Okay, hello everyone.

We are just discussing Nelson's hidden variable theory.

And so for the moment we don't even talk about quantum mechanics, I just wanted to remind

you of the basics of stochastic processes.

And the kinds of stochastic processes we are considering are just Brownian motion, maybe

adding a drift field, so sometimes we call that a drift diffusion process.

And kinds of questions you can ask is, for example, what is the stationary probability

density that develops?

So if I want to draw a picture of such a process, that would be space versus time, and then

by example it has this fractal appearance, typical for Brownian motion, it goes up and

down, but this is already a process that has a tendency to go back to the origin, because

otherwise it would just, it would have a further from the origin.

And so last time I ended with the example of an Einstein-Gulnberg stochastic process,

but more generally the types of processes we describe are always subject to the following

equation where first I will write down the discrete version and then I will write down

the continuum version.

So x at time t plus delta t is just the old position x of t plus some random step in either

direction which I call delta x of t plus then possibly some deterministic drift, so you

have a velocity depending on the current position and multiply that by the time step.

Or you can also write it down in a continuum version if you imagine the idealization that

delta t tends to zero and do the limits correctly, you would say dx over dt equals something

that derives from this noise process, so we have called it v tilde of t, that would be

noisy velocity field which is really white noise that we discussed, plus v of x of t,

plus the deterministic drift, so noise and drift.

We also mentioned that of course somehow you want to characterize the strength of the noise,

for example you can take the variance of delta x and in order to get a reasonable limit you

will take that variance of delta x to be proportional to delta t.

So we said variance of delta x equals 2 times d times delta t and d then would be the diffusion

constant.

Or if you want to describe it in this continuum version you would say v tilde of t, v tilde

of zero, the correlator of this is also just a delta function and that's depending on this

diffusion constant.

Okay, now one of the questions as I said you can ask is if I let the trajectory start at

any point and wait for a long enough time and just register the probability for the

particle to visit any small interval I will then find there will be a steady state probability

density and so one of the tasks is to find the steady state probability density.

So typically in the following I will call this rho, rho of x and t for the moment but

we will then consider stationary processes where this settles down to some steady state

value.

So how does this probability density evolve?

Well, there is the deterministic drift that just carries along the particles of your imaginary

ensemble and it gives rise to a current density which is just rho times the drift velocity

and so that gives the first term minus the divergence of this current density which is

rho times v so that would be the drift plus there is an extra term because even in the

absence of drift of course the probability density does change via the noise term so

it diffuses you start from a very localized probability density and it diffuses outwards

and so that is described by the diffusion term d times the second derivative of rho

you see the second derivative in the center is negative so it gets suppressed and the

outward fringes is positive.

So that is drift and diffusion.