{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.

Dirichlet energy space with zero boundary at infinity

The <Dirichlet form of a Markov chain> is
$$
\mathcal E(f,g)=\frac12\sum_{x,y\in S}
\pi(x)P(x,y)(f(x)-f(y))(g(x)-g(y)).
$$
When $P$ is <reversible Markov chain>[reversible], this equals $\langle f,(I-P)g\rangle_\pi$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 3 / a / i / Solution

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.

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

{title2=$\mathcal E(f,g)$}
{wiki=Dirichlet_form}

For a reversible transition matrix $P$ with invariant distribution $\pi$,
$$
\mathcal E(f,g)=\frac12\sum_{x,y}\pi(x)P(x,y)
(f(x)-f(y))(g(x)-g(y)).
$$


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 Incoming links: Dirichlet form of a Markov chain