Comparison theorem and estimates for transition probability densities of diffusion processes
Consider the solution to an \(n\)-dimensional stochastic differential equation with drift coefficient \(b(x)\) and constant diffusion coefficient given by the identity (called Brownian motion with drift \(b\) by the authors). For such processes several comparison theorems for the transition probability density \(p_b(x,t,y)\) are established. Sharp lower and upper bounds in terms of the norm on the vector field \(b\) with explicitly given constants are established. \(b\) is neither assumed to be symmetrical nor to be the gradient of a function. The main result is obtained through comparison with a Bessel process with constant drift. In order to do this, estimates for some mixed moments of the Bessel process are derived.
scientific article; zbMATH DE number 2055259