An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
type: Item
instance of: scholarly article
title: An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
zbMATH Open document ID: 1327.60122
DOI: 10.4171/JEMS/545
author: Viorel Barbu
author: Michael Röckner
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2015-09-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.4940
review text: The aim of the paper is to study the existence and uniqueness of the solution to a nonlinear monotone stochastic differential equation \[ dX(t)+ A(t)X(t)dt = X(t)dW(t) \] with linear multiplicative noise in a real separable Hilbert space. The approach is to introduce the rescaling transform \(X(t)= e^{W(t)}y(t)\) and to reduce the above equation to the random differential equation \[ \frac{dy}{dt}(t) + e^{-W(t)}A(t)(e^{W(t)}y(t)) +\mu y(t)=0,\qquad t\in [0,T], \] which is treated as an operational equation in a convenient Hilbert space and can be rewritten as a monotone-type random equation in an appropriate space of infinite-dimensional stochastic processes on \([0,T]\). When the nonlinear operator is the subgradient of a convex function, the problem reduces to a convex optimization problem. One of the advantages of the approach adopted in this paper is that it gives sharp pointwise estimates and new pathwise regularity for the solutions.
reviewed by: Ruhollah Jahanipur
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 6483981
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: linear multiplicative noise
zbMATH Keywords: maximal monotone operator
zbMATH Keywords: stochastic integral
zbMATH Keywords: operatorial equations
MaRDI profile type: Publication
arXiv ID: 1402.4940
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OpenAlex ID: W1807107269
description: scientific article