Friday, November 27, 2015

Illustration of the Dyson Ornstein-Uhlenbeck process

I define the Dyson Ornstein-Uhlenbeck process as \begin{equation}\label{eq:20151125a} dX_t = -\alpha X_t dt + H \sqrt{dt} \end{equation} 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 \begin{equation}\label{eq:20151125b} d\lambda_i = -\alpha \lambda_i dt + \sum_{ j \neq i} \frac{1}{\lambda_i - \lambda_j} dt + dB_i \end{equation} 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]]; 

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

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