Weighted averages in harmonic functions.

Question

Is it the case that for a harmonic function on a graph any value of the interior point is the weighted average of the boundary points? I know that for a harmonic function each point is the weighted average of its adjacent neighbors, but can you extend that to the boundary points?

Answer 1

The answer is yes if your graph $G$ is finite (as it appears to be from your post). Let $b_1,\dots,b_m$ be the boundary vertices. For each $k\in\{1,\dots,m\}$ there is a unique harmonic function $u_k$ on $G$ such that $u_k(b_k)=1$ and $u_k(b_j)=0$ for $j\ne k$. (This is just the existence/uniqueness for the Dirichlet problem.) Since $0\le u_k\le 1$ on the boundary, we have $0\le u_k\le 1$ on the entire graph. Also, the function $u_1+\dots+u_m$ is equal to $1$ at every boundary point, and therefore is identically equal to $1$.

Suppose that $u$ is a harmonic function on $G$. Then the equality $$u(x) = \sum_{j=1}^m u(b_j) u_j(x)\tag1$$ holds for all $x$ on the boundary (by construction), and therefore for all $x\in G$ (by the uniqueness for the Dirichlet problem, essentially the maximum principle). The formula (1) represents $u(x)$ as the weighted average of boundary values $u(b_j)$, the weights being $u_j(x)$. Recall that $u_j(x)$ are nonnegative and add up to $1$, so they are indeed weights.

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Some additional remarks.

One key term in this subject is harmonic measure. For each interior point $x$ the harmonic measure $\omega_x$ gives weights $u_j(x)$ to boundary points $b_j$. This is a probability measure supported on the boundary. In terms of this measure, (1) can be written as $$u(x) = \int_{\partial G} u \, d\omega_x\tag2$$ bringing notation in alignment with the harmonic measure for the classical Laplace equation. One can then ask whether (2) holds on infinite graphs. There are simple counterexamples, such as the function $u(n)=n$ on the graph with vertices $0,1,2,3,4,\dots$ where adjacent integers are connected. But if $u$ is assumed to be bounded, then positive results are possible. Or, one can introduce "boundary at infinity" of the graph and re-interpret (2) in terms of a measure on that ideal boundary.

Another key term is random walk. Consider random walk $x_n$ on the graph that starts at $x_0=x$ and stops upon reaching a boundary point. By harmonicity, the value $u(x)$ is equal to the expectation of $u(x_n)$ for any $n$. If (i) the walk reaches boundary almost surely, and (ii) $u$ is reasonable (say, bounded), then we can pass to the limit $n\to\infty$ and obtain (2). For example, if the graph is square grid on half-plane $\{(m,n):m,n\in\mathbb Z, n\ge 0\}$ then $\omega_x$ is distributed along the boundary $\{(m,0):m\in\mathbb Z\}$.