Measure theory and square-mean pseudo almost periodic and automorphic process: application to stochastic evolution equations
type: Item
instance of: scholarly article
title: Measure theory and square-mean pseudo almost periodic and automorphic process: application to stochastic evolution equations
zbMATH Open document ID: 1388.34057
author: Mamadou Abdoul Diop
author: Khalil Ezzinbi
author: Mamadou Moustapha Mbaye
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2018-02-07T00:00:00Z
review text: This paper deals with the existence and uniqueness of square-mean \(\mu\)-pseudo almost periodic and automorphic mild solution to the following nonlinear stochastic evolution equation in a Hilbert space \(H\): \[ dx(t)=Ax(t)dt+h(t,x(t))dt+\theta(t,x(t))dW(t) \text{ for all } t\in \mathbb{R}\quad (1) \] where \(A:D(A)\subset H\) is the infinitesimal generator of a \(C_0\)-semigroup \((T(t))_{t\geq 0}\) on \(H\), \(h,\theta: \,\mathbb{R}\times L^2(P,H)\to L^2(P,H)\) are two stochastic processes and \(W(t)\) is a two-side and standard one-dimensional Brownian motion defined on the filtered probability space \((\Omega,\mathcal{F},P,\mathcal{F}_t)\) with \(\mathcal{F}_t=\sigma\{W(u)-W(v)|\, u,v\leq t\}.\) The authors start by proving that the linear stochastic evolution equation associated to (1) has square-mean \(\mu\)-pseudo almost periodic and automorphic mild solution. Then by means of Banach fixed point theorem they obtain the same results for (1). The proof is achieved by means of a new concept of square-mean pseudo almost periodic and automorphic process using the measure theory. Actually, the spaces of \(\mu\)-pseudo almost periodic and automorphic processes, is defined by the authors by means of the \(\mu\)-ergodic process. Many interesting results like completeness and composition theorems in those space are also established. An example is also given to illustrate the results.
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 35B15
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 43A60
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 6835868
zbMATH Keywords: measure theory
zbMATH Keywords: ergodicity
zbMATH Keywords: pseudo almost periodic solution
zbMATH Keywords: pseudo almost automorphic solution
zbMATH Keywords: stochastic processes
zbMATH Keywords: stochastic evolution equations
reviewed by: Gisèle M. Mophou
MaRDI profile type: Publication
full work available at URL: s40840-015-0278-y
OpenAlex ID: W2286681786
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DOI: 10.1007/S40840-015-0278-Y
description: scientific article; zbMATH DE number 6835868