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Exact Multi-Point Correlations in the Stochastic Heat Equation for Strictly Sublinear Coordinates

Pierre Yves Gaudreau Lamarre, Yier Lin

Abstract

We consider the Stochastic Heat Equation (SHE) in $(1+1)$ dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space coordinates. The sublinear condition is optimal, in the sense that different asymptotics are known to occur when the space coordinates grow linearly [Lin 2023, Theorem 1.1]. Lastly, a notable feature of our result is that it confirms the connection between multi-point correlations in the SHE and the ground state of the Hamiltonian of the delta-Bose gas.

Exact Multi-Point Correlations in the Stochastic Heat Equation for Strictly Sublinear Coordinates

Abstract

We consider the Stochastic Heat Equation (SHE) in dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space coordinates. The sublinear condition is optimal, in the sense that different asymptotics are known to occur when the space coordinates grow linearly [Lin 2023, Theorem 1.1]. Lastly, a notable feature of our result is that it confirms the connection between multi-point correlations in the SHE and the ground state of the Hamiltonian of the delta-Bose gas.
Paper Structure (7 sections, 6 theorems, 75 equations)

This paper contains 7 sections, 6 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Outline of Proof
  5. Contour Integral Representation - Proof of Proposition \ref{['prop:contexpan']}
  6. Leading Order Term - Proof of Proposition \ref{['Proposition: Leading Order']}
  7. Remainder Terms - Proof of Proposition \ref{['Proposition: Remainder Terms']}

Key Result

Theorem 1.2

Let $Z$ be as in Equation: SHE with $Z(0,\cdot)=\delta(\cdot)$. Let $n\in \mathbb{Z}_{\geq 1}$, and for every $1\leq i\leq n$, let $(x_i(t))_{t\geq0}$ be a strictly sublinear sequence of real numbers, in the sense that $x_i(t)=o(t)$ as $t\to\infty$. It holds that where $\mathfrak L_n$ is the Lyapunov exponent in Equation: SHE Moment Lyapunov and $\Psi_n$ is the ground state in Equation: Ground St

Theorems & Definitions (11)

  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • Lemma 3.2
  • ...and 1 more