# Preprint: Weak Existence of a Solution to a Differential Equation Driven by a Very Rough fBm

## Davar Khoshnevisan, Jason Swanson, Yimin Xiao, and Liang Zhang

Abstract. We prove that if $f:{\bf R}\to{\bf R}$ is Lipschitz continuous, then for every $H\in(0\,,1/4]$ there exists a probability space on which we can construct a fractional Brownian motion $X$ with Hurst parameter $H$, together with a process $Y$ that: (i) is Hölder-continuous with Hölder exponent $\gamma$ for any $\gamma\in(0\,,H)$; and (ii) solves the differential equation ${\rm d} Y_t = f(Y_t)\,{\rm d} X_t$. More significantly, we describe the law of the stochastic process $Y$ in terms of the solution to a non-linear stochastic partial differential equation.

Keywords. Stochastic differential equations; rough paths; fractional Brownian motion; fractional Laplacian; the stochastic heat equation.

AMS Classification (2000) 60H10; 60G22; 34F05.

Support. Research supported in part by NSF grant DMS-1307470.

Pre/E-Prints. This paper is available in

 Davar Khoshnevisan Dept. Mathematics Univ. of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090 davar@math.utah.edu Jason Swanson Dept. MathematicsUniv. of Central Florida Orlando, FL 32816-1364 jason@swansonsite.com Yimin Xiao Dept. Statistics & Probability Michigan State Univ. East Lansing, MI 48824-3416 xiao@stt.msu.edu Liang Zhang Dept. Statistics & Probability Michigan State Univ. East Lansing, MI 48824-3416 lzhang81@stt.msu.edu Last Update: September 30, 2013 © 2013 - D. Khoshnevisan, J. Swanson, Y. Xiao, and L. Zhang