×

zbMATH — the first resource for mathematics

Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. (English) Zbl 1075.60072
Let \(x\) be a Markov process on \(\mathbb R^ {d}\) with a transition function \(P_ {t}\). Consider the following two hypotheses: (1) There exist a compact set \(C\subset \mathbb R^ {d}\) with a non-empty interior and a point \(y\in \operatorname {Int}(C)\) such that, for any open neighbourhood \(U\ni y\), there is a time \(\tilde t\) satisfying \(P_ {\tilde t}(x,U)>0\) for all \(x\in C\). Moreover, the function \(P_ {t}\) restricted to \(C\times {\mathcal B}(C)\) admits a density \(p_ {t}\) continuous on \(C\times C\), for all \(t\geq 0\). (Note that this assumption implies the well known minorization condition.) (2) Let \((x_ {n})\) be a Markov chain corresponding to the kernel \(P_ {T}\) for some fixed \(T>0\). Assume there exist a function \(V:\mathbb R^ {d}\to \mathopen [1,\infty \mathclose [\) and constants \(\alpha \in \mathopen ]0,1\mathclose [\), \(\beta \in \mathopen [0,\infty \mathclose [\) such that \(V(x)\to \infty \) as \(| x| \to \infty \) and \(\mathbf E[V(x_ {n+1})\mid {\mathcal F}_ {n}] \leq \alpha V(x_ {n}) + \beta \), \({\mathcal F}_ {n}\) being the \(\sigma \)-algebra generated by \(x_ 1,\ldots ,x_ {n}\). (Note that this assumption is satisfied if \(AV\leq -aV + b\) on \(\mathbb R^ {d}\) for some \(a,b>0\), where \(A\) is the generator of the process \(x\)). Suppose that for some \(T>0\) the embedded chain \((x_ {n})\) obeys Hypotheses (1), (2), with the choice \(C = \{x;\; V(x)\leq 2\beta (\gamma - \alpha )^ {-1} \}\), for some \(\gamma \in \mathopen ]\alpha ^ {1/2},1\mathclose [\). Then there exists a unique invariant measure \(\pi \) for the chain \((x_ {n})\) and the chain is geometrically ergodic, that is, \(| \mathbf E_ {z} f(x_ {n})-\pi (f)| \leq Kr^ {n}V(z)\) for some \(K\geq 0\), \(r\in \mathopen ]0,1\mathclose [\), all \(z\in \mathbb R^ {d}\) and all measurable functions \(f\) such that \(| f| \leq V\) on \(\mathbb R^ {d}\). This theorem follows from more general results which may be found in [S. Meyn and R. Tweedie, “Markov chains and stochastic stability” (1993; Zbl 0925.60001)], but to keep the paper self-contained the authors provide an independent proof. The main aim of the paper is to show that the theorem may be applied to several interesting classes of Markov processes defined by stochastic differential equations whose drift is only locally Lipschitz and diffusion matrix may be degenerate.
First, a second-order stochastic differential equation \(dq = p\,dt\), \(dp = -(\gamma p +\nabla F(q))\,dt + \sigma \,dW\), where \(W\) is a \(d\)-dimensional Wiener process, \(\gamma >0\) and \(\sigma \) is an invertible matrix, is studied. It is proven that the process \((q(t),p(t))\) is geometrically ergodic, provided \(F\) is a nonnegative \(C^ \infty \)-function and \(\langle \nabla F(q),q\rangle /2 \geq \beta F(q) +\gamma ^ 2(8-8\beta )^ {-1} \beta (2-\beta )\| q\| ^ 2 - \alpha \) for some \(\alpha >0\) and \(\beta \in \mathopen ]0,1\mathclose [\). Further, an equation \(dx = Y(x)\,dt + \Sigma \,dW\) is considered, where \(W\) is now an \(m\)-dimensional Wiener process, \(m\leq d\), the columns of the matrix \(\Sigma \) are linearly independent, and the drift is either dissipative, \(\langle Y(x),x\rangle \leq \alpha -\beta \| x\| ^ 2\) for some \(\alpha ,\beta >0\), or of a gradient type, \(Y = \nabla F\). Suitable additional assumptions are found so that geometric ergodicity may be established.
