Lyapunov functions for positive linear evolution problems
- Stephan, Holger
2010 Mathematics Subject Classification
- 47D07 47A63 35B40 35B50 37A35 46E10 82C31
- Markov operator, Lyapunov function, Fokker-Planck equation, positive continuous semigroup, positive minimum principle, Radon measures
We rigorously investigate the time monotonicity of Lyapunov functions for general positive linear evolution problems, including degenerate problems. This can be done by considering the problem in the convex set of probability measures and finding a general inequality for such Radon measures and Markov operators. For linear evolution problems (with discrete or continuous time), the existence of time monotone Lyapunov functions is not a consequence of any physical properties, but of the positivity and norm conservation of the equation. In some special cases the structure of such equations is given. Moreover, we describe completely the case of time constant Lyapunov functions - a property of deterministic dynamical systems.
- ZAMM Z. Angew. Math. Mech., 85 (2005) pp. 766-777