Conditioning a Lifted Stochastic System in a Product Space

Abstract

We study the solution $\xi_t$ to an elliptic stochastic equation and the solution $(\xi_t, \eta_t)$ to a lift of this equation, under a condition on the behaviour of $\xi_t$. We obtain a "conditional" equation which gives the conditional behaviour, and we use it to deduce a conditioned version of the Stroock-Varadhan support theorem.