Rearranged Stochastic Heat Equation: Ergodicity and Related Gradient Descent on ${\mathcal P}({\mathbb R})$

Abstract

This article provides a case study for a recently introduced diffusion in the space of probability measures over the reals, namely rearranged stochastic heat, which solves a stochastic partial differential equation valued in the set of symmetrised quantile functions over the unit circle. This contribution studies probability measure-valued flows perturbed by this noise with a special focus on gradient flows. This is done by introducing a drift to the rearranged stochastic heat equation by means of a vector field from the set of random variables over the unit circle into itself. When the flow is a gradient flow, the vector field may coincide with the Wasserstein derivative of a mean-field potential function. The resulting equation reads as a sort of McKean-Vlasov stochastic differential equation with an infinite dimensional common noise. This article provides conditions on the drift under which solutions exist uniquely for any time horizon and converge exponentially fast towards a unique equilibrium. When the drift derives from a potential, some metastability properties are obtained as the intensity of the noise is tuned to zero: it is shown that under a particular scaling regime, the gradient descent lingers near local minimizers for expected times of the same order as in the finite dimensional setting.