Summary
Let {x t :t≧0} be the solution of a stochastic differential equation (SDE) in ℝd which fixes 0, and let λ denote the Lyapunov exponent for the linear SDE obtained by linearizing the original SDE at 0. It is known that, under appropriate conditions, the sign of λ controls the stability/instability of 0 and the transience/recurrence of {x t :t≧0} on ℝd\{0}. In particular if the coefficients in the SDE depend on some parameterz which is varied in such a way that the corresponding Lyapunov exponentλ z changes sign from negative to positive the (almost-surely) stable fixed point at 0 is replaced by an (almost-surely) unstable fixed point at 0 together with an attracting invariant probability measureμ z on ℝd\{0}. In this paper we investigate the limiting behavior ofμ z asλ z converges to 0 from above. The main result is that the rescaled measures (1/λ z)μ z converge (in an appropriate weak sense) to a non-trivial σ-finite measure on ℝd\{0}.
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Research supported in part by Office of Naval Research contract N00014-91-J-1526