Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 2

The almost sure supermartingale convergence theorem says that a supermartingale whose negative parts have uniformly bounded expectations converges almost surely to a finite integrable limit. In particular, every nonnegative supermartingale converges almost surely.

Here the process is uniformly bounded, say $0\le X_n\le C$. Let $X_{\infty }={\lim }_n{X}_n$, whose existence follows from the theorem. The dominated convergence theorem then givesThe limit on the right also exists directly because the expectations of a supermartingale form a decreasing sequence.

For $x\ne 0$, the Strong Markov property at the first step gives the discrete mean-value identityAt $x=0$, $v(0)=1$ while the same average is at most one. Thus $v$ is a bounded superharmonic function on ${\mathbb{Z}}^3$. Conditioning on the natural filtration and using the one-step Markov property givesTherefore $(v(X_n))$ is a nonnegative supermartingale.

Let $A_n=\{X_m=0\text{ for some }m\ge n\}$. By the Markov property,The events $A_n$ decrease to the event that the walk visits zero infinitely often, which has probability zero by the stated transience assumption. HencePart a and the almost sure supermartingale convergence theorem give an almost-sure limit $V\ge 0$. By Fatou lemma, $\mathbb{E}V\le {\mathrm{liminf}}_n\mathbb{E}[v(X_n)]=0$, so $V=0$ almost surely.