  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.
