#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}
right) = frac{r_{trap}}{d+r_{trap}} , 2Phi

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)
enter image description here

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.

Get this bounty!!!

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