We establish a version of the Feynman-Kac formula for the multidimensional
stochastic heat equation with a multiplicative fractional Brownian sheet. We
use the techniques of Malliavin calculus to prove that the process defined by
the Feynman-Kac formula is a weak solution of the stochastic heat equation.
From the Feynman-Kac formula, we establish the smoothness of the density of the
solution and the H\"{o}lder regularity in the space and time variables. We also
derive a Feynman-Kac formula for the stochastic heat equation in the Skorokhod
sense and we obtain the Wiener chaos expansion of the solution.Comment: Published in at http://dx.doi.org/10.1214/10-AOP547 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
