A lower bound for the transition density function of a stochastic differential equation

Let W t be a one‐dimensional Brownian motion on the probability space (Ω, F,P ), and let dx t = a (x t ) dt + b(x t )dw t , b 2 (x) > 0 , be a one‐dimensional Ito stochastic differential equation. For a(x) = a 0 + a 1 x + … + a n x n on a bounded interval we obtain a lower bound for p(t,x,y) , the transition density function of the homogeneous Markov process x t , depending directly on the coefficients a 0 , a 1 , …, a n , and b(x).