Ergodicity and speed of convergence to equilibrium for diffusion processes. Eva Locherbach¨ ∗ November 14, 2015 Abstract We discuss the long time behavior of diffusion processes (or more general Markov processes). We start by introducing the basic concept of Harris recurrence and es- tablish the link with ergodic theory. We recall classical recurrence conditions from the theory of Markov chains (Doeblin condition, Dobrushin condition, local Doeblin condition). Then we turn to the study of one dimensional diffusions where hitting time moments determine the speed of convergence. We recall Kac’s moments for- mula for hitting times and characterize the speed of convergence to equilibrium under Vertennikov’s drift conditions both in the ergodic theorem and for the total variation distance. In higher dimensional models we show how to use coupling techniques in order to introduce regeneration times that allow to mimic the one dimensional case. Key words : Harris recurrence, polynomial ergodicity, Nummelin splitting. MSC 2000 : 60 J 55, 60 J 35, 60 F 10, 62 M 05 Contents 1 Harris-recurrence and ergodicity: Definitions, basic results and exam- ples 2 1.1 Harris-recurrence ................................. 2 1.1.1 Poincar´einequality for the Ornstein-Uhlenbeck process . 6 1.2 Harris-recurrence, ergodicity and strong mixing . 8 2 Classical recurrence conditions 13 2.1 The case of a finite or countable state space in discrete time . 13 2.2 General state space, discrete time . 17 2.3 Coming back to continuous time . 17 ∗CNRS UMR 8088, D´epartement de Math´ematiques, Universit´ede Cergy-Pontoise, 95 000 CERGY- PONTOISE, France. E-mail: [email protected] 1 3 The case of one-dimensional diffusions 20 3.1 Kac’s moment formula determines the speed of convergence to equilibrium . 22 3.2 Veretennikov’sdriftconditions . 24 3.3 Polynomial ergodicity for one-dimensional diffusions `ala Veretennikov . 26 3.4 Proof of Proposition 3.8 . 29 4 Diffusions in higher dimensions 32 4.1 Regenerative scheme and Harris-recurrence . 32 4.2 Polynomial ergodicity for multidimensional diffusions . 42 5 Recurrence for degenerate diffusions 43 5.1 Local existence of strictly positive transition densities for degenerate diffu- sions having locally smooth coefficients..................... 43 5.2 Control of the return times . 45 6 Appendix : Some basic facts on Harris recurrent Markov processes and Nummelin splitting 46 6.1 Nummelin splitting and regeneration times . 47 6.2 Basic properties of Z ............................... 48 6.3 Regeneration, Chacon-Ornstein’s ratio limit theorem and Harris recurrence of Z ........................................ 50 6.4 Proof of (6.66) . 53 1 Harris-recurrence and ergodicity: Definitions, basic re- sults and examples 1.1 Harris-recurrence Consider a filtered probability space (Ω, , ( t)t 0, (Px)x), where ( t)t 0 is some right A F ≥ F ≥ continuous filtration. Let X =(Xt)t 0 be an ( t)t 0 adapted process defined on (Ω, ). ≥ F ≥ − A We suppose that X takes values in a locally compact Polish space (E, ) and is strong E Markov with respect to ( t)t 0 under Px for any x E. Moreover we assume that X F ≥ ∈ has c`adl`agpaths. The family (Px)x E is a collection of probability measures on (Ω, ) ∈ A such that X = xP almost surely. We write (P ) for the transition semigroup of X. In 0 x− t t particular we assume that the process is homogeneous in time. We start by discussing the main concept of stochastic stability of Markov processes: the recurrence property. Basically, recurrence means that the process “comes back” almost surely and does not escape to as t . In this statement, “coming back” and “not ∞ →∞ escaping to ” have to be defined in a precise manner. But before doing so, let us consider ∞ two very well known examples. Example 1.1 Consider standard Brownian motion (Bt)t 0 in dimension 1,B0 =0. This ≥ 2 process is recurrent in an elementary sense, since lim sup 1 0 (Bt)=1 t { } →∞ almost surely; the process comes back to the point 0 infinitely often and at arbitrary late times. The same is actually true for any other point a IR : Let T =inf t : B = a , ∈ a { t } then we have T < + almost surely and a ∞ 1 a2/t 3/2 Ta e− a t− dt. ∼ √2π | | 2 Proof. Suppose w.l.o.g. that a>0. Let λ>0, then M =exp λB λ t is a martin- t t − 2 gale. Applying the stopping rule, we