In the book "An Introduction to Stochastic Differential Equation" of evans (chapter 3), we try to find the density of ink particules in water at time $t>0$. The experiment goes as follow : Consider a long, thin tube filled with clear water, into which we inject at time $t=0$ a unit amount of ink, at location $x=0$. Let $u=u(x,t)$ denote the density of ink particles at position $x\in\mathbb R$ and time $t\geq 0$. Initially, we have $u(x,0)=\delta _0$ (the Dirac mass at $0$). The probability density of the event that an ink particle moves from $x$ to $x+y$ in time $\tau$ is $f(y,\tau)$. Therefore $$u(x,t+\tau)=\int_{\mathbb R}u(x-y,t)f(y,\tau)dy.\tag{E}$$
Question 1 : First of all, I'm no sure why the equality $(E)$ holds. I can accept it, but I don't really understand. Does someone can provide an explanation ?
Using several hypothesis, we get $$\partial _tu=\frac{D}{2}\partial _{xx}u,$$ and thus $$u(x,t)=\frac{1}{(2\pi Dt)^{1/2}}e^{-\frac{x^2}{2Dt}},$$ and thus the density of diffusing ink is $\mathcal N(0,Dt)$.
Question 2: Can someone explain we hat it really mean ? Does it mean that after time $t$, the particle as traveled $\sqrt{Dt}$ meter ? and so with very high probability, at time $t$ the particle is in $[-\sqrt{Dt},\sqrt{tD}]$ ? I'm never really sure how interpret variance in this context.
The amount of ink that is observed in the infinitesimal region $[x,x+dx]$ at time $t+\tau$ is given by $$u(x,t+\tau)\,dx.$$ That ink, at time $t$, was distributed in other locations and moved into that region at time $t+\tau$. We must consider all possible locations and work out the probability for ink to move to the region. If locations were discrete bins we would add up the probabilities for all possible locations. Since locations are continuous we replace the sum with an integral. The amount of ink that was in the infinitesimal region $[x-y,x-y+dy]$ at time $t$ and moved into the range $[x,x+dx]$ is given by $$u(x-y,t)f(y,\tau)\,dx\,dy.$$ So one integrates over all $y$ and arrives at equation $(E)$.
Regarding the interpretation of variance in this context, after some time has passed each ink particle has undergone a large number of independent scattering events, and its new location is effectively the sum of a large number of random variables. When you add independent random variables you get a new random variable whose variance is the sum of the individual variances.
Because there is a large number of random variables being added, the central limit theorem applies and therefore the ink particles are distributed according to a Gaussian distribution whose variance which is proportional to the number of scattering events (and therefore also with time). Approximately 68% of the ink particles will be located in the range that you give, the remaining will be further from $x=0$.