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. Mathematics
Univ. 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