The Concentration of Invariant Measures for Stochastic Dynamical Systems with Locally Lipschitz Continuous Coefficients in R^d

In this paper, we study the concentration phenomena of the invariant measures for stochastic differential equations with locally Lipschitz continuous coefficients and more than one ergodic state in $\mathbb{R}^d$. Under some dissipative conditions, by using Lyapunov-like functions and the large deviations method, we estimate the invariant measures in the neighborhoods of stable sets, the neighborhoods of unstable sets and their complement, respectively. Our result illustrates that the invariant measures concertrate on the intersection of the stable sets where the $\min W(K_i)$ are attached and the Birkhoff center of the corresponding deterministic system as the noise tends down to zero. Furthermore, we show the large deviations principle of the invariant measures. At the end of this paper, we provide some explicit examples and their numerical simulations.

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• Roland Riachi