Harmonic Analysis and Stochastic Partial Differential Equations: The Stochastic Functional Calculus

Abstract

AbstractIt has been recognised recently that there is a close connection between existence and regularity results for stochastic partial differential equations and functional calculus techniques in harmonic analysis. The connection is made more explicit in this paper with the notion of a stochastic functional calculus.In the deterministic setting, suppose that A1,A2 are 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 \(f\;\mapsto\;f_{\mu_{1},\mu_{2}}(A_1,A_2)\) 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, \(f\;\mapsto\;f_{\mu,\mu}(A_1,A_2)\) is the Weyl functional calculus with equally weighted operator products.Replacing μ1 by Lebesgue measure λ on [0, t] and μ2 by stochastic integration with respect to a Wiener process W, we show that there exists a functional calculus \(f\;\mapsto\;f_{\lambda,W;t}(A+B)\) 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 \(dX_t\;=\;AX_{t}dt+BX_{t}dW_{t},\;X_0\;=\;x\), is represented as \(t\;\mapsto\;e_{\lambda,W;t}^{A+B}x,t\geq\;0.\). We show how to extend some of our results to \(L^{p}-\mathrm{spaces},\;2\leq p < \infty\) and apply them to the regularity of solutions of the Zakai equation.Keywordsfunctional calculusstochastic evolution equationH∞-functional calculussquare functionFeynman’s operational calculus