Towards mesoscopic ergodic theory. (English) Zbl 1508.37023
Summary: The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic differential equations (SDEs) with less regular coefficients and degenerate noises. These equations are often derived as mesoscopic limits of complex or huge microscopic systems. By studying the associated Fokker-Planck equation (FPE), we prove the convergence of the time average of globally defined weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions. In the case where the set of stationary measures consists of a single element, the unique stationary measure is shown to be physical. Similar convergence results for the solutions of the FPE are established as well. Some of our convergence results, while being special cases of those contained in [M. Ji et al., J. Funct. Anal. 277, No. 11, Article ID 108281, 41 p. (2019; Zbl 1428.35600); J. Dyn. Differ. Equations 31, No. 3, 1591–1615 (2019; Zbl 1428.35601)] for SDEs with periodic coefficients, have weaken the required Lyapunov conditions and are of much simplified proofs. Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 35Q84 | Fokker-Planck equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 37B25 | Stability of topological dynamical systems |
| 60J60 | Diffusion processes |
Keywords:
ergodic theory; stochastic differential equation; Fokker-Planck equation; stationary measure; physical measure; mesoscopic limitReferences:
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