nLab harmonic function

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Graph theory

Measure and probability theory

Contents

Idea

A harmonic function is a function which has zero Laplacian.

It can often be interpreted as a function whose value at xx is equal to its average on a “neighborhood” of xx. 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 BB at a certain temperature TT surrounded by bodies of other temperatures, in such a way that the temperature TT is the average of the temperatures of the neighbors. This way, no net flow of heat passes into or out of BB.

For manifolds

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

Δf=0. \Delta f \;=\; 0 .

(…)

Examples

For graphs

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

For finite graphs, ff is equivalently harmonic if and only if

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

If xx has nonzero degree, equivalently,

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

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

If the graph is weighted, the formula becomes

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

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

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

That is, f(x)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, ff is equivalently incoming-harmonic if and only if for all xVx\in V,

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

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

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

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

If the graph is weighted, the formulas become

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

and

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

respectively, where AA is the adjacency matrix.

For Markov chains

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

xXf(y)k(y|x)=f(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)A(x,y)=k(y|x).

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

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

Given an invariant measure pp on XX, we call ff almost surely harmonic if the equation above holds for pp-almost all xx.

Proposition

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

See also

Last revised on June 18, 2025 at 12:34:05. See the history of this page for a list of all contributions to it.