On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

David Criens,Moritz Ritter

doi:10.1007/s10959-021-01107-3

Abstract

AbstractWe consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

Highlights

The classical Yamada–Watanabe theorem [23] for finite dimensional Brownian stochastic differential equations (SDEs) states that weak existence and strong uniqueness implies strong existence and weak uniqueness
Jacod [9] lifted this result to SDEs driven by semimartingales and extended it by showing that weak existence and strong uniqueness is equivalent to strong existence
In [15,21] the theorems were established for mild solutions to semilinear stochastic partial differential equations (SPDEs) and in [17,18] for the variational framework

Summary

Introduction

The classical Yamada–Watanabe theorem [23] for finite dimensional Brownian stochastic differential equations (SDEs) states that weak existence and strong (i.e. pathwise) uniqueness implies strong existence and weak uniqueness (i.e. uniqueness in law). In [15,21] the theorems were established for mild solutions to semilinear stochastic partial differential equations (SPDEs) and in [17,18] for the variational framework In this short article we prove Cherny’s result for analytically weak solutions to the Banach space valued semilinear SPDE d Xt = ( A Xt + μt (X ))dt + σt (X )d Wt , X0 = x0,. We provide martingale characterizations for weak solutions to SPDEs and infinite dimensional Brownian motion, show that the quadratic variations of the corresponding test martingales vanish and deduce the desired independence with help of changes of measure. A detailed construction and standard properties of the stochastic integral can be found in [15]

The Setting and Main Results

A Some Facts for Stochastic Integrals