Energy forms, Hamiltonians, and distorted Brownian paths

Abstract

We study the Hamiltonians for nonrelativistic quantum mechanics—and for the heat equation—in terms of energy forms ∫∇f∇fd’gm, where dμ is a positive, not necessarily finite measure on Rn. We cover the cases of very singular interactions (e.g., N particles in R3 interacting by two-body ’’δ potentials’’). We also exhibit, on the other hand, regularity conditions for μ in order that H be realized as a perturbation of the Laplacian by a measurable or generalized functon. The Hamiltonians defined by energy forms alwasy generate Markov semigroups, and the associated processes are symmetric homogeneous strong Markov diffusion Hunt processes with continuous paths realizations. Ergodicity, transiency, and recurrency are also discussed. The associated stochastic differential equaiton is discussed in the situaion were μ is finite but the drift coefficient is only restricted to be l (Rr,dμ). These results provide a large class of examples where solutions of the heat equaion can be expressed by averages with respect to the constructed Hut processes, rather than with respect to Brownian motion. This is discussed in relation to recent work of Ezawa, Klauder, and Shepp, as well as of Hida and Streit.