# Transition densities for (elliptic) SDEs [or: Fundamental solutions for (elliptic) PDEs]

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.

• Does "Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc." have what you're looking for? – Bananach Apr 29 at 17:03
• Not quite... I cannot find a precise statement about the case of unbounded coefficients. Thanks, though. I'll look through the references in this paper. – Mars Plastic Apr 29 at 17:34
• Take a look at Theorem 2.1 and Theorem 3.1 in the article Absolute continuity of some one-dimensional processes by Fournier and Printems. – saz Apr 29 at 17:38
• @saz Thanks for the link. However, the authors treat only the case where the state space is $\Bbb R$ - neither higher dimensions, nor more general domains. On first glance, the methods seem to be rather specific to the one-dimensional case. – Mars Plastic Apr 29 at 17:47