## Electronic Journal of Probability

### SPDEs with affine multiplicative fractional noise in space with index $\frac{1}{4}\langle H\langle\frac{1}{2}$

#### Abstract

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4 < H < 1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and Hölder continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 54, 36 pp.

Dates
Accepted: 21 May 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067160

Digital Object Identifier
doi:10.1214/EJP.v20-3719

Mathematical Reviews number (MathSciNet)
MR3354614

Zentralblatt MATH identifier
1321.60132

Subjects
Secondary: 60H05: Stochastic integrals

Rights

#### Citation

Balan, Raluca; Jolis, Maria; Quer-Sardanyons, Lluis. SPDEs with affine multiplicative fractional noise in space with index $\frac{1}{4}\langle H\langle\frac{1}{2}$. Electron. J. Probab. 20 (2015), paper no. 54, 36 pp. doi:10.1214/EJP.v20-3719. https://projecteuclid.org/euclid.ejp/1465067160

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