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

Kusuoka-Stroock gradient bounds for the solution of the filtering equation