In the second part of the paper, the effect of time discretization on these stochastic differential equations is investigated and ergodicity for several approximation methods is proved. The main difficulty comes from the fact that the Lyapunov condition (2) is sensitive to the choice of a discretization method if the drift is not globally Lipschitz, and is inherited only by specially constructed implicit discretizations.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents
65C30 Numerical solutions to stochastic differential and integral equations
Citations:
Zbl 0925.60001
Software:
Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, L.; Kliemann, W., On unique ergodicity for degenerate diffusions, Stochastics, 21, 41-61, (1987) · Zbl 0617.60076
[2] Bakry, D., 1997. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In: Elworthy, K.D. (Ed.), New Trends in Stochastic Analysis (Charingworth, 1994). Longman Scientific & Technical, River Edge, NJ, pp. 43-75.
[3] Bell, D.R., The Malliavin calculus, (1987), Longman Scientific & Technical Harlow · Zbl 0678.60042
[4] Down, D.; Meyn, S.P.; Tweedie, R.L., Exponential and uniform ergodicity of Markov processes, Ann. probab., 23, 1671-1691, (1995) · Zbl 0852.60075
[5] Durrett, R., Stochastic calculus, A practical introduction, (1996), CRC Press Boca Raton, FL · Zbl 0856.60002
[6] E, W.; Mattingly, J.C., Ergodicity for the navier – stokes equation with degenerate random forcingfinite dimensional approximation, Comm. pure appl. math., 54, 1386-1402, (2001) · Zbl 1024.76012
[7] Fayolle, G.; Malyshev, V.A.; Menshikov, M.V., Topics in the constructive theory of countable Markov chains, (1995), Cambridge Univ. Press Cambridge · Zbl 0823.60053
[8] Ford, G.W.; Kac, M., On the quantum Langevin equation, J. statist. phys., 46, 803-810, (1987) · Zbl 0682.60069
[9] Grorud, A.; Talay, D., Approximation of Lyapunov exponents of nonlinear stochastic differential equations, SIAM J. appl. math., 56, 627-650, (1996) · Zbl 0855.60057
[10] Hale, J.K., Asymptotic behavior of dissipative systems, (1988), American Mathematical Society Providence, RI · Zbl 0642.58013
[11] Hansen, N.R., 2002. Geometric ergodicity of discrete time approximations to multivariante diffusions. Preprint.
[12] Has’minskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoff and Noordhoff Alphen a/d Rijin
[13] Kifer, Y., Random perturbations of dynamical systems, (1988), Birkhäuser, Boston Inc Boston, MA · Zbl 0659.58003
[14] Kliemann, W., Recurrence and invariant measures for degenerate diffusions, Ann. probab., 15, 690-707, (1987) · Zbl 0625.60091
[15] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1991), Springer New York · Zbl 0731.60050
[16] Kunita, H., 1978. Supports of diffusion processes and controllability problems. In: Itô, K. (Ed.), Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). Wiley, New York, pp. 163-185.
[17] Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 120-216. · Zbl 0957.60016
[18] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050
[19] The MathWorks, Inc., 1992. MATLAB User’s Guide. Natick, MA.
[20] Mattingly, J., Stuart, A.M., in Preparation. Ergodicity for adaptive approximations of SDEs.
[21] Meyn, S.; Tweedie, R.L., Stochastic stability of Markov chains, (1992), Springer New York · Zbl 0925.60001
[22] Meyn, S.P., Tweedie, R.L., 1993. Stability of Markovian processes, I, II and III Adv. Appl. Probab. 24 542-574, 25, 487-517 and 25, 518-548. · Zbl 0757.60061
[23] Norris, J., 1986. Simplified Malliavin calculus. In: Azéma, J., Yor, M. (Eds.), Séminaire de Probabilités, XX, 1984/85, Springer, Berlin, pp. 101-130.
