Parabolic Itô equations

Abstract:

A parabolic Itô equation is an equation of the form \[ (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \alpha (t,\omega ),\quad u(0) = {u_0},{u_0},u \in H.\] $H$ is a Hilbert space with scalar product $u \cdot \upsilon$ and norm $| \cdot |$. $L$ is a linear time-independent negative-definite operator from $H$ to $H$. $f$ is a Lipschitz continuous operator from $H$ to $H$. $\alpha (t,\omega )$ is a white noise process in $H$. Under suitable technical conditions the following results are obtained: I. A unique nonanticipating solution of (1) exists with ${\sup _t}E\{ |u{|^2}\} < \infty$. II. $u(t,\omega ) = R(t,\omega ) + V(t,\omega )$ where $R(t,\omega )$ is a stationary process and \[ \lim \limits _{t \to \infty } E\{ |V(t,\omega ){|^2}\} = 0.\] III. If $L$ is selfadjoint and $f$ is the gradient of a smooth functional then an explicit expression is found for the stationary density of $R(t,\omega )$ on $H$. IV. For the equation $(\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \varepsilon \alpha (t,\omega )$ an asymptotic expansion in $\varepsilon$ is proven which holds uniformly in $t$.