How to find the general solution of a differential equation with a shift, in the form of

$\frac{\partial}{\partial t}g(x,t)=g(x-\Delta,t)+\frac{\partial^2}{\partial x^2} g(x,t)$?

(where $\Delta>0$)

And what about

$\frac{\partial}{\partial t}g(x,t)=g(x,t-\Delta)+\frac{\partial^2}{\partial x^2} g(x,t)$?

Edit1: Here are few follow-up details about my question. Is there a ``nice" way to represent the solution in $x$-space, as opposed to e.g., Fourier? Is the solution real + positive + normalizable? Does it have the correct properties of a probability density function?