The operator-valued Feynman-Kac formula with noncommutative operators

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 - 1980
Externally publishedYes

Fingerprint

Feynman-Kac Formula
Semigroup
Infinitesimal Generator
Operator
Strongly Continuous Semigroups
Commute
Markov Process
Evolution Equation
Initial Value Problem
Contraction
Multiplicative
State Space
Banach space
Restriction

ASJC Scopus subject areas

  • Analysis

Cite this

The operator-valued Feynman-Kac formula with noncommutative operators. / Hagood, John W.

In: Journal of Functional Analysis, Vol. 38, No. 1, 1980, p. 99-117.

Research output: Contribution to journalArticle

@article{a6a1881dd9424dc9b022c38e3d6914d2,
title = "The operator-valued Feynman-Kac formula with noncommutative operators",
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.",
author = "Hagood, {John W}",
year = "1980",
doi = "10.1016/0022-1236(80)90058-0",
language = "English (US)",
volume = "38",
pages = "99--117",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - The operator-valued Feynman-Kac formula with noncommutative operators

AU - Hagood, John W

PY - 1980

Y1 - 1980

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=30244472566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30244472566&partnerID=8YFLogxK

U2 - 10.1016/0022-1236(80)90058-0

DO - 10.1016/0022-1236(80)90058-0

M3 - Article

AN - SCOPUS:30244472566

VL - 38

SP - 99

EP - 117

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -