Variational calculation of Laplace transforms via entropy on Wiener space and applications

Let (W,H,μ) be the classical Wiener space where H is the Cameron-Martin space which consists of the primitives of the elements of L2([0,1],dt)⊗Rd. We denote by La2(μ,H) the equivalence classes w.r.t. dt×dμ whose Lebesgue densities s→u˙(s,w) are almost surely adapted to the canonical Brownian filtration. If f is a Wiener functional s.t. 1E[e−f]e−fdμ is of finite relative entropy w.r.t. μ, we prove thatJ⋆=inf⁡(Eμ[f∘U+12|u|H2]:u∈La2(μ,H))≥−log⁡Eμ[e−f]=inf⁡(∫Wfdγ+H(γ|μ):γ∈P(W)) where P(W) is the set of probability measures on (W,B(W)) and H(γ|μ) is the relative entropy of γ w.r.t. μ. We call f a tamed functional if the inequality above can be replaced with equality. We characterize the class of tamed functionals, which is much larger than the set of essentially bounded Wiener functionals. We show that for a tamed functional the minimization problem of l.h.s. has a solution u0 if and only if U0=IW+u0 is almost surely invertible anddU0μdμ=e−fEμ[e−f] and then u0 is unique. To do this, we prove the theorem which says that the relative entropy of U0μ is equal to the energy of u0 if and only if it has a μ-a.s. left inverse. We use these results to prove the strong existence of the solutions of stochastic differential equations with singular (functional) drifts and also to prove the non-existence of strong solutions of some stochastic differential equations.