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VOL. 44 | 2010 Feynman's Operational Calculus and the Stochastic Functional Calculus in Hilbert Space
Brian Jefferies

Editor(s) Andrew Hassell, Alan McIntosh, Robert Taggart

Abstract

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 $f \rightarrowtail f_{\mu1,|mu2}(A_1, A_2)$ 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 $\mu_2$. For example, $f \rightarrowtail f_{\mu,\mu}(A_1, A_2)$ is the Weyl functional calculus with equally weighted operator products. Replacing $\mu_1$ by Lebesque measure $\lambda$ on $[0,t]$ and $\mu_2$ by stochastic integration with respect to a Winer process $W$, we show that there exists a functional calculus $f \rightarrowtail f_{\lamda,w;t}(A + B)$ 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. By this means, the solution of the stochastic evolution equation $dX_t = AX_tdt + BX_tdW_t, X_0 = x$, is represented as $t \rightarrowtail e_{\lambda,W;t}^{A+B}x, t \geq 0$.

Information

Published: 1 January 2010
First available in Project Euclid: 18 November 2014

zbMATH: 1252.47015
MathSciNet: MR2655383

Rights: Copyright © 2010, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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