Stochastic hybrid parabolic systems under jump Markovian perturbations-I: Convergence and stability via Lyapunov functionals

In this paper, we investigate convergence and stability concepts in the sense of p-th mean and probability of the solution process of stochastic parabolic systems under Markovian structural perturbations. Sufficient conditions are obtained for various kinds of convergence and stability. This is achieved by developing a powerful comparison theorem in the context of vector Lyapunov-like functionals and systems of differential inequalities.