Variational Calculation of Laplace transforms via Entropy on Wiener Space and some Applications

Abstract

Let $(W,H,\mu)$ be the classical Wiener space where $H$ is the Cameron-Martin space which consists of the primitives of the elements of $L^2([0,1],\,dt)\otimes \R^d$, we denote by $L^2_a(\mu,H)$ the equivalence classes w.r.t. $dt\times d\mu$ whose Lebesgue densities $s\to\dot{u}(s,w)$ are almost surely adapted to the canonical Brownian filtration. If $f$ is a Wiener functional s.t. $\frac{1}{E[e^{-f}]}e^{-f}d\mu$ is of finite relative entropy w.r.t. $\mu$, we prove that \beaa J_\star&=& \inf\left(E_\mu\left[f\circ U+\half |u|_H^2\right]: u\in L_a^2(\mu,H)\right)\\ &\geq&-\log E_\mu[e^{-f}]=\inf\left(\int_W fd\ga+H(\ga|\mu):\,\nu\in P(W)\right) \eeaa where $P(W)$ is the set of probability measures on $(W,\calB(W))$ and $H(\ga|\mu)$ is the relative entropy of $\ga$ w.r.t. $\mu$. 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 $u_0$ if and only if $U_0=I_W+u_0$ is almost surely invertible and $$ \frac{dU_0\mu}{d\mu}=\frac{e^{-f}}{E_\mu[e^{-f}]} $$ and then $u_0$ is unique. To do this is we prove the theorem which says that the relative entropy of $U_0\mu$ is equal to the energy of $u_0$ if and only if it has a $\mu$-a.s. left inverse. We use these results to prove the strong existence of the solutions of stochastic differentail equations with singular (functional) drifts and also to prove the non-existence of strong solutions of some stochastic differential equations. \noindent {\sl Keywords:} Invertibility, entropy, Girsanov theorem, variational calculus, Malliavin calculus, large deviations