| Abstract: |
| For non-autonomous linear stochastic differential equations (SDE), we establish that the top Lyapunov exponent is continuous if the coefficients almost uniformly converge. For autonomous SDEs, assuming the existence of invariant measures and the convergence of coefficients and their derivatives in pointwise sense, we get the continuity of all Lyapunov exponents. Furthermore, we talk about the Lipschitz and Holder continuity of Lyapunov exponents. For autonomous SDEs, we establish a relationship between the measure entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the results of Kifer and Yomdin, we obtain that for such systems with $C^\infty$ coefficients, under suitable integrability conditions, the measure entropy is upper semicontinuous with respect to the coefficients. |
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