Hypoellipticity in Infinite Dimensions

TL;DR

This paper investigates the regularity of transition probabilities for semilinear parabolic stochastic PDEs in infinite dimensions, establishing smoothness under Hormander conditions using bounds on Wiener polynomials.

Contribution

It demonstrates that finite-dimensional projections of solutions have smooth densities when Hormander conditions are met, extending regularity results to infinite-dimensional stochastic PDEs.

Findings

Finite-dimensional projections have smooth densities under Hormander conditions

A bound on Wiener polynomials analogous to Norris' lemma is established

Regularity of transition probabilities is confirmed in infinite-dimensional setting

Abstract

We consider semilinear parabolic stochastic PDEs driven by additive noise. The question addressed in this note is that of the regularity of transition probabilities. If the equation satisfies a Hormander 'bracket condition', then any finite-dimensional projection of the solution has a smooth density with respect to Lebesgue measure. One key ingredient in the argument is a bound on 'Wiener polynomials' that plays a role analogue to Norris' lemma.