The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations

Stefan Tappe

doi:10.1214/ecp.v18-2392

Abstract

We prove the Yamada-Watanabe theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called method of the moving frame allows us to reduce the proof to the Yamada-Watanabe theorem for stochastic differential equations in infinite dimensions.

Highlights

The goal of the present paper is to establish the Yamada-Watanabe Theorem – which originates from the paper [17] – for mild solutions to semilinear stochastic partial differential equations (SPDEs) dX(t) = (AX(t) + α(t, X))dt + σ(t, X)dW (t) (1.1)in the spirit of [2, 12, 6] with path-dependent coefficients
We prove the Yamada-Watanabe Theorem for semilinear stochastic partial differential equations with path-dependent coefficients
The so-called “method of the moving frame” allows us to reduce the proof to the Yamada-Watanabe Theorem for stochastic differential equations in infinite dimensions

Summary

Introduction

The goal of the present paper is to establish the Yamada-Watanabe Theorem – which originates from the paper [17] – for mild solutions to semilinear stochastic partial differential equations (SPDEs) dX(t) = (AX(t) + α(t, X))dt + σ(t, X)dW (t). We show that we can reduce the proof to Hilbert space valued SDEs dYt = α(t, Y )dt + σ(t, Y )dWt. We will divide the proof of Theorem 1.1 into two steps: 1. We show that we can reduce the proof to Hilbert space valued SDEs dYt = α(t, Y )dt + σ(t, Y )dWt This is due to the “method of the moving frame”, which has been presented in [5], see [16]. The remainder of this paper is organized as follows: In Section 2 we present the general framework, in Section 3 we provide the proof of Theorem 1.1, and in Section 4 we show an example illustrating Theorem 1.1

Framework and definitions

We have up to indistinguishability

An example