Convergence of the rescaled Whittaker stochastic differential equations and independent sums

TL;DR

This paper investigates the convergence of rescaled Whittaker SDEs derived from a surface growth model, demonstrating Gaussian fluctuations and convergence to a stochastic heat equation through probabilistic representations involving independent sums.

Contribution

It introduces a novel probabilistic approach linking Whittaker SDEs to independent sums, providing new insights into their convergence and fluctuation behavior.

Findings

Proves Gaussian fluctuations exhibit 'coming down from infinity' behavior.

Establishes convergence to the time-inverted stochastic heat equation.

Uses independent sums to explain covariance convergence and process dynamics.

Abstract

We study some SDEs derived from the $q\to 1$ limit of a 2D surface growth model called the $q$-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that "come down from infinity": After rescaling and re-centering, convergence to the time-inverted stationary additive stochastic heat equation holds. The point of view in this paper is a probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations and the in-between slow decorrelation, all in particular integrated forms, explain the convergence of the re-centered covariance functions. With bounds and divergent constants from these approximations, the proof of the process-level convergence identifies additional divergent terms in the dynamics and considers cancellation arguments that treat the independent sums as discrete spin systems.