Non-explosion by Stratonovich noise for ODEs

Mario Maurelli

doi:10.1214/20-ecp347

Abstract

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.

Highlights

Introduction and main result AnODE on Rd, with d ≥ 2, dX = b(X)dt, X0 = x0, (1.1)with locally Lipschitz drift b : Rd → Rd, can exhibit explosion in finite time: this is the case, for example, when b(x) = |x|m−1x, with m > 1, as it can be checked computing the explicit solution
We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension d ≥ 2
We show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE

Summary

First proof

The conditions on m and α in Assumption 1.1 guarantee that this SDE for Y admits a unique solution, whose law is equivalent to the Wiener measure (by Girsanov theorem). For d ≥ 2, the Wiener measure does not see the 0 P-a.s., we conclude that Y does not hit 0 and X does not explode P-a.s. First proof. We are in the position to apply [13, Theorem 1, Corollary 16]: the SDE (2.3) admits a global (strong) solution Ywhose law is equivalent to the d-dimensional Wiener measure starting from φ(x0). The SDE (2.3) admits a unique strong solution before exiting BR−η \ {0}, by the local Lipschitz property of g, and Y = Y on [0, ρY ∧ ρY,R−η ).

Second proof

Invariant measures and geometric ergodicity

Findings

Counterexample in one dimension