Proceedings of the Centre for Mathematics and its Applications

Feynman's Operational Calculus and the Stochastic Functional Calculus in Hilbert Space

Brian Jefferies

Full-text: Open access

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 $t \rightarrowtail e_{\lambda,W;t}^{A+B}x, t \geq 0$.

Article information

Source
The AMSI-ANU workshop on spectral theory and harmonic analysis. Andrew Hassell, Alan McIntosh and Robert Taggart, eds. Proceedings of the Centre for Mathematics and its Applications, v. 44. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2010), 183-210

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416320878

Mathematical Reviews number (MathSciNet)
MR2655383

Zentralblatt MATH identifier
1252.47015

Citation

Jefferies, Brian. Feynman's Operational Calculus and the Stochastic Functional Calculus in Hilbert Space. The AMSI–ANU Workshop on Spectral Theory and Harmonic Analysis, 183--210, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2010. https://projecteuclid.org/euclid.pcma/1416320878


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