  We obtain sharp gradient bounds for perturbed diffusion semigroups. In
contrast with existing results, the perturbation is here random and the bounds
obtained are pathwise. Our approach builds on the classical work of Kusuoka and
Stroock [7],[9],[10],[11], and extends their program developed for the heat
semi-group to solutions of stochastic partial differential equations. The work
is motivated by and applied to nonlinear filtering. The analysis allows us to
derive pathwise gradient bounds for the un-normalised conditional distribution
of a partially observed signal. It uses a pathwise representation of the
perturbed semigroup in the spirit of classical work by Ocone [14]. The
estimates we derive have sharp small time asymptotics.
