Do general additive-noise SDEs on $\mathbb{R}^d$ have finite, strictly positive, transition probability densities?

Let $b \colon \mathbb{R}^d \to \mathbb{R}^d$ be a bounded measurable function.

Let $\{P_t(x,A): t \geq 0, x \in \mathbb{R}^d, A \in \mathcal{B}(\mathbb{R}^d)\}$ denote the family of transition probabilities associated with the SDE $$dX_t \ = \ b(X_t)dt + dW_t$$ where $(W_t)$ is a $d$-dimensional Wiener process.

Is it necessarily the case that the measure $P_t(x,\cdot)$ is equivalent to the Lebesgue measure on $\mathbb{R}^d$ for all $x \in \mathbb{R}^d$ and $t > 0$? If it is not true in general, does it become true when $b$ is taken to be Lipschitz?

References would be very much appreciated. Thank you.

Remark: In the case that $d=1$, the absolute continuity of $P_t(x,\cdot)$ covered by the answer to the question Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

PS: If the general case that $b$ is a bounded measurable function is not known, I would still very much appreciate an answer for the more specific case that $b$ is Lipschitz! (I suppose the case that $b$ is $C^1$ is covered by Hörmander's theorem, but I don't know if this can be extended to more general Lipschitz $b$.)

• For the Lipschitz case, a work by Bouleau & Hirsch might be of interest (eudml.org/doc/113542). – saz Feb 26 '16 at 16:52