Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A stochastic Schauder-Tychonoff type theorem and its applications

Erika Hausenblas, Ankit Kumar, Jonas M. Tölle

TL;DR

The paper develops a stochastic analogue of the Schauder–Tychonoff fixed-point theorem formulated on probability laws and proves the existence of probabilistic weak solutions for nonlinear SPDEs with non-Lipschitz perturbations. The main result identifies a fixed point of a law-map on an invariant, compact set under precise structural assumptions on the drift, diffusion, and nonlinearities, and yields a probabilistic weak solution on an extended probability space. The framework is validated through three applications to stochastic heat and porous medium-type equations, including cases with nonlinear gradient noise, by constructing appropriate energy bounds and using compactness arguments. The proof of the main theorem combines time discretization, tightness and Skorokhod representation, Simon-type compactness, and careful limit identifications of nonlinear terms, ensuring the limit satisfies the SPDE and the fixed-point property on laws. This approach provides a modular and flexible existence theory for broad classes of nonlinear SPDEs without relying on discretization-specific a priori properties.

Abstract

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations

A stochastic Schauder-Tychonoff type theorem and its applications

TL;DR

The paper develops a stochastic analogue of the Schauder–Tychonoff fixed-point theorem formulated on probability laws and proves the existence of probabilistic weak solutions for nonlinear SPDEs with non-Lipschitz perturbations. The main result identifies a fixed point of a law-map on an invariant, compact set under precise structural assumptions on the drift, diffusion, and nonlinearities, and yields a probabilistic weak solution on an extended probability space. The framework is validated through three applications to stochastic heat and porous medium-type equations, including cases with nonlinear gradient noise, by constructing appropriate energy bounds and using compactness arguments. The proof of the main theorem combines time discretization, tightness and Skorokhod representation, Simon-type compactness, and careful limit identifications of nonlinear terms, ensuring the limit satisfies the SPDE and the fixed-point property on laws. This approach provides a modular and flexible existence theory for broad classes of nonlinear SPDEs without relying on discretization-specific a priori properties.

Abstract

One standard way to prove existence for deterministic, highly nonlinear PDEs is to use the Schauder-Tychonoff fixed-point theorem. In what follows, we introduce and verify a stochastic variant of the Schauder-Tychonoff theorem. We apply our existence result to nonlinear stochastic diffusion equations with non-Lipschitz perturbations
Paper Structure (11 sections, 8 theorems, 81 equations)

This paper contains 11 sections, 8 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. The stochastic Schauder-Tychonoff type theorem
  3. Applications
  4. The stochastic heat equation with non-Lipschitz nonlinearity
  5. The stochastic porous media equation with non-Lipschitz nonlinearity
  6. The stochastic porous media equation driven by nonlinear gradient noise
  7. Proof of the stochastic Schauder-Tychonoff theorem \ref{['ther_main']}
  8. Extension of a probability space
  9. $L^p$-spaces and the Haar system
  10. Definition of the shifted Haar projection
  11. Lévy--Ciesielski construction

Key Result

Theorem 2.5

Fix $m\ge 2$. Let $U$ be a separable Hilbert space, $Q: U\to U$ such that $Q$ is linear, symmetric, nonnegative definite, and of trace class, let $H$ be another Hilbert space, and let us assume that we have a compact and dense embedding $\mathbb{X}_1\hookrightarrow\mathbb{X}$. Suppose that for any f Define the operator where $w$ is a solution to the system spdes. Now, we assume that the operator

Theorems & Definitions (23)

  • Definition 2.1: Probabilistic strong solution
  • Definition 2.2: Probabilistic weak solution
  • Remark 2.4
  • Theorem 2.5
  • Corollary 3.1
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.5
  • proof
  • ...and 13 more