Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to

$$\frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$

for some elliptic differential operator $L$ where $f$ is not continuous?

Or about the stochastic representations in terms of a diffusion $X$,

$$u(t,x) = E^{t,x}(f(X_T)),$$

where $f$ is not continuous?

What about the case where the set of discontinuities has measure zero?

• What is $L$? $~~~~$ – Andrew Mar 14 '15 at 21:23
• What exactly do you want to know? If $f$ is bounded and $L$ is nice enough then the solution can be represented via the Poisson potential as usual. In all the points of continuity of $f$ there will be limit $u(x,t)\to f(x)$ as $t\to0+$. The proof is the same as in the case of continuous initial function. – Andrew Mar 16 '15 at 9:28
• My question is "Where can I read more about ...", so it would be great if you could give some references for learning and understanding what you just wrote :-) – JSG Mar 16 '15 at 16:51