; moreover, we show the existence of a regular version in t. Under an additional assumption we show the existence of a continuous version of U (.,.) in the space of bounded operators on the state space. Finally, we analyse a p.d.e. model of a simply supported beam to illustrate the applicability of our results to modelling uncertainty in large flexible space structures     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.     

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C ₀ semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.      

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.     

Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities

We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {P_t, t ≥ 0}, acting on the space of bounded measurable functions from H to , transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given by the semigroup {P_t, t ≥ 0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions to Hamilton - Jacobi - Bellman equations arising in control theory. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

Abstract

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ε > 0. As ε vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ε ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L ²(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.     

On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (Stochastic Partial Differential Equations and Applications, Series of Lecture Notes in Pure and Appl. Math., vol. 245, pp. 253–260, Chapman and Hall, London, 2006) and Goreac (Applied Analysis and Differential Equations, pp. 153–164, World Scientific, Singapore, 2007) from the finite dimensional to the infinite dimensional case.     

一类时滞方程的谱与解展开

该文研究一类时滞方程解的展开问题. 研究的模型来自于实际高精密切割过程中具有时间延迟的机床振动问题. 对此模型,借助于泛函分析方法,将其写成抽象发展方程. 对系统确定的算子给出了较细致的谱分析,得到本征值的渐近表达式.同时证明相应的本征向量不能构成状态空间基, 但给出方程解的展开式.     

On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

Stability Conditions for Abstract Functional Differential Equations in Hilbert Space

Abstract. Stability conditions for functional differential equations of the form: du (t)/ dt = Au(t)+ bAu(t-h)+(a^\ast Au)(t) are studied, where A is the infinitesimal generator of an analytic semigroup in a Hilbert space, b\neq 0 and the convolution term contains a square integrable real function a\neq 0 . Norm discontinuity of the solution semigroup of the equation with discrete delay is avoided by studying the inverse of the characteristic operator. Sufficient and necessary conditions for the uniform exponential stability of the solution semigroup are obtained. The results are applied to a retarded partial integrodifferential equation.     

Topological properties of solution sets for stochastic evolution inclusions

In this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and an R_δ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.     

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.

     

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.     

Logarithmic derivatives of invariant measure for stochastic differential equations in hilbert spaces

We consider a process X solution of a semilinear stochastic evolution equation in a Hilbert space. Assuming that X has an invariant measure ν, we investigate its regularity properties. Logarithmic derivatives of ν in certain directions, are shown to exist under appropriate conditions on the nonlinear term in the equation. A set of directions of differentiability for ν is explicitly described in terms of the coefficients of the equation. In some cases, logarithmic derivatives are represented as conditional expectations of random variables related to an appropriate stationary process. An application to a system of stochastic partial differential equations in one space variable is given