# Fr\'echet differentiability of mild solutions to SPDEs with respect to the initial datum

## TL;DR

This paper proves high-order Fréchet differentiability of mild solutions to jump-diffusion SPDEs in Hilbert spaces, enabling the construction of classical solutions to related non-local Kolmogorov equations.

## Contribution

It establishes n-th order Fréchet differentiability of solutions with respect to initial data for a broad class of jump-diffusion SPDEs, extending previous differentiability results.

## Key findings

- Proved well-posedness of the SPDEs in the mild sense.
- Established first-order Gâteaux differentiability of solutions.
- Demonstrated higher-order Fréchet differentiability of solutions.

## Abstract

We establish n-th order Fr\'echet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order G\^ateaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.