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Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder

Hongyi Chen

Abstract

We study the nonlinear multiplicative stochastic heat equation on Dirichlet spaces with white in time noise under weak disorder. We show that positive invariant random fields with uniformly bounded second moments are in one-to-one correspondence with bounded positive harmonic functions. The proof combines a remote past pullback construction with a uniqueness argument that applies a contraction inspired by chaos expansion. As a consequence, the space of invariant measures inherits geometric structure from the Martin Boundary. We further establish a small-noise Gaussian fluctuation result within each harmonic sector and show that the long-time behavior of solutions is completely determined by the Martin boundary data of the initial condition. These results reveal a direct connection between stochastic PDE dynamics and boundary theory in potential analysis.

Martin Boundary and the Nonlinear Multiplicative Stochastic Heat Equation in Weak Disorder

Abstract

We study the nonlinear multiplicative stochastic heat equation on Dirichlet spaces with white in time noise under weak disorder. We show that positive invariant random fields with uniformly bounded second moments are in one-to-one correspondence with bounded positive harmonic functions. The proof combines a remote past pullback construction with a uniqueness argument that applies a contraction inspired by chaos expansion. As a consequence, the space of invariant measures inherits geometric structure from the Martin Boundary. We further establish a small-noise Gaussian fluctuation result within each harmonic sector and show that the long-time behavior of solutions is completely determined by the Martin boundary data of the initial condition. These results reveal a direct connection between stochastic PDE dynamics and boundary theory in potential analysis.
Paper Structure (10 sections, 7 theorems, 64 equations)

This paper contains 10 sections, 7 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Acknowledgements
  4. Preliminaries
  5. Proof of Main Theorem
  6. Existence by Backward Time Shifts
  7. Uniqueness by Chaos Expansion
  8. Proof of Theorem \ref{['thm:fluc']}
  9. Proof of Theorem \ref{['thm:IC determine long time']}
  10. Appendix: Connections to Martin Boundary

Key Result

Theorem 1.2

Let $u_0\in L^\infty(M)$ and suppose $R$ satisfies Assumption assump:WD. Let $u$ be the solution of eq:SHE starting from $u_0$. The following statements hold.

Theorems & Definitions (18)

  • Theorem 1.2
  • Remark 1.3: Nontriviality of the assumption on $R$
  • Remark 1.4: On the moment assumption
  • Theorem 1.5: First order Gaussian fluctuation in $\beta\to 0$ limit of invariant fields
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1: Pullback construction from harmonic initial condition
  • proof
  • Theorem 3.2: Uniqueness of stationary solutions in the weak-disorder class
  • ...and 8 more