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We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semi-groups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.
Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-va
In this paper we address an open question formulated in [17]. That is, we extend the It{^o}-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker-Planck equation
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}rightarrowmathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gau
Let ${u(t,, x)}_{t >0, x inmathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $delta_0$ and driven by space-time white noise on $mathbb{R}_+timesmathbb{R}$, and let $bm{p}_t(x):= (2pi t)^{-1/2}exp{-x^2/(2t)}$ denot
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}tomathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian no