Ornstein-Uhlenbeck process

A stationary Ornstein-Uhlenbeck process is a Gaussian Markov process with exponential covariance $K(s,t)=A^2{e}^{-|t-s|/\tau }$.

A multivariate Ornstein-Uhlenbeck process obeys $dX=-AXdt+bdW$. If every eigenvalue of $A$ has positive real part, its stationary covariance $\Sigma$ is the unique solution of $A\Sigma +\Sigma A^T=b{b}^T$.

The stationary spectral-density matrix of $dX=-AXdt+bdW$ is

Colored noise has nonzero temporal correlation away from equal times. An Ornstein-Uhlenbeck noise has exponential covariance and approaches white noise when its correlation time tends to zero at fixed integrated covariance.