Almost global stochastic stability

We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability. The method can be viewed as a dual to Lyapunov's second method for stochastic differential equations and extends the deterministic result of [A. Rantzer, Syst. Control Lett., 42 (2001), pp. 161-168]. The result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization. The main technical tool is the theory of stochastic flows of diffeomorphisms.