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Sharp Riemannian heat kernel estimates on the cut locus and the Parabolic Anderson model

Hongyi Chen, Robert Neel, Cheng Ouyang

Abstract

Using sharp global heat kernel bounds and geodesic comparison geometry, we show that the Dalang condition for well-posedness of the parabolic Anderson model with measure-valued initial conditions, first introduced on Euclidean space, holds on general compact Riemannian manifolds. We furthermore establish upper and lower moment bounds for all such solutions, providing evidence for intermittency in this generality. This extends and simplifies earlier work that required non-positive curvature.

Sharp Riemannian heat kernel estimates on the cut locus and the Parabolic Anderson model

Abstract

Using sharp global heat kernel bounds and geodesic comparison geometry, we show that the Dalang condition for well-posedness of the parabolic Anderson model with measure-valued initial conditions, first introduced on Euclidean space, holds on general compact Riemannian manifolds. We furthermore establish upper and lower moment bounds for all such solutions, providing evidence for intermittency in this generality. This extends and simplifies earlier work that required non-positive curvature.

Paper Structure

This paper contains 15 sections, 23 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Upper Bounds of the heat kernel
  5. Colored Noise and Mild Solution on Compact Riemannian Manifolds
  6. Proof of Theorem \ref{['thm: main result']}
  7. Properties of the Three Distances Function $F$
  8. Some Useful Integral Estimates
  9. Proof of Theorem \ref{['thm: L1 bd']} for $t\ge1$
  10. Proof of Theorem \ref{['thm: L1 bd']} for $0<t<1$
  11. Bounding $\mathcal{L}_n$
  12. Well-Posedness and Moment Upper Bound
  13. Comparison Principles and Moment Lower Bound
  14. Comparison Principles
  15. Exponential lower bound for Moments

Key Result

Theorem 1.1

Let $M$ be any compact manifold and $W=W_{\alpha,\rho}$ be the colored noise of Definition def: G_alpha etc. For any $\alpha>{(d-2)}/{2}$ and any finite measure $\mu$ on $M$, equation eq: SHE admits a random field solution $\{u(t,x)\}_{t>0,x\in M}$ in the sense of Definition def: SHE mild. The solut Here $J_\mu(t,x):=\int_M P_t(x,y) \mu(dy)$ is the homogeneous solution to the heat equation startin

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Proposition 2.2: Li-Yau Bound
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • ...and 37 more