Stochastic Logarithmic Lipschitz Constants: A Tool to Analyze Contractivity of Stochastic Differential Equations

TL;DR

This paper introduces stochastic logarithmic Lipschitz constants to analyze the contractivity of Itô stochastic differential equations with multiplicative noise, providing bounds and insights into noise-induced synchronization.

Contribution

It proposes a new stochastic Lipschitz constant concept and applies it to characterize and analyze stochastic contractivity and synchronization in SDEs.

Findings

Stochastic logarithmic Lipschitz constants can bound stochastic contractivity.

Noise can induce contractivity in non-contracting deterministic systems.

Illustration on a noisy Van der Pol oscillator demonstrates the theory.

Abstract

We introduce the notion of stochastic logarithmic Lipschitz constants and use these constants to characterize stochastic contractivity of It\^o stochastic differential equations (SDEs) with multiplicative noise. We find an upper bound for stochastic logarithmic Lipschitz constants based on known logarithmic norms (matrix measures) of the Jacobian of the drift and diffusion terms of the SDEs. We discuss noise-induced contractivity in SDEs and common noise-induced synchronization in network of SDEs and illustrate the theoretical results on a noisy Van der Pol oscillator. We show that a deterministic Van der Pol oscillator is not contractive. But, adding a multiplicative noise makes the underlying SDE stochastically contractive.