Stability Along Trajectories at a Stochastic Bifurcation Point

We consider a particular class of multidimensional nonlinear stochastic differential equations with 0 as a fixed point. The almost sure stability or instability of 0 is determined by the Lyapunov exponent λ for the associated linear system. If parameters in the stochastic differential equation are varied in such a way that λ changes sign from negative to positive then 0 changes from being (almost surely) stable to being (almost surely) unstable and a new stationary probability measure μ appears. There also appears a new Lyapunov exponent $$ \tilde \lambda $$ , say, corresponding to linearizing the original stochastic differential equation along a trajectory with stationary distribution μ. The value of $$ \tilde \lambda $$ determines stability or instability along trajectories. We show that, under appropriate conditions, the ratio $$ \tilde \lambda $$ /λ has a limiting value Γ at a bifurcation point, and we give a Khasminskii-Carverhill type formula for Γ. We also provide examples to show that Γ can take both negative and positive values.