Strong completeness and semi-flows for stochastic differential equations with monotone drift

TL;DR

This paper investigates the conditions under which solutions to stochastic differential equations with monotone drift generate continuous semiflows and addresses strong Δ-completeness, extending classical results to less restrictive assumptions.

Contribution

It demonstrates that under a slightly stronger one-sided Lipschitz condition, solutions form a stochastic semiflow that is jointly continuous, even without being invertible or surjective.

Findings

Solutions generate a stochastic semiflow under monotone drift conditions.

The semiflow is jointly continuous in all variables.

Addresses strong Δ-completeness for solution modifications.

Abstract

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong $\Delta$-completeness which means that there exists a modification of the solution which if restricted to any set of dimension $\Delta$ is almost surely continuous with respect to the initial condition.