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

Zahra Aminzare

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

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

Zahra Aminzare

ArXiv.org

11/04/2021

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https://arxiv.org/abs/2111.03155View

Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

We introduce the notion of stochastic logarithmic Lipschitz constants and use these constants to characterize stochastic contractivity of Itô 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.

Details

Title: Subtitle: Stochastic Logarithmic Lipschitz Constants: A Tool to Analyze Contractivity of Stochastic Differential Equations
Creators: Zahra Aminzare
Resource Type: Preprint
Publication Details: ArXiv.org
ISSN: 2331-8422
Language: English
Date posted: 11/04/2021
Academic Unit: Mathematics; Iowa Neuroscience Institute
Record Identifier: 9984197393002771

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