  We assume that we observe $N$ independent copies of a diffusion process on a
time interval $[0,2T]$. For a given time $t$, we estimate the transition
density $p_t(x,y)$, namely the conditional density of $X_{t + s}$ given $X_s =
x$, under conditions on the diffusion coefficients ensuring that this quantity
exists. We use a least squares projection method on a product of finite
dimensional spaces, prove risk bounds for the estimator and propose an
anisotropic model selection method, relying on several reference norms. A
simulation study illustrates the theoretical part for Ornstein-Uhlenbeck or
square-root (Cox-Ingersoll-Ross) processes.
