Rearranged Stochastic Heat Equation: Ergodicity and Related Gradient Descent on the Space of Probability Measures

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. Probability measure-valued flows perturbed by this noise are studied, 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, as defined in Lions' approach to the differential calculus on the space of probability measures. The resulting equation reads as a sort of McKean-Vlasov stochastic differential equation with an infinite dimensional common noise. Conditions on the drift are provided, under which solutions exist uniquely for any time horizon and converge exponentially fast towards a unique equilibrium. When the drift derives from a potential on the space of probability measures, 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. Interestingly, in order to accommodate a wider class of potentials, such as the square of the second Wasserstein distance, a weaker notion of derivative is defined over the subspace of symmetrised quantile functions.