nLab harmonic function

Idea

A harmonic function is a function which has zero Laplacian.

It can often be interpreted as a function whose value at $x$ is equal to its average on a “neighborhood” of $x$. A similar interpretation is as a function which is invariant on average under the transition given by diffusion?, by a Markov chain, or by the edges of a graph.

In physics, a harmonic function represents a quantity which, in some sense, is spatially at equilibrium. For example, a body $B$ at a certain temperature $T$ surrounded by bodies of other temperatures, in such a way that the temperature $T$ is the average of the temperatures of the neighbors. This way, no net flow of heat passes into or out of $B$.

For manifolds

Given a Riemannian manifold $M$, a smooth function $f:M\to\mathbb{R}$ or $M\to\mathbb{C}$ is called harmonic if and only if its Laplacian is zero:

$$\Delta f \;=\; 0 .$$

(…)

Examples

• Every holomorphic function $\mathbb{C}\to\mathbb{C}$ is harmonic.
• In physics, the electric potential is harmonic when there are no electric charges.

For graphs

Given an undirected graph $G=(V,E)$, a function $f:V\to\mathbb{R}$ is called harmonic if and only if its discrete Laplacian $\Delta f$ is zero.

For finite graphs, $f$ is equivalently harmonic if and only if

$$\sum_{y\sim x} f(y) \;=\; f(x) \, deg(x) .$$

If $x$ has nonzero degree, equivalently,

$$\frac{1}{deg(x)}\sum_{y\sim x} f(y) \;=\; f(x) .$$

That is, $f(x)$ is equal to the average over the adjacent? vertices.

If the graph is weighted, the formula becomes

$$\sum_{y\in V} f(y) \, A(y,x) \;=\; \sum_{y\in V} f(x) \, A(y,x) ,$$

where $A$ denotes the adjacency matrix. Again, for nonzero degree, this can be equivalently written as

$$\frac{\sum_{y\in V} f(y) \, A(y,x)}{\sum_{y\in V} A(y,x)} \;=\; f(x) .$$

That is, $f(x)$ is equal to the weighted average over the adjacent? vertices.

Directed graphs

For directed graphs there is a difference between incoming and outcoming Laplacians. Accordingly, a function is incoming-harmonic if its incoming Laplacian is zero, and outcoming-harmonic of its outcoming Laplacian is zero.

For finite graphs, $f$ is equivalently incoming-harmonic if and only if for all $x\in V$,

$$\sum_{y\in in(x)} f(y) \;=\; f(x)\,|in(x)|$$

where $in(x)$ denotes the set of vertices $y$ with edges $y\to x$, and outcoming-harmonic if and only if for all $x\in V$,

$$\sum_{y\in out(x)} f(y) \;=\; f(x)\,|out(x)|$$

where $out(x)$ denotes the set of vertices $y$ with edges $x\to y$.

If the graph is weighted, the formulas become

$$\sum_{y\in V} f(y) \, A(y,x) \;=\; \sum_{y\in V} f(x) \, A(y,x)$$

and

$$\sum_{y\in V} f(y) \, A(x,y) \;=\; \sum_{y\in V} f(x) \, A(x,y)$$

respectively, where $A$ is the adjacency matrix.

For Markov chains

Given a finite-state Markov chain? (or equivalently a stochastic matrix) with transitions $k(y|x)$ on a finite set $X$, a function $f:X\to\mathbb{R}$ is called harmonic if and only if for all $x\in X$,

$$\sum_{x\in X} f(y)\,k(y|x) \;=\; f(x) .$$

Notice that this is the same as an outgoing-harmonic function on the weighted graph whose adjacency matrix is $A(x,y)=k(y|x)$.

More generally, given a Markov kernel $k$ on a measurable space $X$, a measurable function $f:X\to\mathbb{R}$ is called harmonic if and only if for all $x\in X$,

$$\int_X f(y) \; k(d y|x) \;=\; f(x) .$$

Given an invariant measure $p$ on $X$, we call $f$ almost surely harmonic if the equation above holds for $p$-almost all $x$.

Proposition

Given a Markov kernel on $X$ with an invariant measure, the following conditions for a function $f:X\to\mathbb{R}$ are equivalent:

• $f$ is almost surely harmonic;

• $f$ is almost surely invariant;

• $f$ is measurable for the almost surely invariant sigma-algebra.

See also

• harmonic map, harmonic form

• Laplace operator

• discrete Laplace operator

• harmonic form