The comparability of motions in dynamical systems and recurrent solutions of (S)PDEs

David Cheban ; Zhenxin Liu

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 53-83

Résumé

Shcherbakov's comparability method is very useful to study recurrent solutions of differential equations. In this paper, we extend the method from metric spaces to uniform spaces, which applies well to dynamical systems in infinite-dimensional spaces. This generalized comparability method can be easily used to study recurrent solutions of (stochastic) partial differential equations under weaker conditions than in earlier results. We also show that the distribution of solutions of SDEs naturally generates a semiflow or skew-product semiflow on the space of probability measures, which is interesting in itself. As illustration, we give an application to semilinear stochastic partial differential equations.