Stochastic evolution equations in Hilbert spaces
type: Item
instance of: scholarly article
title: Stochastic evolution equations in Hilbert spaces
zbMATH Open document ID: 0836.60070
DOI: 10.1007/BF01187902
published in: Applied Mathematics and Optimization
publication date: 1996-04-09T00:00:00Z
review text: Let \(H\) be a separable Hilbert space, \(A\) a single-valued maximal monotone operator on \(H\) and \(W\) a Wiener process in a separable Hilbert space \(K\) with nuclear covariance operator. Consider the stochastic differential equation \[ dx(t) + Ax(t)dt = Bx(t) dw(t), \quad x(0) = x \in H, \tag{1} \] and a time discretization scheme of it, \[ x_k - x_{k - 1} + \Delta t Ax_k = Bx_{k - 1} \bigl( w(k \Delta t) \bigr) - w \bigl( (k - 1) \Delta t \bigr). \tag{2} \] The existence of a solution \(x(t)\) to (1) is established by using the theory of monotone operators on the Hilbert space \(H\). Moreover, the weak convergence of the sequence defined by (2) to \(x(t)\), in an appropriate state space, is proved. Approximation theorems are given. Also, regularity results of solutions of (1) for the operator \(A\) being the subdifferential of a lower semicontinuous convex function on \(H\) are obtained. Stochastic evolution equations under Gelfand triple and hyperbolic stochastic evolution equations are considered. Examples of applications of these results are given.
reviewed by: Jacek Jakubowski
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 806343
zbMATH Keywords: stochastic evolution equations
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: discretization scheme
zbMATH Keywords: weak convergence
zbMATH Keywords: Gelfand triple
zbMATH Keywords: hyperbolic stochastic evolution equations
author: Kazufumi Ito
MaRDI profile type: Publication
cites work: Approximation of some stochastic differential equations by the splitting up method
cites work: Generation of Semi-Groups of Nonlinear Transformations on General Banach Spaces
cites work: Nonlinear semigroups in Hilbert space
cites work: Stochastic evolution equations
cites work: Q4189698
cites work: Q3207428
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description: scientific article; zbMATH DE number 806343