[Paper Review] Moments of the SHE under delta initial measure

Promit Ghosal|arXiv (Cornell University)|Aug 13, 2018

Stochastic processes and statistical mechanics31 references5 citations

TL;DR

This paper provides a rigorous proof of contour integral formulas for the one-point moments of the stochastic heat equation (SHE) with delta initial measure at the origin. Using a correspondence between the SHE and the Airy point process—established via a Laplace transform identity—and a limiting argument from semi-discrete directed polymers, the authors derive the conjectured moment formulas without relying on prior moment formulas, confirming their validity for all orders k ∈ ℕ.

ABSTRACT

We give a rigorous proof of the contour integral formulas of the moments of the stochastic heat equation (SHE) started from the delta initial measure at the origin. These formulas were conjectured in [BC14] (see also [CDR10, Dot10]). Our proof is based on a correspondence between the SHE and the Airy point process which was proved in [BG16, Theorem 1] using the formula of [ACQ11, Theorem 1.1].

Motivation & Objective

• To provide a rigorous proof of the contour integral formulas for the one-point moments of the stochastic heat equation (SHE) with delta initial measure, which were previously conjectured in [8] and supported by heuristic arguments.
• To establish a correspondence between the moments of the SHE at the origin and the complete homogeneous symmetric functionals of the Airy point process, without assuming the moment formulas as input.
• To derive the Airy point process moments rigorously using a limiting procedure from semi-discrete directed polymers, avoiding reliance on divergent series expansions.
• To confirm the convergence of moments of the semi-discrete polymer model to those of the SHE, ensuring the validity of the resulting integral formulas.
• To establish uniform integrability and tail bounds for the SHE solution, ensuring the convergence of moments under the limit process.

Proposed method

• Utilizes a Laplace transform identity from [10, Theorem 1], relating the Laplace transform of the SHE solution at the origin to a multiplicative functional of the Airy point process.
• Employs a Taylor expansion of the Laplace transform on the Airy side, carefully controlling divergent series by truncating and bounding remainder terms.
• Applies a limiting procedure from the semi-discrete directed polymer model, where the partition function moments are known to satisfy contour integral formulas.
• Uses a critical point analysis (Laplace method) to deform the integration contours in the polymer moment integrals, transforming them into the desired form for the SHE moments.
• Applies Markov’s inequality and uniform integrability to control tail probabilities and justify the interchange of limits and expectations.
• Combines the convergence of the polymer model to the SHE with the convergence of their moments, using the dominated convergence theorem to conclude the result.

Experimental results

Research questions

• RQ1 Can the contour integral formulas for the one-point moments of the SHE with delta initial data be rigorously derived without assuming the formulas a priori?
• RQ2 Is there a direct correspondence between the moments of the SHE and the Airy point process that does not rely on prior moment formulas?
• RQ3 Can the moment formulas be derived via a limiting procedure from the semi-discrete directed polymer model, and does this limit preserve the moment structure?
• RQ4 What is the nature of the convergence of the moments of the semi-discrete polymer model to those of the SHE, and how can uniform integrability be established?
• RQ5 How can divergent series arising from the Laplace transform expansion of the Airy functional be handled rigorously to extract moments?

Key findings

• The paper establishes the rigorous validity of the contour integral formula for the k-th moment of the SHE at the origin, confirming the conjecture in [8] for all k ∈ ℕ.
• The formula is given by: E[(Z(T,X))^k] = (1/(2πi)^k) ∫⋯∫ ∏_{1≤A<B≤k} (z_A - z_B)/(z_A - z_B - 1) × exp( (T/2)∑z_j² + X∑z_j ) ∏dz_j, with contours C_j = α_j + iℝ and α_1 > α_2 + 2 > ⋯ > α_k + (k-1).
• The correspondence between the SHE moments and the Airy point process is proven independently of the moment formulas, using only the Laplace transform identity from [10, Theorem 1].
• The Airy point process moments are derived rigorously via a limiting argument from the semi-discrete polymer model, avoiding divergent series expansions.
• The moments of the semi-discrete polymer model converge to those of the SHE, and the convergence is uniform in N, ensuring the limit of moments equals the moment of the limit.
• Tail bounds are established via Markov’s inequality and uniform integrability, proving that E[(Z(T,X))^k] < ∞ and justifying the limit interchange in the moment convergence.