Does anyone know the distribution for the first passage time of a Gaussian random walk i.e.
$$ S_n = \sum_{i=1}^n X_i $$
where $X_i$ are iid normally distributed random variables. The first passage time is $$ \tau = inf\{n: S_n \geq C\} $$ where $C$ is a constant. The literature I have come across mostly deal with expectations and even then is more focused on trying to bound the expectations and examine the limiting behaviour as $C \rightarrow \infty.$ Really appreciate any help you can provide.
For every $|z|\leqslant1$, the generating function $u(x)=E_x(z^\tau)$ for the random walk starting at $x$ is the unique solution of the integral identity $$ u(x)=z\cdot\int_\mathbb Ru(x+y)g(y)\mathrm dy, $$ where $g$ is the standard normal density, with the boundary condition that $u(x)=1$ for every $x\geqslant C$.