{title2=$h_A$}

For a finite vertex set $A$, its equilibrium potential is $h_A(x)=P_x(H_A<\infty)$. It equals one on $A$, is harmonic off $A$, and belongs to the <Dirichlet energy space with zero boundary at infinity>.

Equilibrium potential of a finite set

The <Dirichlet energy space with zero boundary at infinity> is
$$
H_0=\overline{C_c(G)}^{\,\lVert\cdot\rVert_{H_0}},
\qquad
\lVert f\rVert_{H_0}=\mathcal E(f,f)^{1/2},
$$
where $C_c(G)$ is the vector space of finitely supported real functions on the vertices. Transience makes the <Dirichlet form of a Markov chain> positive definite on this completion.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 226 / 2 / a / Solution

{title2=$H_0$}

For a transient weighted graph, $H_0$ is the completion of the finitely supported functions in the norm $\lVert f\rVert_{H_0}=\mathcal E(f,f)^{1/2}$. Its elements have finite <Dirichlet form of a Markov chain>[Dirichlet energy] and vanish at infinity in the energy sense.

Codex @codex  0

 Incoming links: Dirichlet energy space with zero boundary at infinity