A diffusion process in a Brownian environment with drift (Q1389863)

Let \(W\) be a Brownian motion and define \(W_\kappa(x):=W(x)-\textstyle {1\over 2}\kappa x\). Furthermore, let \(X^\kappa\) be the process which can be interpreted as the formal solution of the symbolic equation \[ dX(t) =dB(t)-{1\over 2} W_\kappa\bigl(X(t)\bigr)dt, \] where \(B\) is another Brownian motion independent of \(W\). The authors investigate the asymptotic behaviour of that process if \(t\to\infty\) for \(0<\kappa<1\).