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

Abstract. Let A1, A2 be bounded linear operators acting on a Banach space E. A pair (μ1, μ2) of continuous probability measures on [0, 1] determines a functional calculus f 7−→ fμ1,μ2(A1, A2) for analytic functions f by weighting all possible orderings of operator products of A1 and A2 via the probability measures μ1 and μ2. For example, f 7−→ fμ,μ(A1, A2) is the Weyl functional calculus with equally weighted operator products. Replacing μ1 by Lebesgue measure ∏ on [0, t] and μ2 by stochastic integration with respect to a Wiener process W , we show that there exists a functional calculus f 7−→ f∏,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 dXt = AXtdt + BXtdWt, X0 = x, is represented as t 7−→ e ∏,W ;tx, t ≥ 0.