**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 
<cite cite="ISBN: 0-387-96535-1" mrnumber="1241411" authors="Ioannis Karatzas and Steven E. Shreve">_Ioannis Karatzas and Steven E. Shreve_, MR 1241411 [**Brownian motion and stochastic calculus**](http://dx.doi.org/10.1007/978-1-4684-0302-2)</cite>, 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*][1], 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.

  [1]: http://www.hairer.org/notes/Convergence.pdf