Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 27 / 3 / ii / Solution

Assume $\kappa >0$ and $x\ne 0$, since the initial driving point does not admit the displayed ordinary boundary flow. The sign of the driver in this part is negative, soAfter division by $\sqrt{\kappa }$, the Boundary-point Bessel flow for SLE isFor $x>0$ this is a Bessel process of dimension $\delta =1+2a$. For $x<0$, $-X_t$ obeys the same equation driven by $-W_t$ until hitting zero. It suffices to treat a positive initial value.

The infinitesimal generator is $\mathcal{L}f=\frac{1}{2}{f}^{\prime \prime }+(a/u)f^{\prime }$. An increasing scale function of a one-dimensional diffusion isIt satisfies $\mathcal{L}s=0$. For $0<\epsilon <x<b$, optional stopping theorem gives the boundary hitting probability from a diffusion scale functionWhen $a<1/2$, put $\beta =1-2a>0$. The inner boundary is reached in finite time: the nonnegative functionvanishes at $0,b$ and satisfies $\mathcal{L}v=-1$. Applying the Itô formula before exiting $(\epsilon ,b)$ gives $\mathbb{E}({\tau }_{\epsilon }\wedge {\tau }_b)\le v(x)$. As $\epsilon \downarrow 0$, these times increase to a finite limiting exit time almost surely. Thus the scale limit is an actual hitting event, not just asymptotic approach to zero, andLet $b\uparrow \infty$ to obtain ${\mathbb{P}}_x({\tau }_0<\infty )=1$.

If $a>1/2$, then $s(\epsilon )\to -\infty$, and the same formula makes the probability of hitting zero before any fixed $b$ equal to zero. A finite-time hit would occur before reaching some integer upper level, because the stopped path is continuous and bounded on a finite interval. Taking the countable union over those levels proves that no hit occurs. At $a=1/2$, $s(u)=\log u$ gives the same conclusion. ThereforeThe source's $x=0$ case must be excluded: zero is then already the initial value, and the displayed singular flow is undefined.