We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov-Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by Hérau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.
We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).
We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $α$-stable processes. For SDE driven by $α$-stable process ($α\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $α$.
In this paper we study the ergodicity and the related semigroup property for a class of symmetric Markov jump processes associated with time changed symmetric $α$-stable processes. For this purpose, explicit and sharp criteria for Poincaré type inequalities (including Poincaré, super Poincaré and weak Poincaré inequalities) of the corresponding non-local Dirichlet forms are derived. Moreover, our main results, when applied to a class of one-dimensional stochastic differential equations driven by symmetric $α$-stable processes, yield sharp criteria for their various ergodic properties and corre…
We establish a variational formula for the exponential decay rate of the Green function of Brownian motion evolving in a random stationary and ergodic nonnegative potential. Such a variational formula is established by Schroeder in 'Green's Functions for the Schrödinger Operator with Periodic Potential', J. Funct. Anal. 77 (1988), for potentials on compact spaces and is generalised in the present article to a non-compact setting. We show exponential decay of the Green function implicitly. This formula for the Lyapunov exponent has several direct implications. It allows to compare the influence…
This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincaré inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.
Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergod…
We develop a theory of weak Poincaré inequalities to characterize convergence rates of ergodic Markov chains. Motivated by the application of Markov chains in the context of algorithms, we develop a relevant set of tools which enable the practical study of convergence rates in the setting of Markov chain Monte Carlo methods, but also well beyond.
The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4, 30] that there is an equivalence between a Poincar{é} inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by $\bullet$ showing that strong functional inequalities are equivalent to Lyapunov type conditions; $\bullet$ showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential momen…
Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for logconcave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial, ...). In particular, in the one dimensional case, ultracontractivity is equivalent to a bounded Lyapuno…
Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodi…
By using Lyapunov conditions, weak Poincaré inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.
We investigate ergodic properties of generalized Ornstein--Uhlenbeck processes. In particular, we provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster--Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.
We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = κΔu(x,t) + ξ(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, and $ξ$ plays the role of a \emph{dynamic random environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$, $x\in\Z^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2dκ$, split into…
We review here some recent results by the authors, and various coauthors, on (weak,super) Poincaré inequalities, transportation-information inequalities or logarithmic Sobolev inequality via a quite simple and efficient technique: Lyapunov conditions.
New relations between ergodic rate, L_p convergence rates, and asymptotic behavior of tail probabilities for hitting times of a time homogeneous Markov process are established. For L_p convergence rates and related spectral and functional properties (spectral gap and Poincare inequality) sufficient conditions are given in the terms of an exponential ϕ-coupling. This provides sufficient conditions for L_p convergence rates in the terms of appropriate combination of `local mixing' and `recurrence' conditions on the initial process, typical in the ergodic theory of Markov processes. The range of …
Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincaré Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Cesàro mean associated with the iterates of the process. Applied to processes in $L^1(Ω,{\mathcal A},μ)$, where $μ$ is a probability measure, new conditional versions…
In this paper we establish the ergodicity of Langevin dynamics for simple two-particle system involving a Lennard-Jones type potential. To the best of our knowledge, this is the first such result for a system operating under this type of potential. Moreover we show that the dynamics are {\it geometrically} ergodic (have a spectral gap) and converge at a geometric rate. Methods from stochastic averaging are used to establish the existence of a Lyapunov function. The existence of a Lyapunov function in this setting seems resistant to more traditional approaches. This is a corrected version of th…
In this paper we introduce and study the $α$-Farey map and its associated jump transformation, the $α$-Lüroth map, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $α$…
Ergodic properties of the signal-filtering pair are studied for continuous time finite Markov chains, observed in white noise. The obtained law of large numbers is applied to the stability problem of the nonlinear filter with respect to initial conditions. The Furstenberg-Khasminskii formula is derived for the top Lyapunov exponent of the Zakai equation and is used to estimate the stability index of the filter.
The aim of this note is to present an elementary proof of a variation of Harris' ergodic theorem of Markov chains. This theorem, dating back to the fifties essentially states that a Markov chain is uniquely ergodic if it admits a ``small'' set which is visited infinitely often. This gives an extension of the ideas of Doeblin to the unbounded state space setting. Often this is established by finding a Lyapunov function with ``small'' level sets. This topic has been studied by many authors (cf. Harris, Hasminskii, Nummelin, Meyn and Tweedie). If the Lyapunov function is strong enough, one has a …
