We give lower bounds for the density pT(x, y) of the law of Xt, the solution of dXt=σ(Xt) dBt+b(Xt) dt, X0=x, under the following local ellipticity hypothesis: there exists a deterministic differentiable curve xt, 0≤t≤T, such that x0=x, xT=y and σσ*(xt)>0, for all t∈[0, T]. The lower bound is expressed in terms of a distance related to the skeleton of the diffusion process. This distance appears when we optimize over all the curves which verify the above ellipticity assumption.
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The arguments which lead to the above result work in a general context which includes a large class of Wiener functionals, for example, Itô processes. Our starting point is work of Kohatsu-Higa which presents a general framework including stochastic PDE’s.