François Delarue

The rearranged stochastic heat equation is a version of the 1d stochastic heat equation that is reflected on the boundary of quantile functions on the torus by means of a rearrangement procedure. By isometry between the space of quantile functions equipped with L^2 distance and the space of probability measures (on R) equipped with the 2-Wasserstein distance, it provides a diffusion process on the space of 1d probability measures. In a recent work with W. Hammersley (just appeared in PTRF), we explored the construction and the properties of this process. In an going work with R. Likibi, we explore the situation when the dynamics on the space of probability measures are also subjected to a convolution operation: at infinitesimal time step, the probability valued process moves according to the rearranged dynamics and then is convoluted by an infinitesimal Gaussian kernel. The latter convolution accounts for the impact of idiosyncratic noises in large particle systems, whist the rearranged dynamics would correspond to the impact of an infinite dimensional common noise.

Rearranged Stochastic Heat Equation

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