## Electronic Journal of Probability

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

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

#### 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