Stability and convergence via Lyapunov-like functionals of stochastic parabolic partial differential equations
The authors consider Itô-type stochastic parabolic partial differential equations with a one-dimensional Wiener process as a driving process. A comparison theorem is established, employing vector Lyapunov-like functionals and ordinary differential equations, and further, a theorem is given which provides a construction of the comparison system. The authors present several notions of convergence of the solution process to zero, e.g., convergence in \(p\)th moment or convergence in probability of the \(p\)th moment, and of stability of the zero solution of the stochastic parabolic partial differential equation. Appropriate notions for the deterministic comparison system of ordinary differential equations are also given. Then results stating that a convergence or stability property of the deterministic system implies a corresponding property of the stochastic system are provided. The results are illustrated with an example.
scientific article; zbMATH DE number 2117986