Computation of the Lyapunov measure for almost everywhere stochastic stability

In our recent work [1], we introduced Lyapunov measure as a new tool to verify weaker set-theoretic notion of almost everywhere stability of stochastic nonlinear systems. A Linear transfer Perron-Frobenius operator for stochastic systems was used to provide an explicit formula for the Lyapunov measure, verifying almost everywhere almost sure stability of stochastic systems. The focus of this paper is on the computational aspect of the Lyapunov measure for stochastic systems. We used set-oriented numerical methods for the finite dimensional approximation of the linear operator and the Lyapunov measure. Stability results in the finite dimensional approximation space are also presented. In particular, we show the finite dimensional approximation leads to a further weaker notion of stability referred to as coarse stability.