Tuesday, January 23, 2018

The Brownian motion as scaling limit of a random walk in discrete time

I start with an infinite set of random variables $X_n$, with $n: 1,2, \ldots$ and probability density function proportional to \begin{equation*} \exp -\frac{1}{2} \sum_{n =0}^{+\infty} (x_{n+1} - x_n)^2 \end{equation*} with $x_0 =0$. Random paths of $X_n$ can be generated by \begin{equation*} X_{n+1} = X_n + G_n \end{equation*} starting at $X_0 =0$ and with $G_n$ independent standard Gaussian random variables. The $X_n$ form a random walk in discrete time ("the lattice"). The two-point correlation function is \begin{equation}\label{eq:20180123} \mathbb{E} [ X_m X_n ] = n \quad\text{if}\quad m \ge n \end{equation} Now I move to the real line: I imagine that the distance between the lattice points is $a$. Therefore, the ratio $t/a$ is the number of lattice points between $0$ and $t$, approximated as an integer. I define $B_t = a^h X_{t/a}$ for $a$ very small. The factor $a^h$ is an extra scaling to make everything work, see below. I calculate the limit \begin{equation*} \mathbb{E} [ B_s B_t] = \lim_{a \to 0}\mathbb{E} [ a^h X_{s/a} \ a^h X_{t/a}] \end{equation*} Using \eqref{eq:20180123}, this is equal to $\lim_{a \to 0} a^{2 h} \frac{t}{a}$ for $s \ge t$. This limit exists if $h = 1/2$ and is then equal to $t$. $B_t$ is the standard Brownian motion: it is a Gaussian process and has the correct two-point correlation function.

From the definition of the Brownian motion as scaling limit, one can also prove the following formula. Suppose $b > 0$, then \begin{equation}\label{eq:20180122} \mathbb{E}[B_{b t_1}\cdots B_{b t_n}] = b^{n/2} \mathbb{E}[B_{t_1}\cdots B_{t_n}] \end{equation} Indeed, the left hand side is equal to \begin{equation*} \lim_{a \to 0}\mathbb{E} \left[ a^{1/2} X_{b t_1/a}\ \cdots\ a^{1/2} X_{bt_n/a}\right] \end{equation*} Write $a = b a'$, then the limit is equal to \begin{equation*} \lim_{a' \to 0}\mathbb{E} \left[ (ba')^{1/2} X_{t_1/a'}\ \cdots \ (ba')^{1/2} X_{t_n/a'}\right] \end{equation*} This is equal to the right hand side of \eqref{eq:20180122}.

This is all quite similar to the scaling limit discussed in conformal field theory (CFT), see for example [1]. $h$ plays the role of the scaling dimension, formula \eqref{eq:20180122} is the analogue of scale invariance in CFT. For my job I use the Brownian motion and more general stochastic processes. As a hobby I wanted to do something different and study CFT. These topics are related after all: CFT seems to be some kind of two-dimensional generalization of stochastic processes.

References and comments

[1] Conformal field theory and statistical mechanics (Lecture - 01) by John Cardy

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