## Proceedings of the Centre for Mathematics and its Applications

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

Brian Jefferies

#### 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