The Random-Walk

Suppose you have a particle which undergoes kicks at fixed discrete instants of time. Each kick moves it along a real axis of an unknown quantity s. This quantity is uncertain but different values are not equally probable; on the contrary, they are distributed with a certain PD


which is defined on the entire real axis and whose integral on a certain interval is the probability that the particle has moved of a quantity between the lower and the upper integration limits.
Even if we know the position of the particle at the initial time, as soon as it is first moved, we can no longer know its position in a deterministic way. This means that, as the first random event takes place, all the process becomes stochastic.
Let us suppose however that at the t=1 the particle is in a fixed position at x=x0 and let us write the equation for its PD. At the initial time the PD for the particle is


since we exactly know the position.
Now, if the particle undergoes an event that moves it of a quantity s in the positive direction of the real axis then its PD becomes


which is still a deterministic knowledge since the particle continues to be localized.
However we do not know exactly the amount of the displacement and so the PD for the particle at instant t=1 has to be evaluated as an average on all possible events of motion being the weight given by the function K. In formulae this reads

and in general we have that


By using the property of commutation of the convolution product we now obtain


which seems a sort of Boltzmann equation.
Therein, indeed, we have a quantity changing at discrete instants of time by the action of an integral operator.
I have intentionally used discrete-time to evidence once more how the link between stochastic processes and integral operators goes much beyond transport theory.
Moreover, notice that the equation above is "exact" since in deducting it I made no approximations.
I have illustrated a discrete-time processes in which one writes down the PD at a certain time as a function (functional is the correct wording and will soon be explained) of the PD at the previous instant.
The underlying process here is in truth somewhat "artificial" since it is hard to imagine something in nature that acts with such precise clocking.
Suppose, on the contrary, as it mostly happens in nature, that your PD changes due to events whose taking place is not assured but also has some probability per unit time nu of happening. In this case the correct equation would have been


If, for example, the particle had to be moved by wind blows, then the last equation would have been a more accurate description than the discrete-time one since wind blows in a non-deterministic way also and so it is impossible to exactly know when a blow will arrive.
Let us now write the last equation by using again the commutation of the convolution product obtaining


Now, by taking the power series in s for phi in this equation and truncating it to the second order you will obtain a Fokker-Planck’s which is so understood to be a physical approximation. I will not do this since my intention here is to continue on the way of integral physics.