Quadratic forms corresponding to the generalized schrödinger semigroups

J. Glover, M. Rao, H. Ŝikić, R. Song

Research output: Contribution to journalArticlepeer-review

Abstract

Suppose that Xt is the standard Brownian motion in Rd, d ≥ 3, that ρ ∈ H1(Rd) is a bounded continuous function such that |∇ρ|2 belongs to the Kato class and μ is a measure belonging to the Kato class. Let A[ρ]t be defined as A[ρ]t = ρ(Xt) - ρ(X0) - ∫t0 ∇ρ(Xs) · dXs, and let Aμt be the continuous additive functional with μ as its Revuz measure. Define At as the sum of the two additive functionals above. Then the semigroup defined as Ttf(x) = Ex(eAtf(Xt)) is called a generalized Schrödinger semigroup. In this paper we identify the quadratic form corresponding to (Tt) as (E ̃, H1(Rd)) with E ̃(u, v) = 1 2 ∫ ∇u(x) · ∇v(x) dx + ∫Rd ∇(uv)(x) · ∇ρ(x) dx - ∫Rdu(x) v(x) μ(dx).

Original languageEnglish (US)
Pages (from-to)358-378
Number of pages21
JournalJournal of Functional Analysis
Volume125
Issue number2
DOIs
StatePublished - Nov 1 1994
Externally publishedYes

ASJC Scopus subject areas

  • Analysis

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