Is there a way to compute transition operators for Markov processes? To ask something much more tractable, suppose I have an Ito diffusion $$dX_t \ = \ \sigma(X_t) dB_t \ + \ b(X_t) dt$$

(or given by its generator). Is there any way to write down the transition kernels for the process $X_t$ from the above SDE? Also, maybe with assumptions on the coefficients $\sigma, b$, is there any way to establish any niceness in the sense of $L^p(\mathbb{R})$ for the transition kernels given just the diffusion? I'm guessing the drift component gives Dirac masses along some curve defined by $b(x)$ and the diffusion component gives a Gaussian process, but I'm not seeing quite how to write down the kernels themselves...

Thanks!!

Quick edit: I know one can solve the corresponding Fokker-Planck equation (with a given initial condition) with Green's functions, Fourier series, etc., but is there another way to detect any properties like $L^p$-regularity without PDE methods?