Computing transition operators for Markov processes

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?

1) Explicit expressions for transition densities

In the case of linear systems with additive noise of the form

$d X(t) = (A X_t + b) \, d t + \sigma \, d B_t$

it is possible to obtain an explicit solution, see Ioannis Karatzas and Steven E. Shreve, MR 1241411 Brownian motion and stochastic calculus, Section 5.6.

There are a few other explicitly solvable cases (in terms of transition density), for example the SDE for the Bessel process

$d X_t = (d-1)/(2X_t) \, d t + B_t$,

and the exponential Brownian motion,

$d X_t = a X_t \, d t + \sigma X_t \, d B_t$,

and quite a few other examples. In general finding the transition density of an SDEs means that you have solved the associated Fokker-Planck equation (equivalently known as forward Kolmogorov equation) which is a parabolic PDE. So finding transition densities is in general of the same difficulty as solving a linear PDE of a particular type, and solving such PDEs explicitly is typically not possible.

Also noteworthy is the Lamperti transform to change from general one-dimensional diffusion coefficients to a diffusion $Z$ with unit diffusion coefficient, by defining $Z_t = \varphi(X_t)$ where $\varphi(x) = \int^x \frac{1}{\sigma(\xi)} \ d \xi$.

I would recommend to consult a basic textbook focussing on explicit computations such as Øksendal's and look at the exercises to get a feel for what is possible in this direction.

2) Regularity of solutions

Of course, if $\sigma = 0$, your solution will be a deterministic trajectory, with corresponding Dirac measure as transition kernel, i.e. not belonging to any $L^p$. A simple and well known result is that if $\sigma(x)$ is globally bounded away from 0 (known as ellipticity), you will have smooth transition densities. The important generalization of this result for multi-dimensional SDEs where some of the components of $\sigma$ may be zero is given by Hörmander's theorem. I recommend Martin Hairer's lecture notes, Convergence of Markov processes, for a gentle introduction.

Once you establish the probability density is smooth (using ellipticity/Hörmander) and vanishing at infinity (corresponding to boundedness in probability), it will be bounded and therefore in $L^p(\mathbb R)$ for any $p \geq 1$. Boundedness in probability can be established using e.g. bounds on $\mathbb E|X(t)|$, see the book by Xuerong Mao, or various other sources.