A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems

Given the linear stochastic system $\dot x = A(\xi (t))x,x \in \mathbb{R}^d ,\xi (t)$ some nice diffusion. Sample and moment stability of $x \equiv 0$ are studied via the sample Lyapunov exponent $\lambda $ and the Lyapunov exponent of the pth mean $g(p)$ of a solution. Generalizing a result of Molchanov for the undamped linear oscillator, it is shown that $g(p)$ is a smooth convex function of $p \in \mathbb{R}$ with $g(0) = 0$ and $g'(0) = \lambda $. For trace $A = 0$ a second zero of g at $p = - d$ is detected. This allows one to draw simple, but quite complete conclusions about the relation between sample and moment stability.