Generalized solutions of a stochastic partial differential equation
type: Item
instance of: scholarly article
title: Generalized solutions of a stochastic partial differential equation
zbMATH Open document ID: 0802.60056
DOI: 10.1007/BF02214271
author: Hiroshi Kunita
published in: Journal of Theoretical Probability
publication date: 1994-12-19T00:00:00Z
review text: The author considers the Cauchy problem of a certain stochastic parabolic partial differential equation, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The existence and uniqueness theorem for the generalized (distributional) solution is proved and an analogue of the Feynman-Kac formula is obtained. Applying the partial Malliavin calculus of Kusuoka- Stroock, the author shows that any generalized solution is a \(C^ \infty\)-function under a condition similar to Hörmander's hypoellipticity condition.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 566135
zbMATH Keywords: Cauchy problem
zbMATH Keywords: stochastic parabolic partial differential equation
zbMATH Keywords: nonhomogeneous noise
zbMATH Keywords: Schwartz distributions
zbMATH Keywords: analogue of the Feynman-Kac formula
zbMATH Keywords: partial Malliavin calculus
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
Wikidata QID: Q115392707
cites work: Q4073917
cites work: Diffusions conditionnelles. I. Hypoellipticité partielle
cites work: Hypoelliptic second order differential equations
cites work: Q3992729
cites work: Stochastic partial differential equations and diffusion processes
cites work: Stochastic Integrals Based on Martingales Taking Values in Hilbert Space
cites work: Q3995203
cites work: Stochastic flows acting on Schwartz distributions
cites work: Q3684932
cites work: The partial malliavin calculus and its application to non-linear filtering
cites work: Stochastic partial differential equations and filtering of diffusion processes
description: scientific article; zbMATH DE number 566135