We consider the classical Foster–Lyapunov condition for the existence of an invariant measure for a Markov chain when there are no continuity or irreducibility assumptions. Provided a weak uniform countable additivity condition is satis/ed, we show that there are a /nite number of orthogonal invariant measures under the usual drift criterion, and give conditions under which the invariant measur...

Let P → M be a principal G-bundle. We construct well-defined substitutes for “Lebesgue measure” on the space A of connections on P and for “Haar measure” on the group G of gauge transformations. More precisely, we define algebras of “cylinder functions” on the spaces A, G, and A/G, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Bo...

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is Gibbs associated with Morse potential. In companion paper (Lee and Seo in Probab Theory Relat Fields 182:849–903, 2022), we proved the Eyring–Kramers formula for corresponding processes. this article, further develop result by proving that suitably time-rescaled process converges to Markov...

<p style='text-indent:20px;'>In this work we study the long time behavior of nonlinear stochastic functional-differential equations neutral type in Hilbert spaces with non-Lipschitz nonlinearities. We establish existence invariant measures shift for such equations. Our approach is based on Krylov-Bogoliubov theorem tightness family measures.</p>

The current paper is devoted to stochastic damped higher order KdV equation driven by Poisson process. We establish the well-posedness of higher-order equation, and prove that there exists an unique invariant measure for non-random initial conditions. Some discussion on general pure jump noise case are also provided. numerical simulations provided support theoretical results.