Multivariate Ornstein-Uhlenbeck process

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 $A\Sigma +\Sigma A^T=b{b}^T$.

Let $X$ be the centered stationary solution of $d{X}_t=-A{X}_{t}dt+Bd{W}_t$, where $A$ is a real $d\times d$ matrix with eigenvalues of positive real part and $W$ is an $m$-dimensional Brownian motion. Use the two-sided angular-frequency conventionThe Ornstein-Uhlenbeck power spectrum isThe stationary linear filter has transfer matrix $H(\omega )=(A+i\omega I{)}^{-1}$, so the output spectral matrix is $HB{B}^T{H}^{\ast }$, explaining the adjoint and sign of the second factor. This matrix is Hermitian positive semidefinite for every frequency.

For scalar $dX=-\lambda Xdt+\sigma dW$ with $\lambda >0$, the spectrum is ${\sigma }^2/({\lambda }^2+{\omega }^2)$, proportional to a Cauchy distribution probability density function. Its inverse-frequency integral gives variance ${\sigma }^2/(2\lambda )$, checking the normalization. With ordinary frequency $f$ and inverse transform $\int e^{2\pi ift}{S}_f(f)df$, use $S_f(f)=S(2\pi f)$. A one-sided positive-frequency spectrum doubles this density away from zero. These conventions must be matched when comparing a theoretical spectrum with measured spectral power.

In the classical thermal model, a coordinate in a harmonic potential obeys $\gamma \dot{x}+kx=\xi$, where $\gamma$ is the drag coefficient and $⟨\xi (t)\xi (t^{\prime })⟩=2\gamma k_{B}T\delta (t-t^{\prime })$. Its stationary one-sided spectral density per hertz is the displayed Lorentzian line shape. The integral over positive frequencies equals $k_{B}T/k$, consistent with the equipartition theorem. On a log-log plot the spectrum is flat for $f\ll f_c$ and has slope $-2$ for $f\gg f_c$. The crossover is determined by the restoring stiffness and drag; an added detector response multiplies the mechanical spectrum by the squared magnitude of its transfer function.

For stationary covariance $C(t)=(k_{B}T/k)e^{-|t|/{\tau }_k}$, the mean over duration $\mathcal{T}$ has variance $2(k_{B}T/k)[{\tau }_k/\mathcal{T}-({\tau }_k/\mathcal{T}{)}^2(1-e^{-\mathcal{T}/{\tau }_k})]$. At $\mathcal{T}\gg {\tau }_k$ this becomes $2(k_{B}T/k){\tau }_k/\mathcal{T}$. Closely sampled positions remain correlated and cannot be counted as independent.

For power spectral density $P(f)=C/(f^2+f_c^2)$, the low-frequency plateau is $P_0=C/f_c^2$ and the high-frequency asymptote is $C/f^2$. Their intersection on a log-log plot gives the displayed corner frequency. Equivalently $P(f_c)=P_0/2$. Use the range unaffected by additional low-frequency noise or high-frequency filtering; a measurement chain can otherwise shift an apparent corner.