## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 5 (2017), 3336-3384.

### The Feynman–Kac formula and Harnack inequality for degenerate diffusions

Charles L. Epstein and Camelia A. Pop

#### Abstract

We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [*SIAM J. Math. Anal.* **42** (2010) 568–608; *Degenerate Diffusion Operators Arising in Population Biology* (2013) Princeton University Press; *Applied Mathematics Research Express* (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman–Kac formulas concerning the assumptions on the degeneracy of the diffusion matrix, the boundedness of the drift coefficients and the a priori regularity of the weak solutions.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 5 (2017), 3336-3384.

**Dates**

Received: May 2015

Revised: July 2016

First available in Project Euclid: 23 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1506132040

**Digital Object Identifier**

doi:10.1214/16-AOP1138

**Mathematical Reviews number (MathSciNet)**

MR3706745

**Zentralblatt MATH identifier**

06812207

**Subjects**

Primary: 35J90

Secondary: 60J60: Diffusion processes [See also 58J65]

**Keywords**

Degenerate elliptic equations degenerate diffusions generalized Kimura diffusions Markov processes Feynman–Kac formulas Girsanov formula weighted Sobolev spaces anisotropic Hölder spaces

#### Citation

Epstein, Charles L.; Pop, Camelia A. The Feynman–Kac formula and Harnack inequality for degenerate diffusions. Ann. Probab. 45 (2017), no. 5, 3336--3384. doi:10.1214/16-AOP1138. https://projecteuclid.org/euclid.aop/1506132040