General question: When are the transition kernels corresponding to the SDE
$$ dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \quad t\in[0,1], \tag{$\heartsuit$}$$
absolutely continuous w.r.t. Lebesgue's measure? Or, if you prefer to talk about PDEs, when is there a fundamental solution for the equation $(\partial_t+\mathcal{L})u=f$ with
$$ \mathcal{L}= \frac12\sum_{i,j=1}^N (\sigma\sigma^\top)_{ij}\partial_{ij}+\sum_{i=1}^N b_i\partial_i \quad ? $$
More nuanced question: Let $U\subset\Bbb R^N$ and assume that $b\colon [0,1]\times U\to\Bbb R^N$ and $\sigma\colon [0,1]\times U\to\Bbb R^{N\times N}$ are locally Lipschitz-continuous. Assume that for given $X_0\in U$ the SDE ($\heartsuit$) has a unique strong solution taking values in $U$. Which additional assumptions secure the existence of transition densities, i.e. measurable functions $p_{s,t}\colon U\times U \to[0,\infty)$ with
$$ E[f(X_t)|X_s=x]=\int_U p_{s,t}(x,y)f(y)dy \quad \forall\; f\in L^\infty(U), \, x\in U,\, 1\ge t\ge s \ge 0 \quad ?$$
Does it help to simply assume $\sigma$ to be uniformly elliptic in the sense that
$$ \exists \; \sigma_0 >0 : \quad \xi^\top (\sigma\sigma^\top)(t,x) \xi \ge \sigma_0 |\xi|^2 \quad \forall \; x \in U, \xi \in \Bbb R^N, t\in[0,1] \quad?$$
Context: Assuming $U=\Bbb R^N$, higher regularity (maybe boundedness) of the coefficient functions and some form of ellipticity for $\sigma$ indeed yields the existence of transition densities. The usual modern approach to questions in this direction involves (variants of) Hörmander's condition and methods from Malliavin calculus. I wonder to which extend the regularity conditions can be relaxed when one has uniform ellipticity which is a lot stronger than Hörmander's condition. I feel like this should be known rather well, but I just can't find results in the literature where the above setting is discussed.