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Quantification of ergodicity for Hamilton--Jacobi equations in a dynamic random environment

Xiaoqin Guo, Wenjia Jing, Hung Vinh Tran, Yuming Paul Zhang

Abstract

We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as a Hamilton--Jacobi equation with random forcing. Understanding the large-time averaged behavior of solutions is closely connected to fundamental questions about fluctuations and scaling in such growth processes.

Quantification of ergodicity for Hamilton--Jacobi equations in a dynamic random environment

Abstract

We study quantitative large-time averages for Hamilton--Jacobi equations in a dynamic random environment that is stationary ergodic and has unit-range dependence in time. Our motivation comes from stochastic growth models related to the tensionless (inviscid) KPZ equation, which can be formulated as a Hamilton--Jacobi equation with random forcing. Understanding the large-time averaged behavior of solutions is closely connected to fundamental questions about fluctuations and scaling in such growth processes.

Paper Structure

This paper contains 18 sections, 31 theorems, 316 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. Settings
  4. Main results
  5. Regularity of the metric distance and the optimal curves
  6. Basic bounds
  7. Rough metric and curve regularity
  8. Improved regularity via iterations
  9. Concentration estimates of the metric distance
  10. Quantification of the error between the mean and the ergodic average of the metric
  11. An iterative argument
  12. Good skeletons
  13. Proof of Theorem \ref{['thm:error-deterministic']}
  14. Proof of the main results
  15. Proof of Theorem \ref{['thm:main-1']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Assume that, for some $c,C_0>0$, Let $u$ be the solution eq:HJ and $R>0$. Then, for $t$ large enough depending only on $c,d,q,R,C_0, N_1$, Here, are slow-varying functions defined for $t>e$. And $c_1>0$ is a constant depending only on $c,d,q,N_1$, which is given in Theorem thm:path-reg. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1.1: Some admissible curves connecting $(x_1,t_1)$ to $(x_2,t_2)$
  • Figure 2.1: Curves $\gamma$ and $\beta$
  • Figure 4.1: An example of $Q_{x,t}$ and $F_{x,t}$

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 1
  • Lemma 2.5
  • proof
  • ...and 58 more