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.
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.
"Lower bounds for the density of locally elliptic Itô processes." Ann. Probab. 34 (6) 2406 - 2440, November 2006. https://doi.org/10.1214/009117906000000458