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?