Regularity and convergence of stochastic convolutions in duals of nuclear Fréchet spaces
type: Item
instance of: scholarly article
title: Regularity and convergence of stochastic convolutions in duals of nuclear Fréchet spaces
zbMATH Open document ID: 0764.60056
DOI: 10.1016/0047-259X(92)90033-C
author: Constantin Tudor
author: Víctor Pérez-Abreu
published in: Journal of Multivariate Analysis
publication date: 1993-04-01T00:00:00Z
review text: \(\Phi\) is nuclear Fréchet space, \(\Phi'\) its dual, and \(D_ \infty(\Phi')\) the space (for an appropriate topology) of maps defined on \({\mathfrak R}_ +\), with values in \(\Phi'\), which are right-continuous, with left limits (càdlàg). \(M\) is a martingale with values in \(\Phi'\) and \(S(u,v)\) a family (depending on \((u,v)\)) of continuous linear operators which forms a reverse evolution system of contraction type. A stochastic convolution process is a process of the form \(Y_ t=\int_ 0^ t S'(t,u)dM_ u\), \(S'\) being the adjoint of \(S\). The paper provides results on three topics: weak convergence on \(D_ \infty(\Phi')\), càdlàg versions of \(Y\), convergence of \(Y^{(n)}\) to \(Y\) when \(M^{(n)}\) converges to \(M\). The technical requirements for the results to be true are too lengthy to be stated in a review.
reviewed by: A. F. Gualtierotti
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 148928
zbMATH Keywords: nuclear space
zbMATH Keywords: stochastic evolution equation
zbMATH Keywords: weak convergence
zbMATH Keywords: Fréchet space
zbMATH Keywords: stochastic convolution process
MaRDI profile type: Publication
cites work: Langevin equations for S'-valued Gaussian processes and fluctuation limits of infinite particle systems
cites work: Inhomogenous infinite dimensional langevin equations
cites work: Q3468396
cites work: Q3995021
cites work: Stability of a class of transformations of distribution-valued processes and stochastic evolution equations
cites work: On the Skorokhod topology
cites work: Q4698005
cites work: Infinite dimensional stochastic differential equation models for spatially distributed neurons
cites work: Stochastic evolution equations driven by nuclear-space-valued martingales
cites work: Q3471286
cites work: A submartingale type inequality with applicatinos to stochastic evolution equations
cites work: A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations
cites work: Law of large numbers and central limit theorem for linear chemical reactions with diffusion
cites work: Linear parabolic differential equations as limits of space-time jump Markov processes
cites work: MARTINGALES OF RANDOM DISTRIBUTIONS
cites work: Tightness of probabilities on C([0,1];\(S_ p\)) and D([0,1];\(S_ p\))
cites work: An \(\infty\)-dimensional inhomogeneous Langevin's equation
cites work: Stochastic evolution equations in locally convex space
description: scientific article; zbMATH DE number 148928