[Paper Review] Pathwise stationary solutions of stochastic Burgers equations with $L^2[0,1]$-noise and stochastic Burgers integral equations on infinite horizon

Yong Liu, Huaizhong Zhao|ArXiv.org|Sep 13, 2006

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper establishes the existence and uniqueness of pathwise stationary solutions for the stochastic Burgers equation with $L^2[0,1]$-valued noise and large viscosity. Using a random dynamical system framework, it proves that the stationary solution $Y(\omega) \in H^1[0,1]$ satisfies a stochastic integral equation on the infinite horizon, representing the solution as a Wiener pathwise limit of a Volterra-type integral involving the heat semigroup and noise increments.

ABSTRACT

In this paper, we show the existence and uniqueness of the stationary solution $u(t,ω)$ and stationary point $Y(ω)$ of the differentiable random dynamical system $U:R imes L^2[0,1] imes Ω o L^2[0,1]$ generated by the stochastic Burgers equation with $L^2[0,1]$-noise and large viscosity, especially, $u(t,ω)=U(t,Y(ω),ω)=Y(θ(t,ω))$, and $Y(ω) \in H^1[0,1]$ is the unique solution of the following equation in $L^2[0,1]$ $$ Y(ω)={1/2}\int_{-\infty}^0T_ν(-s)\frac{\partial (Y(θ(s,ω))^2}{\partial x}ds +\int_{-\infty}^0T_ν(-s)dW_s(ω), $$ where $θ$ is the group of $P$-preserving ergodic transformation on the canonical probability pace $(Ω, {\cal F}, P)$ such that $θ(t,ω)(s)=W(t+s)-W(t)$.

Motivation & Objective

• To establish the existence and uniqueness of pathwise stationary solutions for the stochastic Burgers equation with $L^2[0,1]$-valued noise.
• To characterize the stationary solution as a solution to a stochastic integral equation on the infinite time horizon $(-\infty, 0]$.
• To show that the stationary solution lies in the Sobolev space $H^1[0,1]$, ensuring sufficient regularity.
• To demonstrate that the solution is invariant under the group of $P$-preserving ergodic transformations $\theta(t,\omega)$, confirming pathwise stationarity.
• To provide a constructive representation of the stationary solution via a Wiener integral involving the heat semigroup $T_\nu(-s)$ and the noise process $dW_s(\omega)$.

Proposed method

• Formulate the stochastic Burgers equation on $[0,1]$ with Dirichlet boundary conditions and $L^2[0,1]$-valued white noise.
• Define a random dynamical system $U(t, \cdot, \omega)$ generated by the SPDE, satisfying cocycle and continuity properties.
• Represent the stationary solution $Y(\omega)$ as the solution to a Volterra-type stochastic integral equation on $(-\infty, 0]$: $Y(\omega) = \frac{1}{2}\int_{-\infty}^{0} T_\nu(-s) \frac{\partial}{\partial x}(Y(\theta(s,\omega))^2) ds + \int_{-\infty}^{0} T_\nu(-s) dW_s(\omega)$.
• Use the heat semigroup $T_\nu(t)$ to propagate the noise and nonlinear terms backward in time, ensuring convergence in $L^2[0,1]$.
• Apply ergodic theory and the group action $\theta(t,\omega)$ to ensure the solution satisfies $U(t, Y(\omega), \omega) = Y(\theta(t,\omega))$ for all $t \geq 0$.
• Leverage estimates from stochastic calculus and PDE theory (e.g., $L^2$-boundedness of derivatives, heat kernel bounds) to prove existence and integrability of the integral equation.

Experimental results

Research questions

• RQ1 Does a pathwise stationary solution exist for the stochastic Burgers equation with $L^2[0,1]$-valued noise and large viscosity?
• RQ2 Can the stationary solution be represented as a solution to a stochastic integral equation on the infinite time horizon $(-\infty, 0]$?
• RQ3 Is the stationary solution unique and regular enough to lie in $H^1[0,1]$?
• RQ4 Does the solution satisfy the cocycle identity $U(t, Y(\omega), \omega) = Y(\theta(t, \omega))$ for all $t \geq 0$?
• RQ5 Can the stationary solution be constructed as a limit of Wiener-type integrals involving the heat semigroup and noise increments?

Key findings

• The stationary solution $Y(\omega)$ exists and is unique in $L^2[0,1]$ for the stochastic Burgers equation with $L^2[0,1]$-valued noise and large viscosity.
• The stationary solution lies in the Sobolev space $H^1[0,1]$, indicating higher regularity than just $L^2$ integrability.
• The solution is characterized as the unique solution to the infinite-horizon stochastic integral equation: $Y(\omega) = \frac{1}{2}\int_{-\infty}^{0} T_\nu(-s) \frac{\partial}{\partial x}(Y(\theta(s,\omega))^2) ds + \int_{-\infty}^{0} T_\nu(-s) dW_s(\omega)$.
• The solution satisfies the pathwise stationarity condition $U(t, Y(\omega), \omega) = Y(\theta(t, \omega))$ for all $t \geq 0$, confirming invariance under the shift $\theta$.
• The existence of the solution is established via convergence of Wiener integrals and estimates on the heat kernel and noise increments, ensuring integrability in $L^2$ and $H^1$.
• The solution is shown to be a perfect cocycle, and its existence implies the existence of an invariant measure for the stochastic semiflow, with pathwise invariance providing stronger structural insight than measure-theoretic invariance.