The stable manifold theorem for stochastic differential equations
Consider a stochastic differential equation on \(\mathbb{R}^d\) driven by a Kunita-type semimartingale with stationary ergodic increments, which may be either of Itô or of Stratonovich type. Suppose that the stochastic flow induced by the equation has a hyperbolic stationary (possibly non-adapted) trajectory, with hyperbolicity meaning that none of the associated Lyapunov exponents vanishes. Under these conditions existence of stable and unstable manifolds around the hyperbolic stationary trajectory is proved. The proof rests on arguments invoked in the approach of \textit{D. Ruelle} [Publ. Math., Inst. Hautes Étud. Sci. 50, 27-58 (1979; Zbl 0426.58014)], which are adapted for stochastic flows in continuous time. A key ingredient is the verification of integrability conditions for the derivative of the flow around the stationary solution.
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