Time reversal of infinite-dimensional diffusions

Abstract

Let I be a countable index set, and let P be a probability measure on C[0, 1] I such that the coordinate process satisfies an infinite-dimensional stochastic differential equation dX = dW+ b( X, t) dt. In contrast to the finite-dimensional case, the time reversed process X ̂ t=X 1−t cannot always be described by a stochastic differential equation d X ̂ =d W ̂ + b ̂ ( X ̂ ,t)dt ; some bounds on the interaction are needed. We introduce a condition of locally finite entropy which implies such bounds and also smoothness of the conditional densities. This allows us to derive an infinite-dimensional analogue of the classical duality equation b+ b ̂ =∇ logϱ . It is also shown that locally finite entropy holds under some growth and locality conditions on the forward drift which are close to the usual conditions for existence and uniqueness of strong solutions.