Suppose one considers some non-autonomous SDE thereby the Markov transition function is not homogeneous. In order to "recover" some homogeneity, one can consider the "spacetime" or "lifted" coordinates $(t,X_t)$ where $X_t$ solves the SDE where now the Markov transition is now time-homogeneous. My question concerns the spacetime generator and if one solves a PDE with it, due to its structure, is it a parabolic PDE thus classical parabolic results and methods apply?
Let's put some notation and symbols to this. Let $L_X$ denote the generator for this process. Due to the connection SDE has with PDEs, one may wish to solve $$L_Xu = f$$ with some Dirichlet boundary conditions say $u=0$ on $\partial\Omega$ for some open bounded $\Omega$. For simplicity, suppose we are considering the one-dimensional SDE with one-dimensional Brownian motion $W_t$ with sufficiently regular coefficients and non-degenerate $\sigma$
$$dX_t = b(t,X_t) dt + \sigma(t,X_t)dW_t$$ Then the PDE reads explicitly as (retaining partial derivative notation)
$$L_Xu = b\partial_xu + \sigma^2 \partial^2_xu = f $$ In this framework, one would be solving in general an elliptic PDE.
In the world of spacetime coordinates $(t,X_t)$, one has the PDE for $\hat u: R^+ \times R^1 \rightarrow R$ $$L\hat u = \partial_s \hat u + b\partial_x \hat u + \sigma^2 \partial^2_x \hat u = \hat f$$ As this PDE would somewhat reflect the original PDE, one imposes $\hat u(s,x) = u(x) = 0$ if $x \in \partial \Omega$ amongst others (can fill in more if needed) My question is whether this last PDE parabolic or (degenerate) elliptic (although in this spacetime coordinate case, "time" is treated as a spatial variable)? As a follow up, if parabolic, classical parabolic results would apply? And if, degenerate elliptic, any good references on how to solve such PDEs (as in many PDE books one assumes possibly uniformly ellipticity)?