There are several ways to define "diffusion process", and one of the most frequently cited definitions uses the "transition density". However, in some cases, it seems that the transition density is hard to derive. For example, consider the following SDE $$d X_t=\mu(X_t)dt+dB_t,$$ where $\mu:\Re\to\Re$ is Borel and bounded, and $B_t$ is the standard Brownian motion. We know that this SDE has a unique strong solution $\{X_t\}_{t\ge 0}$.
My question is: How can we confirm that $\{X_t\}_{t\ge 0}$ is a diffusion process? Or more specifically, even assume that $\mu$ takes a very simple form: $\mu(x)=\mathbf 1_{\{x>0\}}(x)$, it still seems that the transition density is hard to derive.
I hope someone can give me some advices or references. Thanks.