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Infinite-dimensional paracontrolled distributions: the Burgers generator

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SRQW01 - Renormalisation in quantum field theory and in stochastic partial differential equations: a gentle introduction and some recent developments

Regularity structures, paracontrolled distributions and all that provide pathwise, deterministic tools to solve and study singular stochastic PDEs over finite-dimensional spaces. From a probabilistic point of view we would also like to understand the associated Kolmogorov backward equations, which can be interpreted as infinite-dimensional singular SPD Es. I will discuss on the example of the conservative stochastic Burgers equation how to construct a space of (para-) paracontrolled distributions in which the backward equation is well posed. As an application we obtain a martingale formulation and an alternative proof for the well-posedness of “energy solutions”, without using the Cole-Hopf transform. The approach extends to some other singular SPD Es with Gaussian invariant measures and quadratic nonlinearities. This is joint work with Massimiliano Gubinelli.

This talk is part of the Isaac Newton Institute Seminar Series series.

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