The operator-valued Feynman-Kac formula with noncommutative operators

Let X(t) be a right-continuous Markov process with state space E whose expectation semigroup S(t), given by S(t) φ(x) = E_x[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 T_x(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 {T_x(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.