## Friday, November 27, 2015

### Illustration of the Dyson Ornstein-Uhlenbeck process

I define the Dyson Ornstein-Uhlenbeck process as $$\label{eq:20151125a} dX_t = -\alpha X_t dt + H \sqrt{dt}$$ with $\alpha > 0$ and $H$ a random matrix from the Gaussian Unitary Ensemble of $n \times n$ Hermitian matrices. The eigenvalues $\lambda_i(t)$ of $X_t$ then have the following dynamics $$\label{eq:20151125b} d\lambda_i = -\alpha \lambda_i dt + \sum_{ j \neq i} \frac{1}{\lambda_i - \lambda_j} dt + dB_i$$ where $B_1, \ldots, B_n$ are independent Brownian processes. In this post I illustrate the process \eqref{eq:20151125b} numerically.
Process \eqref{eq:20151125b} is a random process with mean version to zero because of the $-\alpha \lambda_i dt$ term. The eigenvalues $\lambda_i(t)$ also repel one another because of the term $\sum_{ j \neq i} \dfrac{1}{\lambda_i - \lambda_j}$. I now illustrate this behaviour for $\alpha = 20$ and $n = 5$. The code is quite similar to the code I used in the previous blog post. The only difference is that I use the following code to simulate $X_t$

 (*Simulates a path of the Dyson Ornstein-Uhlenbeck process
in: The dimension of the matrices is size x size. 
T is the maximum time to simulate.
NSteps is in how many steps you simulate.
out: a list of matrices X_t*) 
SimulateOnePathOU[alpha_, Size_, T_, NSteps_] := Module[{dt = T/NSteps, t, H, path}, 
path = ConstantArray[0, NSteps + 1];
path[[1]] = ConstantArray[0, {Size, Size}];
For[t = 1, t <= NSteps, t++, H = RandomMatrixGUE[Size];
path[[t + 1]] = - alpha path[[t]] dt + path[[t]] + H Sqrt[dt]];
path]

This produces the following graph
 The eigenvalues  $\lambda_1, \lambda_2, \ldots, \lambda_5$ as function of time
The paths $\lambda_i(t)$ do not cross one another; they also do not spread out because each one tries to get back to zero. If I use the same value of $\alpha$ to simulate five independent Ornstein-Uhlenbeck processes, I get a picture like this
Here, the paths also do not spread out because they are driven back to zero, but the paths cross many times.

Related: Illustration of Dyson Brownian motion