I am dealing with the simulation of particles exhibiting Brownian motion without drift, currently by updating the position in given time steps $\Delta t$ by random displacement in each direction drawn from $N(0,k\Delta t)$ (k being a particle-specific constant). 

Now, I would rather fix the displacement to a specific value and calculate random numbers representing the time it would take a particle to first travel that far from its previous position (no matter whether that is in the positive or negative direction). What is the appropriate distribution to draw these $\Delta t$ from?

I learned (from wikipedia, since I do not have much of a background here) that for Brownian motion with drift the first passage time follows an Inverse Gaussian distribution and for zero drift a Lévy distribution, but do not see how to arrive at the appropriate parameters. As a test, I generated random numbers from N(0,1) and summed them up until |sum| exceeded some threshold $\alpha$ (I tested 3 to 7). The number of summands needed seemed to be distributed according to an Inverse Gaussian distribution with the parameters roughly following $\mu = \alpha^2+\alpha+1$ and $\lambda = 1.5\alpha^2+2\alpha+1$. I cannot see it matching any Lévy distribution, and it is not intuitively clear to me why the values I am looking for should have an expected value of $\infty$ anyway.