Comparison theorem and estimates for transition probability densities of diffusion processes

We establish several comparison theorems for the transition probability density p b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.