Stochastic heat equation with white-noise drift
type: Item
instance of: scholarly article
title: Stochastic heat equation with white-noise drift
zbMATH Open document ID: 0970.60068
DOI: 10.1016/S0246-0203(00)00122-9
author: Elisa Alòs
author: David Nualart
author: Frederi Viens
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2001-06-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77656
review text: The authors study the existence and uniqueness of solutions for a one-dimensional anticipative stochastic evolution equation on the real line \[ u(t,x)=\int_{\mathbb R} p(0,t,y,x)u_0(y) dy + \int_{\mathbb R} \int_0^t p(s,t,y,x)F(s,y,u(s,y)) dW_{s,y} \] driven by a two-parameter Wiener process \(W_{t,x}\) and based on a stochastic semigroup defined by the kernel \(p(s,t,y,x)\). This kernel is supposed to be measurable w.r.t. the increments of the Wiener process on \([s,t]\times\mathbb R\). The results are based on \(L^p\)-estimates for the Skorokhod integral. As an application they establish the existence of a weak solution for the following heat equation on the real line subject to white noise drift, \[ \partial_t u(t,x)=\partial_x^2 u(t,x)+\dot{v}(t,x)\partial_x u(t,x)+F(t,x,u)\partial_t\partial_x W(t,x), \] where \(\dot{v}\) is a white noise in time.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 1453271
zbMATH Keywords: anticipative stochastic evolution equation
zbMATH Keywords: Skorokhod integral
zbMATH Keywords: stochastic semigroup
zbMATH Keywords: white noise drift
zbMATH Keywords: real line
zbMATH Keywords: backward heat kernel
MaRDI profile type: Publication
OpenAlex ID: W1495174296
description: scientific article; zbMATH DE number 1453271