Electronic Journal of Probability

Long time asymptotics of unbounded additive functionals of Markov processes 

Fuqing Gao

Full-text: Open access

Abstract

Under hypercontractivity and $L_p$-integrability of transition density for some $p>1$, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on the coefficients and prove that these asymptotics solve the related ergodic Hamilton-Jacobi-Bellman equation with nonsmooth and quadratic growth cost in viscosity sense.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 94, 21 pp.

Dates
Received: 29 March 2017
Accepted: 6 September 2017
First available in Project Euclid: 1 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1509501717

Digital Object Identifier
doi:10.1214/17-EJP104

Mathematical Reviews number (MathSciNet)
MR3724562

Zentralblatt MATH identifier
06827071

Subjects
Primary: 60F10: Large deviations 60J55: Local time and additive functionals 60H10: Stochastic ordinary differential equations [See also 34F05] 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]

Keywords
additive functional hypercontractivity long time asymptotics moderate deviation perturbation theory

Rights
Creative Commons Attribution 4.0 International License.

Citation

Gao, Fuqing. Long time asymptotics of unbounded additive functionals of Markov processes. Electron. J. Probab. 22 (2017), paper no. 94, 21 pp. doi:10.1214/17-EJP104. https://projecteuclid.org/euclid.ejp/1509501717


Export citation

References

  • [1] Bitseki Penda, V., Djellout, H., Dumaz, L., Merlevede, F., Proia, F.: Moderate deviations of functional of Markov processes. ESAIM: Proceedings and Surveys 44, (2014), 214–238.
  • [2] Bouleau, N. and Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. Walter de Gruyter and Co., Berlin, 1991.
  • [3] Cattiaux, P., Dai Pra, P. and Rœlly, S.: A constructive approach to a class of ergodic HJB equatons with unbounded and nonsmooth cost, SIAM J. Control Optim. 47, (2008), 2598–2615.
  • [4] Chen, X.: Limit Theorems for Functionals of Ergodic Markov Chains with General State Space. Memoirs of AMS 139, (1999), No. 664.
  • [5] Deuschel, J. D. and Stroock, D. W. Large Deviations, volume of Pure and Applied Mathematics 137, Academic Press, 1989.
  • [6] Feng, J. and Kurtz, T. G.: Large Deviations for Stochastic Processes. Math. Surveys Monogr. 131, American Mathematical Society, Providence, RI, 2006.
  • [7] Fleming, W.H. and McEneaney, W.M.: Risk sensitive control on an infinite time horizon. SIAM J. Control Optim. 33, (1995), 1881–1915.
  • [8] Gao, F. Q.: Moderate deviations for martingales and mixing random processes. Stochastic Processes and their Applications 61, (1996), 263–275.
  • [9] Ichihara, N. and Sheu, S.-J.: Large time behavior of solutions of Hamilton-Jacobi-Bellman equations with quadratic nonlinearity in gradients. SIAM J. Math. Anal.45, (2013), 279–306.
  • [10] Ikede, N. and Watanable,S.: Stochastic Differential Equation and Diffusion Processes, North-Holland, Amsterdam, 1989.
  • [11] Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2012.
  • [12] Kato, T.:Perturbation Theory for Linear Operators. Springer, 1984.
  • [13] Kontoyiannis, I. and Meyn, S. P.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab., 13, (2003), 304–362.
  • [14] Kontoyiannis, I. and Meyn, S. P.: Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10, (2005), 61–123.
  • [15] Kotecký, R. and Preiss, D.: Cluster expansions for abstract polymer models. Comm. Math. Phys. 103, (1986), 491–498.
  • [16] H. Kaise and S.J. Sheu, On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2006), 284–320.
  • [17] Lezaud, P.: Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8, (1998), 849–867.
  • [18] Lezaud, P.: Chernoff and Berry-Essen’s inequalities for Markov processes. ESAIM: Probability and Statistics 5, (2001), 183–201
  • [19] McEneaney, W.M. and Ito, K.: Infinite time-horizon risk-sensitive systems with quadratic growth. in Proceedings of the 36th IEEE Conference on Decision and Control. IEEE Press, Piscataway, NJ, 1997, pp. 3413–3418.
  • [20] Nualart, D.: The Malliavin Calculus and Related Topics. Springer Verlag, Berlin, 2006.
  • [21] Stroock, D. W. and Varadhan, S. R. S.: Multidimentional Diffusion Processes. Springer, Berlin, 1979.
  • [22] Robertson, S. and Xing, H.: Large time behavior of solutions to semiLinear equations with quadratic growth in the gradient. SIAM J. Control Optim. 53, (2015), 185–212.
  • [23] Wang, F. Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109, (1997), 417–424.
  • [24] Wang, F. Y.: Functional Inequalities, Markov Processes, and Spectral Theory. Science Press, Beijing, 2004.
  • [25] Wang, F. Y.: Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex mainfolds. Ann. Probab. 39, (2011), 1449–1467.
  • [26] Wu, L.M.: Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23, (1995), 420–445.
  • [27] Wu, L.M.: Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172, (2000), 301–376.