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

by A. Andersson and A. Jentzen and R. Kurniawan and T. Welti

(Report number 2016-47)

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.

Keywords: stochastic evolution equations, differentiability, initial values

@Techreport{AJKW16_684,
  author = {A. Andersson and A. Jentzen and R. Kurniawan and T. Welti},
  title = {On the differentiability of solutions of stochastic evolution equations with respect to their initial values},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-47.pdf },
  year = {2016}
}

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