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Feynman’s operational calculus and the stochastic functional calculus in Hilbert space. (English) Zbl 1252.47015
Hassell, Andrew (ed.) et al., The AMSI-ANU workshop on spectral theory and harmonic analysis. Proceedings of the workshop, Canberra, Australia, July 13–17, 2009. Canberra: Australian National University, Centre for Mathematics and its Applications (ISBN 978-0-7315-5208-5). Proceedings of the Centre for Mathematics and its Applications, Australian National University 44, 183-210 (2010).
Let $$A_1, A_2$$ be bounded linear operators acting on a Banach space $$E$$. A pair $$(\mu_1,\mu_2)$$ of continuous probability measures on $$[0,1]$$ determines a functional calculus $$f \to f_{\mu_1,\mu_2}(A_1,A_2)$$ for analytic functions $$f$$ by weighting all possible orderings of operator products of $$A_1$$ and $$A_2$$ via the probability measures $$\mu_1$$ and $$\mu_2$$. Replacing $$\mu_1$$ by Lebesgue measure $$\lambda$$ on $$[0,t]$$ and $$\mu_2$$ by stochastic integration with respect to a Wiener process $$W$$, the author shows that there exists a functional calculus $$f \to f_{\lambda,W;t}(A+B)$$ for bounded holomorphic functions $$f$$ if $$A$$ is a densely defined Hilbert space operator with a bounded holomorphic functional calculus and $$B$$ is small compared to $$A$$ relative to a square function norm.
For the entire collection see [Zbl 1218.47003].
##### MSC:
 47A60 Functional calculus for linear operators 47D06 One-parameter semigroups and linear evolution equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)