Professor Palle Jorgensen; University of Iowa

Stochastic Processes and Dual Pairs of Operators

Abstract. A new harmonic analysis for Krein-Feller operators is presented.  We first show that a Krein-Feller operator is associated to pairs of measures assumed positive, sigma-finite, and non-atomic. Our approach to the problem is via dual pairs of operators, referring to the corresponding pairs of L^2 Hilbert spaces.
The operator pairs used for our Krein-Feller analysis consist of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. We show how this approach yields a rigorous analysis of the corresponding Krein-Feller operators as closable quadratic forms. For given measures, including the case of fractal measures, we compute the associated diffusion, Markov processes, semigroups, Dirichlet forms, and generalized heat equations. Key tools for our analysis are the use of associated reproducing kernel Hilbert spaces (RKHSs), time-change, and Gaussian fields.

ZOOM ID: 965 4759 0309