have for any N 1, ≥ λ2 λBTa N Ta N E0 e ∧ − 2 ∧ =1. Since eλBTa N eλa, dominated convergence gives as N , ∧ ≤ →∞ λ2 λa 2 Ta E0 e − 1 Ta< =1. (1.1) { ∞} Letting λ 0 implies that T < almost surely. (1.1) is the Laplace transform of T → a ∞ a and gives easily the density. For a<0, the result follows by symmetry. Let us consider a second example. Example 1.2 2 dimensional standard Brownian motion (Bt)t 0 is still recurrent, but − ≥ does not come back to points any more. Actually, the following is true: For all sets 2 A (IR ) having positive Lebesgue measure λ(A) > 0, we have lim supt 1A(Bt)=1 ∈B →∞ almost surely. Thus, all Lebesgue positive sets are visited infinitely often by the process. The same is true for one-dimensional Brownian motion. For dimension d 3, Brownian ≥ motion is not recurrent any more. These considerations lead to the following definition. Recall that a measure µ on (E, ) E is called invariant if µP = µ for all t 0, i.e. µ(dx)P (x, dy)=µ(dy). t ≥ E t Definition 1.3 X is called recurrent in the sense of Harris if X possesses a σ finite − invariant measure µ such that for all A , ∈E µ(A) > 0 implies lim sup 1A(Xt)=1Px almost surely for all x E. (1.2) t − ∈ →∞ A deep result of Az´ema, Duflo and Revuz (1969) implies that for a Harris recurrent process, the measure µ is necessarily unique, up to multiplication with a constant. Therefore we can introduce the following classification of Harris recurrent processes. Definition 1.4 Let X be Harris recurrent, with invariant measure µ. If µ(E) < , then ∞ X is called positive recurrent, else null recurrent. 3 Hence, Brownian motion in dimension d = 1 and d = 2 is null recurrent, since Lebesgue measure is its invariant measure. We consider another well known example of a one-dimensional diffusion process which is positive recurrent. It is the recurrent Ornstein Uhlenbeck process. This process is even recurrent in a much stronger sense, we will see that it is exponentially recurrent or exponentially ergodic in a sense that will be defined precisely later. The presentation of the next example is widely inspired by a course given by Reinhard H¨opfner in Bonn in 1995/96, see also H¨opfner (2008), http://www.informatik.uni-mainz.de/ hoepfner/Material- MathStat.html, Kap IX. Example 1.5 Let B be one-dimensional Brownian motion and dX = aX dt + dB ,X = x ,a>0. t − t t 0 0 at Apply the Itˆo-formula to e Xt in order to obtain the explicit solution of the above equation t at a(t s) Xt = e− x0 + e− − dBs. (1.3) 0 In particular, at 1 2at X (e− x , (1 e− )). t ∼N 0 2a − This implies that for any bounded function f, 1 at 2 1 2 (y xe− ) 1 1 2 ∞ − 2σ − ∞ 2 y Ptf(x)= e t f(y)dy e− 2σ f(y)dy, 2πσ2 → √2πσ2 t −∞ −∞ as t , where σ2 = 1 (1 e 2at) and σ2 = 1 . Therefore, X L µ := (0, 1 ), as →∞ t 2a − − 2a t → N 2a t . It is immediate to see that µ is an invariant measure of X. 1 →∞ We show that X is recurrent in the sense of Harris. We have to show that µ(A) > 0 implies that X returns to A almost surely. But X is recurrent in an elementary sense: Suppose that x0 > 0 and take Brownian motion starting from x0 : X0 = B0 = x0. Then b(X )= aX < 0 on [0,τ[, where τ =inf t : X =0. Using the comparison theorem t − t { t } for diffusions, this implies X B for all t<τ.Therefore, τ inf t : B =0 =L T , t ≤ t ≤ { t } x0 where Tx0 is the hitting time of x0 of a standard Brownian motion starting from 0 at time 0. Hence τ< almost surely. The same argument applies to a starting point x < 0. ∞ 0 As a consequence, we can cut the trajectory of X into i.i.d. excursions out of 0. Put R =inf t : X =0 ,R =inf t>R +1:X =0 ,n 1. 1 { t } n+1 { n t } ≥ In the above definition, we take the infimum over all t>Rn +1 (we could as well take t>Rn + ε for any fixed ε>0) in order to ensure that the sequence (Rn)n does not have accumulation points. 1 First proof: We have to show that µPt = µ. But µ =lims Ps, hence for any bounded measurable →∞ function f, µPtf =lims PsPtf =lims Ps+tf = µf, where we have used the strong Feller property →∞ →∞ at 2at 1 of the process. Second proof: Choose X0 independently of (Bt)t,X0 µ. Then e− X0 (0,e− 2a ). 2at 1 1 2at ∼ ∼N This implies by (1.3) that Xt (0,e− ) (0, (1 e− )=µ. ∼N 2a ∗N 2a − 4 Clearly, R < almost surely for all n, R almost surely as n . Moreover, the n ∞ n →∞ →∞ trajectories (XRn+s)s Rn+1 Rn ,n 1, are i.i.d.