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Invariant measures for stochastic Burgers equation in unbounded domains with space-time white noise

Zhenxin Liu, Zhiyuan Shi

TL;DR

The paper analyzes the stochastic damped Burgers equation on the real line driven by multiplicative space-time white noise. It develops a well-posedness theory in weighted spaces $L^2_{\rho}$, proves boundedness in probability, and establishes the existence of invariant measures via the Krylov-Bogoliubov framework. Core methods include fixed-point arguments for local solutions, a Gronwall-based global extension leveraging the damping term, stochastic convolution bounds, and Ornstein-Uhlenbeck regularization to obtain tightness. The results extend invariant-measure theory to SPDEs on unbounded domains with rough noise, providing a rigorous basis for long-time behavior in Burgers-type/damped models with spatially homogeneous noise.

Abstract

In this paper, we investigate the stochastic damped Burgers equation with multiplicative space-time white noise defined on the entire real line. We prove the existence and uniqueness of a mild solution of the stochastic damped Burgers equation in the weighted space and establish that the solution is bounded in probability. Furthermore, by using the Krylov-Bogolioubov theorem, we obtain the existence of invariant measures.

Invariant measures for stochastic Burgers equation in unbounded domains with space-time white noise

TL;DR

The paper analyzes the stochastic damped Burgers equation on the real line driven by multiplicative space-time white noise. It develops a well-posedness theory in weighted spaces , proves boundedness in probability, and establishes the existence of invariant measures via the Krylov-Bogoliubov framework. Core methods include fixed-point arguments for local solutions, a Gronwall-based global extension leveraging the damping term, stochastic convolution bounds, and Ornstein-Uhlenbeck regularization to obtain tightness. The results extend invariant-measure theory to SPDEs on unbounded domains with rough noise, providing a rigorous basis for long-time behavior in Burgers-type/damped models with spatially homogeneous noise.

Abstract

In this paper, we investigate the stochastic damped Burgers equation with multiplicative space-time white noise defined on the entire real line. We prove the existence and uniqueness of a mild solution of the stochastic damped Burgers equation in the weighted space and establish that the solution is bounded in probability. Furthermore, by using the Krylov-Bogolioubov theorem, we obtain the existence of invariant measures.
Paper Structure (10 sections, 16 theorems, 114 equations)

This paper contains 10 sections, 16 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Preliminaries
  4. Main results
  5. Solution: existence and uniqueness
  6. Local existence and uniqueness
  7. Global existence and uniqueness.
  8. Existence of invariant measures
  9. Boundedness in probability
  10. The existence of invariant measures

Key Result

Theorem 2.2

Assume that $(A1)$ holds and $k\geq\frac{C^*}{3}+\delta$. If the initial condition $u_{0}$ belongs to $L^2_{\rho}({\mathbb R})$, then there is a unique mild solution $u(t,x)$ to equation 2.1. Moreover, for any $T>0$, there exists a constant $C(T)>0$ such that for $p\geq 2$

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 23 more