In the problem of continuous time random walk, I have a summation of the form $$ P(x,t)=\sum_{N=0}^{\infty} \frac{\exp\left[-x^2/(2N\sigma^2)\right]}{\sqrt{2\pi N \sigma^2}} \left(\frac{t}{\tau}\right)^N \frac{e^{-t/\tau}}{N!} $$
While this summation can't be directly evaluated, is it possible to evaluate the integral where I consider N to vary continuously
$$ P(x,t)=\int_{0}^{\infty}\;dN\; \frac{\exp\left[-x^2/(2N\sigma^2)\right]}{\sqrt{2\pi N \sigma^2}} \left(\frac{t}{\tau}\right)^N \frac{e^{-t/\tau}}{\Gamma(N+1)} $$
As a matter of fact, for large values of $x$ and $t$, the integral can be estimated to be, $$ P(x,t) \sim \exp\left[-\frac{x^2}{4(\sigma^2/\tau) t} \right] $$
So, can we evaluate the integral through some asymptotics, such that I get the corrections to the Gaussian form given for large $x$ and $t$?