I have been working on this for a while, but my incompetency in mathematics seemed to be taking me nowhere.
Suppose I have a system characterised by a 2D Brownian motion model. A "particle" that starts at the origin and takes a step of size $s$ with probability $P_s(s),\,s\in[s_l,\,s_h]$, in $\vec{n}$, a unit vector that can point in any direction with equal probability.
So it follows that the position after $N$ steps is $\vec{R}_N$, and the next step is
$$\vec{R}_{N+1} = \vec{R}_{N} + s\vec{n}.$$
The average over some steps later is then
$$\langle \vec{R}_{N+1} \rangle =\langle \vec{R}_{N} \rangle + \langle s \rangle \langle \vec{n} \rangle.$$
Using the fact that we started at the origin ($\vec{R}_0=0$) and the average of directional vector is 0 ($\langle\vec{n}\rangle=0$), we have the following
$$\langle \vec{R}_{N} \rangle = 0.$$
Now, we compute the squared distance of this "particle" from the origin, after $N+1$ steps, we have
$$\langle \lvert \vec{R}_{N+1} \rvert^2 \rangle = \langle \lvert \vec{R}_{N} \rvert^2 \rangle + \langle s^2 \rangle + 2 \langle s \rangle \langle \vec{n} \rangle \langle \vec{R}_N \rangle.$$
Using the fact that $\langle\vec{n}\rangle=0$ and $\vec{R}_0=0$, we arrive at
$$\langle \lvert \vec{R}_{N}\rvert^2 \rangle=N\langle s^2 \rangle.$$
Here is my proposition:
Now, let us define a parameter $D$, such that
$$D=\frac14 \frac{\langle \lvert \vec{R}_{N}\rvert^2 \rangle}{t} \equiv \frac14\frac{N\langle s^2 \rangle}{N \langle \tau \rangle} = \frac14\frac{\langle s^2 \rangle}{\langle \tau \rangle}.$$
Where $t$ is the observation time, and $\tau$ is the time it takes for the particle to take each step, and the following is also said to be true $$ \tau=\frac14 \frac{s^2}{D}.$$
When (remember that $s$ is bounded by $[s_l,\,s_h]$)
$$t\gg\frac14 \frac{{s_h}^2}{D},$$
the probability $P(R)$ for the total displacement in the interval $(R, R+\Delta R)$, with $R = \lvert \vec{R} \rvert$, measured from the origin, is given by
$$P(R) \,\mathrm{d}R = \frac{R}{2Dt}\exp\left(-\frac{R^2}{4Dt}\right)\,\mathrm{d}R.$$
This PDF certainly holds when $s$ takes on a discrete value (the classic random walk), but I can't show or convince myself the above PDF also holds true for a range of $s$ drawn from the probability density function $P_s(s)$.
Feel free if you want to have a go at proving my proposition, or if you know something similar to what I'm talking about, please point me in that direction. I am very much not familiar with stochastic mechanics.
Side note:
The only physical parameter that I am able to measure is $D$, which makes relaxing the assumption that $s$ is a discrete value a very reasonable proposition. The relaxation of $s$ being a discreet value does not change the measurable outcome $D$, but changes other measurable properties that are associated with the system when $\langle \vec{n} \rangle \neq 0$ (in a biased condition).