Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 3

Set $\kappa =4/(d-1)>4$. For chordal Schramm–Loewner evolution, the centered image of a real boundary point, divided by $\sqrt{\kappa }$, follows the Boundary-point Bessel flow for SLE; changing $B$ to $-B$ matches the sign convention in the question. Thus ${\tau }_x$ and ${\tau }_y$ are the times at which the marked boundary points $x$ and $y$ are swallowed, or equivalently disconnected from infinity, by the Loewner chain.

ConsequentlyIt is the event that the negative marked point is swallowed before the positive marked point.

For every $a>0$, defineThe Brownian scaling theorem makes $\hat{B}$ a standard Brownian motion, and substitution shows that $(\hat{X},\hat{Y})$ satisfies the same coupled Bessel process equations from initial values $(x/a,y/a)$. Both hitting times are divided by $a^2$, so their order is unchanged. Taking $a=x$ or comparing any two pairs with the same ratio proves that $\mathbb{P}({\tau }_x>{\tau }_y)$ depends only on $x/y$.

The Strong Markov property and part (b) show that, before $\tau ={\tau }_x\wedge {\tau }_y$,Thus $F(V_{t\wedge \tau })$ is a bounded martingale. From the given stochastic differential equation,The Itô formula says that the drift of $F(V_t)$ isIt must vanish. Dividing by $(1-v{)}^2/2$ and using the algebraic identity supplied in the question gives

Because $\tilde{F}$ solves the differential equation from part (c), the Itô formula makes $\tilde{F}(V_t)$ a local martingale before $\tau$. The defining improper integral converges at both endpoints: its integrand is asymptotic to $(-u{)}^{1-d}$ near zero and to $|u{|}^{d-3}$ at minus infinity. Since $1<d<2$, both exponents are integrable. Henceso the stopped local martingale is a bounded martingale.

If ${\tau }_x<{\tau }_y$, then $V_t\to 0$ and $\tilde{F}(V_t)\to 0$. If ${\tau }_y<{\tau }_x$, then $V_t\to -\infty$ and $\tilde{F}(V_t)\to I$. The hitting times cannot coincide because $X_t-Y_t$ stays positive. The optional sampling theorem for a supermartingale and bounded convergence theorem therefore giveConsequentlywhich is the Two-sided SLE boundary swallowing probability.