## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 22 (2017), paper no. 52, 40 pp.

### Construction and Skorohod representation of a fractional $K$-rough path

#### Abstract

We go ahead with the study initiated in [7] about a heat-equation model with non-linear perturbation driven by a space-time fractional noise. Using general results from Hairer’s theory of regularity structures, the analysis reduces to the construction of a so-called $K$-rough path (above the noise), a notion we introduce here as a compromise between regularity structures formalism and rough paths theory. The exhibition of such a $K$-rough path at order three allows us to cover the whole roughness domain that extends up to the standard space-time white noise situation. We also provide a representation of this abstract $K$-rough path in terms of Skorohod stochastic integrals.

#### Article information

**Source**

Electron. J. Probab., Volume 22 (2017), paper no. 52, 40 pp.

**Dates**

Received: 21 July 2016

Accepted: 20 May 2017

First available in Project Euclid: 21 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1498010465

**Digital Object Identifier**

doi:10.1214/17-EJP69

**Mathematical Reviews number (MathSciNet)**

MR3666015

**Zentralblatt MATH identifier**

1368.60066

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G22: Fractional processes, including fractional Brownian motion 60H07: Stochastic calculus of variations and the Malliavin calculus

**Keywords**

stochastic PDEs fractional noise rough paths theory regularity structures theory Malliavin calculus

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Deya, Aurélien. Construction and Skorohod representation of a fractional $K$-rough path. Electron. J. Probab. 22 (2017), paper no. 52, 40 pp. doi:10.1214/17-EJP69. https://projecteuclid.org/euclid.ejp/1498010465