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An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries. (English) Zbl 1512.60049

Summary: This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function \(f\), the time average \(\frac{1}{t}\int_0^tf(X_s)ds\) converges in \(\mathbb{L}^2\) towards a limiting distribution, starting from any initial distribution for the process \((X_t)_{t\geq 0}\). This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result is then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.

MSC:

60J25 Continuous-time Markov processes on general state spaces
60F25 \(L^p\)-limit theorems
60J55 Local time and additive functionals
60J60 Diffusion processes
60J65 Brownian motion
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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