On the differentiability of solutions of stochastic evolution equations with respect to their initial values

TL;DR

This paper proves that solutions to certain stochastic evolution equations are differentiable with respect to initial conditions, given smooth nonlinear coefficients, and establishes enhanced regularity properties of their derivatives.

Contribution

It demonstrates that if the nonlinear coefficients are sufficiently smooth, then the solutions are also smoothly differentiable with respect to initial values, with improved regularity of derivative processes.

Findings

Solutions are n-times Fréchet differentiable w.r.t. initial values.

Derivative processes exhibit enhanced regularity due to smoothing effects.

Results apply to parabolic semilinear stochastic evolution equations.

Abstract

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are $n$-times continuously Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also $n$-times continuously Fr\'{e}chet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes.