Stability of dynamical systems in the presence of noise

Pinsky's research is concerned with the exponential growth rate (= Lyapunov exponent) of solutions of stochastic differential equations. In a paper to appear in the Annals of Applied Probability, a formula is obtained for the quadratic Lyapunov exponent of the simple harmonic oscillator in the presence of a finite-state Markov noise process. In case the noise process is reversible, the quadratic Lyapunov exponent is strictly less than for the corresponding white noise process obtained from the central limit theorem. An example is presented of a non-reversible Markov noise process for which this inequality is reversed. In another article, to appear in the volume 'Stochastic Partial Differential Equations and their Applications' in the Springer Verlag Lecture Notes in Control and Information Sciences (Proceedings of the 1991 Charlotte NC Conference on SPDE, ed. B. Rozovskii), the Lyapunov exponent is computed for the solution of a hyperbolic partial differential equation with damping. In this case, one studies the exponential growth rate of the energy of the solution with Dirichlet boundary conditions. The detailed results depend on the size of the damping constant (overdamped vs. underdamped case). To our knowledge, this is the first study ever of the Lyapunov exponent for a partial differential equation.