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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].
47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)