Exponential and Uniform Ergodicity of Markov Processes

Immortal particle for a catalytic branching process

“…The set D δ is a small set for the process due to the fact that (x(t)) has a density bounded below by the density function of the process killed at the boundary; in its turn, this density function has a uniform lower bound on D δ for any t > 0. Exponential ergodicity is guaranteed [4] by the sufficient condition (3.4) that there exists an exponential moment of the time to reach D δ , uniformly over all x ∈ D = G N .…”

Large Deviation Multifractal Analysis of a Class of Additive Processes With Correlated Nonstationary Increments

“…In the proofs below, we will however need a bit more than ergodicity: we will require that convergence in (5) takes place at an exponential rate. To check this, we make use of powerful results proved in [10] for general Markov processes. Again, it is easy to verify that Theorem 7.1 of [10] applies, to the effect that:…”

“…To check this, we make use of powerful results proved in [10] for general Markov processes. Again, it is easy to verify that Theorem 7.1 of [10] applies, to the effect that:…”

A sufficient condition for the existence of an invariant probability measure for Markov processes

“…Following the framework of [3], [6], and [7], and as defined in Section 2, {X t } t∈R + is a time-homogeneous Markov process with state space (X, B(X)) and associated transition semigroup {P t } t∈R + . Recall that, for every t ∈ R + , we have P t (x, A) = P x (X t ∈ A), where x ∈ X is the initial condition, A represents the extended generator of {X t } t∈R + , D(A) represents the domain of the extended generator (see Definition 2.7), and R represents the resolvent associated with the transition semigroup {P t } t∈R + (see (2.1) and Definition 2.4).…”