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Put $\rho =e^{-\Delta t/\tau }=e^{-x}$ and retain $R=k(0,0)$, which equals one for the stated exponential covariance function. The covariance matrix of $(y_1,y_2,y_3)$ isConditioning a multivariate normal distribution and simplifying givesThe absence of $y_1$ from both conditional moments shows that $y_3$ and $y_1$ are conditionally independent given $y_2$. Equivalently, the exponential-kernel Gaussian process is the stationary Ornstein-Uhlenbeck process, which is Markov.

Stack $y=(y_1^{\top },y_2^{\top }{)}^{\top }$. After integrating out the Ornstein-Uhlenbeck process,Writing $k(a,b)=A^2{e}^{-|a-b|/\tau }$, the covariance blocks areand $({\Sigma }_{12}{)}_{jk}=k(t_j,t_k-\Delta t)$, with ${\Sigma }_{21}={\Sigma }_{12}^{\top }$. Therefore

For an Ornstein-Uhlenbeck process $dX=\sigma dB-\lambda Xdt$ under a reference measure, this is a positive supermartingale when $\alpha a\ge {\sigma }^2/2$. Its short rate is $[\alpha a-{\sigma }^2/2+(\lambda +\alpha /2)X^2]/(a+X^2/2)$. Conditional zero-coupon bond prices are ratios of the OU second moments of the quadratic factor. The money-market density has diffusion coefficient $\sigma X/(a+X^2/2)$, bounded by $\sigma /\sqrt{2a}$, so the Novikov condition validates the pricing measure. The reference OU dynamics must not be silently treated as the risk-neutral dynamics.

Two Ornstein-Uhlenbeck processes with the same driving Brownian motion have the displayed difference. If $Y_0$ is standard normal and independent of future Brownian increments, replacing it by $-Y_0$ produces another stationary process with the same covariance and second moment, but a different pathwise coupling. Equal initial values are necessary for indistinguishability of stochastic processes.

For $d{C}_t=a(b-C_t)dt+\sigma \sqrt{C_t}d{W}_t$, the Itô formula applied to $Z=\sqrt{C}$ gives the displayed dynamics before hitting zero. In particular $2ab={\sigma }^2$ does not eliminate the reciprocal drift. The cancellation condition is $4ab={\sigma }^2$. Under that condition $C$ can be represented as the square of a signed Ornstein-Uhlenbeck process with drift $-aY/2$ and volatility $\sigma /2$; its principal square root is the reflected process $|Y|$, not an unrestricted signed OU process. The mean of $C$ is $b+(C_0-b)e^{-at}$ for all admissible positive parameters.

If the displayed covariance is finite and the mean obeys $m^{\prime }(x)+A(x)=0$, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over $[t,\infty )$ and fresh noise contributes the integral over $[0,t]$, so their sum is $K$. A deterministic initial curve generally is not stationary, and nondecaying volatility may make $K(x,x)$ infinite. Exponentially decaying finite-factor volatilities give stationary Ornstein-Uhlenbeck processes.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.

Close to the upward pole, the two Cartesian tilt components of bottom-heavy spherical-cell orientation dynamics obey the displayed Ornstein-Uhlenbeck process. Thus $d⟨|\mathbf{q}{|}^2⟩/dt=-2⟨|\mathbf{q}{|}^2⟩/B+4D_r$. The stationary spread is $2B{D}_r$, and late axisymmetric variance perturbations relax on $B/2$ in this weak-noise approximation. The zero-noise mean from deterministic alignment of an initially isotropic orientation distribution has the same exponential scale but a different transient prefactor. Reaching a tiny diffusion-supported mean-deficit floor $B{D}_r$ from an initially isotropic population takes order $B\log [(B{D}_r{)}^{-1}]$, with logarithmic corrections. Equality of limiting mean directions does not make the full transients identical, and the alignment time is not automatically the pure rotational-diffusion time $D_r^{-1}$.

For $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$ with ${\sigma }_0=0$, the explicit stochastic integral is ${\sigma }_t=\sqrt{\kappa }{\int }_0^t{e}^{-(t-u)/\tau }d{W}_u$. The Itô isometry on the overlap of the two integration intervals gives the displayed covariance. The second exponential records the zero initial condition; removing it gives the stationary Ornstein-Uhlenbeck process covariance.