Continuity of measure-theoretic entropy for stochastic differential equations

For stochastic differential equations, we establish a relationship between the measure-theoretic entropy of the stochastic flow and the rate of volume growth of stable submanifolds under iteration. By combining the result of Kifer and Yomdin (1988), we show that under a suitable integrability condition, for systems with $C^{\mathrm{\infty }}$ coefficients, the measure-theoretic entropy is upper semicontinuous with respect to the coefficients of SDEs.