Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Stefan Tappe

TL;DR

This paper establishes a dual Yamada–Watanabe theorem for mild solutions to semilinear SPDEs with path-dependent coefficients by employing the method of moving frame to lift the problem to an infinite-dimensional SDE on a dilated Hilbert space $\mathscr{H}$. It analyzes the connections between SPDE mild solutions and SDE solutions via the dilation, projections, and a lift mapping $F \in \hat{\mathscr{E}}(H)$, enabling a unified treatment of uniqueness notions. The main results show that, under joint uniqueness in law for a given initial distribution, pathwise uniqueness follows, and under delta-uniqueness in law plus the group extension, delta-pathwise uniqueness holds; a Yamada–Watanabe-type equivalence for mild solutions is derived. The framework encompasses cylindrical and trace-class noise and provides a versatile tool for proving existence and uniqueness results for a broad class of SPDEs with path-dependent coefficients.

Abstract

We provide the dual result of the Yamada-Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called "method of the moving frame", which allows us to reduce the proof to infinite dimensional stochastic differential equations.

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

TL;DR

This paper establishes a dual Yamada–Watanabe theorem for mild solutions to semilinear SPDEs with path-dependent coefficients by employing the method of moving frame to lift the problem to an infinite-dimensional SDE on a dilated Hilbert space . It analyzes the connections between SPDE mild solutions and SDE solutions via the dilation, projections, and a lift mapping , enabling a unified treatment of uniqueness notions. The main results show that, under joint uniqueness in law for a given initial distribution, pathwise uniqueness follows, and under delta-uniqueness in law plus the group extension, delta-pathwise uniqueness holds; a Yamada–Watanabe-type equivalence for mild solutions is derived. The framework encompasses cylindrical and trace-class noise and provides a versatile tool for proving existence and uniqueness results for a broad class of SPDEs with path-dependent coefficients.

Abstract

We provide the dual result of the Yamada-Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called "method of the moving frame", which allows us to reduce the proof to infinite dimensional stochastic differential equations.

Paper Structure

This paper contains 5 sections, 22 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Framework and definitions
  3. Infinite dimensional stochastic differential equations
  4. Proofs of the main results
  5. Examples

Key Result

Theorem 1.1

Let $\mu$ be a probability measure on $(H,\mathscr{B}(H))$. Suppose there exists a mild solution $(X,W)$ to the SPDE (SPDE) such that $\mu$ is the distribution of $X(0)$, and that joint uniqueness in law given $\mu$ holds for (SPDE). Then pathwise uniqueness given $\mu$ holds for (SPDE) as well.

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.2: Martingale solution
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 49 more