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A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients

Thomas Hughes, Marcel Ortgiese

Abstract

We consider non-negative solutions to some infinite-dimensional SDEs on $\mathbb{Z}^d$ with Hölder continuous noise coefficients. We prove that if the Hölder 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.

A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients

Abstract

We consider non-negative solutions to some infinite-dimensional SDEs on with Hölder continuous noise coefficients. We prove that if the Hölder exponent is less than , 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.

Paper Structure

This paper contains 16 sections, 35 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Existence of solutions
  5. The half-space process
  6. A one-dimensional coupling result
  7. A class of semimartingales
  8. The behaviour of class-$\mathscr{M}_{\gamma,\delta}$ semimartingales near $0$
  9. Exit times and exit distributions for intervals
  10. Some discrete operators and approximate harmonic functions
  11. Exponential tails for the exit time of an interval
  12. The zero sets of class-$\mathscr{M}_{\gamma,\delta}$ semimartingales
  13. Time spent going up
  14. Time spent going down
  15. Proof of Proposition \ref{['prop_downcrosscount']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

Under Assumptions assumption1 and assumption1.5, for any $X_0 \in \mathbb{X}_{\mathrm{rap}}^+$, there exists a continuous $\,\mathbb{X}_{\mathrm{rap}}^+$-valued solution to e_sdesystem started from $X_0$.

Theorems & Definitions (69)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Lemma 3.2
  • Corollary 3.3
  • proof : Proof of Lemma \ref{['lemma_bddweak']}
  • Remark 3.4
  • Lemma 3.5
  • ...and 59 more