# Renormalization and Brownian Motion

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka Brownian motion. I'd be particularly interested in references exploring analogies between stochastic processes and (statistical or quantum) field theory, and the renormalization group.

My understanding of Brownian motion, is that the "IR" or "coarse grained theory" is well described by a diffusion equation:

$${\partial \over \partial t} f(x, t) - D {\partial^2 \over \partial^2 x} f(x, t) = 0$$

where $f(x,t)$ would represent some probability density function. The constant D is some finite value "coupling constant", measured in the IR in lab, and perfectly models the IR physics.

Random walk as a model for Brownian Motion

Here's a discrete time model for the above diffusion equation:

$$x(t_{n+1}) = v \times n(t_n) \Delta t + x(t_n)$$

$v$ is the UV "coupling" and $n(t_n)$ is a set of IID random variables.

In the model above, as we go deep into the UV, the velocity to correctly model a finite $D$ in the IR scales as ${D \over \sqrt{\Delta t}}$, because of the central limit theorem. We therefore know that our model of physics cannot be correct to arbitrary time scales, hence the velocity generating brownian motion in the IR (or coupling constant) being an "irrelevant operator."

Question(s)

Is this model an example of the renormalization group at work?
If yes, what is the irrelevant operator? How does the degree of superficial divergence show up? (my gut says the dimensionality of the velocity in the discrete time model). What kind of "loop corrections" to the RG flow can we incorporate in such a simple model of physics at different time scales (aka anomalous dimensions)?

Comment & Discussion

I am aware the random walk has a path integral solution in the continuum. If I generalize the random walk of a point particle to a random of a field (the observable now being a statistical field in space/time rather than a coordinate). How does this relate to the renormalization group?

Thank you!

## 1 Answer

I do not know about the renormalization group but I think what you are talking about is the scale invariance/fractale structure of the Brownian motion. It is the reason why you do not need to specify the time step when you write the stochastic equation for the Brownian motion. If you look on this wikipedia page https://en.wikipedia.org/wiki/Wiener_process you can even see that there is a definition of the Brownian motion that almost only depends on thoses scale invariances properties