[24] Orey, S., 1971. Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Co., London, Van Nostrand Reinhold Mathematical Studies, No. 34. · Zbl 0295.60054
[25] Pages, G., Sur quelques algorithmes recursifs pour LES probabilities numeriques, ESAIM: probab. statist., 5, 141-170, (2001) · Zbl 0998.60073
[26] Roberts, G.; Rosenthal, J.S., Optimal scaling of discrete approximations to Langevin diffusions, J. roy. statist. soc., 60, 255-268, (1998) · Zbl 0913.60060
[27] Roberts, G.O.; Tweedie, R.L., Exponential convergence of Langevin diffusions and their discrete approximations, Bernoulli, 2, 341-363, (1996) · Zbl 0870.60027
[28] Roberts, G.O.; Tweedie, R.L., Bounds on regeneration times and convergence rates for Markov chains, Stochastic proc. applic., 80, 211-229, (1999) · Zbl 0961.60066
[29] Rogers, L.C.G., Williams, D., 2000. Diffusions, Markov processes and Martingales, Vol. 2, 2nd Edition. Cambridge University Press, Cambridge, reprinted. · Zbl 0949.60003
[30] Rosenthal, J.S., Minorization conditions and convergence rates for Markov chain Monte Carlo, J. amer. statist. assoc., 90, 558-566, (1995) · Zbl 0824.60077
[31] Sanz-Serna, J.M.; Stuart, A.M., Ergodicity of dissipative differential equations subject to random impulses, J. differential equations, 155, 262-284, (1999) · Zbl 0934.34047
[32] Shardlow, T.; Stuart, A.M., A perturbation theory for ergodic Markov chains with application to numerical approximation, SIAM J. numer. anal., 37, 1120-1137, (2000) · Zbl 0961.60068
[33] Sparrow, C., The Lorenz equations, bifurcations, chaos and strange attractors, (1982), Springer Berlin · Zbl 0504.58001
[34] Stramer, O.; Tweedie, R.L., Langevin-type models idiffusions with given stationary distributions, and their discretizations, Method. comput. appl. probab., 1, 283-306, (1999) · Zbl 0947.60071
[35] Stramer, O.; Tweedie, R.L., Langevin-type models iiself-targeting candidates for MCMC algorithms, Methodol. comput. appl. probab., 1, 307-328, (1999) · Zbl 0946.60063
[36] Stroock, D.W., 1982. Lectures on Topics in Stochastic Differential Equations. Tata Institute of Fundamental Research, Bombay (With notes by Satyajit Karmakar). · Zbl 0516.60065
[37] Stroock, D.W., Varadhan, S.R.S., 1972. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Math. Stat. and Prob., Vol. III, pp. 333-360. · Zbl 0255.60056
[38] Stuart, A.M.; Humphries, A.R., Dynamical systems and numerical analysis, (1996), Cambridge University Press Cambridge · Zbl 0869.65043
[39] Talay, D., Second-order discretization schemes for stochastic differential systems for the computation of the invariant law, Stochastics stochastics rep., 29, 13-36, (1990) · Zbl 0697.60066
[40] Talay, D., Approximation of upper Lyapunov exponents of bilinear stochastic differential systems, SIAM J. numer. anal., 28, 1141-1164, (1991) · Zbl 0738.65106
[41] Talay, D., 1999. Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz co-efficients. Appears In: Bouc, R., Soize, C. (Eds.), Progress in Stochastic Structural Dynamics, Vol. 152. Publication du L.M.A.-CNRS.
[42] Talay, D., 2002. Stochastic Hamiltonian dissipative systems with non globally Lipschitz co-efficients: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Proc. Rel. Fields, submitted. · Zbl 1011.60039
[43] Tropper, M.M., Ergodic properties and quasideterministic properties of finite-dimensional stochastic systems, J. statist. phys., 17, 491-509, (1977) · Zbl 1255.60174
[44] Veretennikov, A.Y., On polynomial mixing bounds for stochastic differential equations, Stochastic process. appl., 70, 115-127, (1997) · Zbl 0911.60042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.