# #StackBounty: #distributions #brownian #diffusion What is the distribution of the peak time of the first hitting time process

### Bounty: 50

I need to find the distribution of the random variable $$T_{peak}$$ where $$T_{peak}$$ represents the peak time of the first hitting time process.

Detailed Explanation of the System:
There are emitted molecules from a specific point in 3D environment. Molecules diffuse in the environment according to the followings:

$$r[t] = r[t-1] + (Delta r_1, Delta r_2, Delta r_3)$$
$$Delta r_i sim mathcal{N}(0,, 2DDelta t)$$
where $$r[t]$$, $$r_i$$, $$D$$, and $$Delta t$$ are the location vector at time $$t$$, $$i$$-th component of the location vector, diffusion coefficient, and the time step, respectively.

If there is an absorbing spherical trap at a distance $$d$$, the mean number of arriving/hitting molecules until time $$t$$ is:

$$E[N^{absorb}(t)] = frac{r_{trap}}{d+r_{trap}} , text{erfc} left( frac{d}{sqrt{4Dt}} right) = frac{r_{trap}}{d+r_{trap}} , 2Phi left( frac{-d}{sqrt{2Dt}} right)$$
where $$r_{trap}$$ is the radius of the absorbing spherical trap.

When you plot $$N^{absorb}(t)$$ in small intervals, you get something like the following figure (Scaled Inverse Gaussian distribution)

And the expected value of the peak time of hitting time histogram is
$$E[T_{peak}] = frac{d^2}{6D}$$

When simulating this diffusion process and the focusing on the absorbing times, $$T_{peak}$$ differs from simulation instance to instance, $$T_{peak}$$ is a random variable and I need to find the distribution of $$T_{peak}$$.

P.S. These absorbed molecules are considered as the received signal in molecular communications and the peak time distribution of the received signal is important for many applications.

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