Let $(X_t, t ≥ 0)$ be a 1-dimensional diffusion process with generator Af(x) =\frac{1}{2}a(x)f''(x)+b(x)f'(x), \mathcal{D}(A)=C^2({\mathbb{R}}) where $b$ and $a=\sigma^2$ are continuous functions of $x$ and $\sigma(x)> 0$ for all $x$. Let $\tau_0=\inf\{t\ge 0: X_t=0\}.$
1) Suppose that for some $x_0$, $b(x)\leq 0,$ for all $x\ge x_0$. Show that for any $x> 0$: $\mathcal{P}^{x}[\tau_0< \infty]=1.$
2) Suppose that for some $x_0,$ $\frac{b(x)}{a(x)}>\epsilon> 0,$ for all $x\ge x_0.$ Show that for any $x> 0$: $\mathcal{P}^x[\tau_0< \infty]<1$