## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- 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

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

#### 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

**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