## Electronic Journal of Probability

### Long time asymptotics of unbounded additive functionals of Markov processes

Fuqing Gao

#### 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
Accepted: 6 September 2017
First available in Project Euclid: 1 November 2017

https://projecteuclid.org/euclid.ejp/1509501717

Digital Object Identifier
doi:10.1214/17-EJP104

Mathematical Reviews number (MathSciNet)
MR3724562

Zentralblatt MATH identifier
06827071

#### 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

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