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We prove existence, uniqueness and Lipschitz dependence on the initial datumfor mild solutions of stochastic partial differential equations with Lipschitzcoefficients driven by Wiener and Poisson noise. Under additional assumptions,we prove Gateaux and Frechet differentiability of solutions with respect to theinitial datum. As an application, we obtain gradient estimates for theresolvent associated to the mild solution. Finally, we prove the strong Fellerproperty of the associated semigroup.

Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise