Strangely sweeping one-dimensional diffusion

Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = Prob{X(t) ≥ 0} as t → ∞ and construct an equation such that ${\displaystyle lim{\supset }_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=1}$ and ${\displaystyle lim\in f_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=0}$.