Stochastic Wright's Equation: Existence of Invariant Measures

Abstract

Wright's delay differential equation is one of the prime examples of a fully nonlinear equation without an explicit solution and whose dynamics can be understood by analytic means. In this paper, we introduce stochastic perturbations by adding Brownian noise with a bounded Lipschitz noise coefficient to a transformed version of Wright's equation. The transformation considered plays an important role in the deterministic theory as well. We demonstrate that this stochastically perturbed equation has (at least) two invariant measures: a trivial measure concentrated at $-1$ and a nontrivial measure on $(-1,\infty)$. The crucial and most challenging step of the proof is showing that every solution is bounded away from $-1$ in probability. In addition, a major part of our analysis is devoted to deriving detailed estimates for It\^o processes with a negative drift.