Asymptotic behavior of transition density of diffusion markov process with small diffusion †

We consider the asymptotic behavior of transition density p,x,y) of an n-dimensional diffusion x(), where dx(t) = s dw(t) + b(x(t)) dt. Girsanov transformation is used to show that pE(t,x,y) is equal to the expectation of certain functional of a diffusion process depending on a small parameter. Through machinery developed by Varadhan and Azencott for the large deviation theory, we first obtain a Ventsel-Friedlin type result, i.e. lim e2pe(t,x,y)=V(t,x,y) exists. Then, under further condition on t,x,y, we get the expansion p (t,x,y)=(2nte2)~n/2expl-(V(t,x,y))/e2)2 (T0(t, x, y) + s2T1(t, x, y) +... +e2NTN(U x, y) + 0(e2N+2))