A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces

Jan van Neerven

A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces

Probability 2026-08-03 v1 Classical Analysis and ODEs Functional Analysis

Authors: Jan van Neerven

Abstract

We prove a Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces. The admissible directions are those for which the deterministic orbit is square integrable with values in the Hilbertian range of the noise coefficient. Under this square function condition, the derivative of the semigroup is represented by a stochastic integral against an explicit deterministic control. The result is a concrete control realisation of the Cameron--Martin formula for the fixed-time Gaussian transition measures.

Keywords

Bismut-Elworthy-Li formula stochastic evolution equations Banach spaces semigroup derivative Cameron-Martin formula

Cite

@article{arxiv.2608.02417,
  title  = {A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces},
  author = {Jan van Neerven},
  journal= {arXiv preprint arXiv:2608.02417},
  year   = {2026}
}

Comments

18 pages, to appear in Czech. Math. J