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 in [A. Rantzer, Syst. Contr. 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.Comment: Submitte
