Unbounded Domains

Abstract

We recall that the Ornstein-Uhlenbeck process EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WG5bWaaSbaaSqaaaqabaaakiaawUhacaGL9baadaWgaaWcbaGaamiD % aiabgwMiZkaaicdaaeqaaaaa!3CFF!]]<>/EquationSource><EquationSource Format="TEX"><![CDATA[$${\left\{ {{y_{}}} \right\}_{t \geqslant 0}}$$ is the diffusion on the line specified by the differential generator

$$Af\left( x \right) = \frac{1}{2}''\left( x \right) - yx'\left( x \right),f \in D\left( A \right)$$

(1.1)

(y>0).An alternative description is that ${\left\{ {{y_t}} \right\}_{t \geqslant 0}}$ is a Gaussian Markov process with continuous paths and the distribution of ${y_{t + s}}$ give y_s = y Gaussian with mean $y{e^{ - yt}}$ and variance $\left( {1 - {e^{ - 2yt}}} \right)/{2_y}$.