The operator-valued Feynman-Kac formula with noncommutative operators

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Abstract

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = Ex[φ(X(t))] for functions φ mapping E into a Banach space L, has the infinitesimal generator A. For each x ϵ E, let V(x) generate a strongly continuous semigroup Tx(t) on L. An operator-valued Feynman-Kac formula is developed and solutions of the initial value problem ∂u/∂t = Au + V(x)u, u(0) = φ are obtained. Fewer conditions are assumed than in known results; in particular, the semigroups {Tx(t)} need not commute, nor must they be contractions. Evolution equation theory is used to develop a multiplicative operative functional and the corresponding expectation semigroup has the infinitesimal generator A + V(x) on a restriction of the domain of A.

Original languageEnglish (US)
Pages (from-to)99-117
Number of pages19
JournalJournal of Functional Analysis
Volume38
Issue number1
DOIs
StatePublished - Jan 1 1980

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ASJC Scopus subject areas

  • Analysis

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