## The Annals of Probability

- Ann. Probab.
- Volume 40, Number 3 (2012), 1041-1068.

### Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2

Yaozhong Hu, Fei Lu, and David Nualart

#### Abstract

In this paper, a Feynman–Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter *H* < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman–Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malliavin calculus is used to show that the Feynman–Kac integral is the weak solution to the stochastic partial differential equation.

#### Article information

**Source**

Ann. Probab., Volume 40, Number 3 (2012), 1041-1068.

**Dates**

First available in Project Euclid: 4 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1336136058

**Digital Object Identifier**

doi:10.1214/11-AOP649

**Mathematical Reviews number (MathSciNet)**

MR2962086

**Zentralblatt MATH identifier**

1253.60074

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G22: Fractional processes, including fractional Brownian motion 60H05: Stochastic integrals 60H30: Applications of stochastic analysis (to PDE, etc.) 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

**Keywords**

Feynman–Kac integral Feynman–Kac formula stochastic partial differential equations fractional Brownian field nonlinear stochastic integral fractional calculus

#### Citation

Hu, Yaozhong; Lu, Fei; Nualart, David. Feynman–Kac formula for the heat equation driven by fractional noise with Hurst parameter H < 1/2. Ann. Probab. 40 (2012), no. 3, 1041--1068. doi:10.1214/11-AOP649. https://projecteuclid.org/euclid.aop/1336136058