A compact support property for infinite-dimensional SDEs with H\"older continuous coefficients

TL;DR

This paper proves that solutions to certain infinite-dimensional SDEs with H"older noise coefficients are compactly supported for almost all times when the H"older exponent is less than 1/2, revealing sharp support propagation properties.

Contribution

It introduces a novel approach to analyze support properties of infinite-dimensional SDEs with H"older continuous coefficients, extending classical SPDE support results.

Findings

Solutions are compactly supported for almost all times when H"older exponent < 1/2.

Support propagation for associated superprocesses is effectively sharp.

Zero sets of certain non-negative solutions have positive Lebesgue measure.

Abstract

We consider non-negative solutions to some infinite-dimensional SDEs on $\mathbb{Z}^d$ with H\"older continuous noise coefficients. We prove that if the H\"older exponent is less than $1/2$, solutions are compactly supported for almost all times, a variant of the classical compact support property for SPDEs. Our results imply that the instantaneous propagation of supports for superprocesses associated to discontinuous spatial motions is effectively sharp. We also show in a special case that the set of times when the support is arbitrarily large is dense. The proof uses a general approach which we expect can be applied to prove similar results for non-local SPDEs. It is based on an analysis of the excursions and zero sets of semimartingales whose quadratic variation satisfies a certain lower bound. As a corollary of our method, we show that the zero sets of non-negative solutions to some simple one-dimensional SDEs have positive Lebesgue measure, despite the absence of "sticky" dynamics.