On the stochastic Benjamin--Ono equation (Q852582)

This paper deals with the Cauchy problem for the stochastic Benjamin-Ono equation \[ u_{t}+uu_{x}+{\mathcal H}(u_{xx})=\sum_{j=1}^{\infty}g_{j}{dB_{j}\over dt}, \quad (t,x)\in (0,\infty)\times \mathbb R,\quad u(0,x)=u_0(x), \] \(x\in \mathbb R\), where \({\mathcal H}\) is the Hilbert transform and \(g_{j}=g_{j}(t,x),\;j=1,2,\ldots\). The right-hand side of equation corresponds to a random noise which is white in the time variable. It is established an existence and uniqueness result for solution of considered Cauchy problem in the function Sobolev class \(H^{s}(\mathbb R), s>3/2\). The author also obtains estimates of the mean energy for \(s=2\). The existence of an invariant measure, when considered equation includes an additional term of zero-order dissipation, is proved.