The wave equation is usually expressed in the form
where is a function of both time and space , with being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain with some other manifold and replacing the Laplacian with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on . We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy
which we can rewrite using integration by parts and the inner product on as
A key feature of the wave equation is finite speed of propagation: if, at time (say), the initial position and initial velocity are both supported in a ball , then at any later time , the position and velocity are supported in the larger ball . This can be seen for instance (formally, at least) by inspecting the exterior energy
and observing (after some integration by parts and differentiation under the integral sign) that it is non-increasing in time, non-negative, and vanishing at time .
The wave equation is second order in time, but one can turn it into a first order system by working with the pair rather than just the single field , where is the velocity field. The system is then
Finite speed of propagation then tells us that if are both supported on , then are supported on for all . One also has time reversal symmetry: if is a solution, then is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times using this symmetry.
If one has an eigenfunction
of the Laplacian, then we have the explicit solutions
of the wave equation, which formally can be used to construct all other solutions via the principle of superposition.
When one has vanishing initial velocity , the solution is given via functional calculus by
and the propagator can be expressed as the average of half-wave operators:
One can view as a minor of the full wave propagator
which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that
Viewing the contraction as a minor of a unitary operator is an instance of the “dilation trick“.
It turns out (as I learned from Yuval Peres) that there is a useful discrete analogue of the wave equation (and of all of the above facts), in which the time variable now lives on the integers rather than on , and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form
where is now an integer, take values in some Hilbert space (e.g. functions on a graph ), and is some operator on that Hilbert space (which in applications will usually be a self-adjoint contraction). To connect this with the classical wave equation, let us first consider a rescaling of this system
where is a small parameter (representing the discretised time step), now takes values in the integer multiples of , and is the wave propagator operator or the heat propagator (the two operators are different, but agree to fourth order in ). One can then formally verify that the wave equation emerges from this rescaled system in the limit . (Thus, is not exactly the direct analogue of the Laplacian , but can be viewed as something like in the case of small , or if we are not rescaling to the small case. The operator is sometimes known as the diffusion operator)
Assuming is self-adjoint, solutions to the system (3) formally conserve the energy
This energy is positive semi-definite if is a contraction. We have the same time reversal symmetry as before: if solves the system (3), then so does . If one has an eigenfunction
to the operator , then one has an explicit solution
to (3), and (in principle at least) this generates all other solutions via the principle of superposition.
Finite speed of propagation is a lot easier in the discrete setting, though one has to offset the support of the “velocity” field by one unit. Suppose we know that has unit speed in the sense that whenever is supported in a ball , then is supported in the ball . Then an easy induction shows that if are supported in respectively, then are supported in .
where and are the Chebyshev polynomials of the first and second kind, thus
As before, is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers and are discrete analogues of the heat propagators and wave propagators respectively.
One nice application of all this formalism, which I learned from Yuval Peres, is the Varopoulos-Carne inequality:
Theorem 1 (Varopoulos-Carne inequality) Let be a (possibly infinite) regular graph, let , and let be vertices in . Then the probability that the simple random walk at lands at at time is at most , where is the graph distance.
This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers . Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length concentrate on the ball of radius or so centred at the origin of the random walk.
Proof: Let be the graph Laplacian, thus
for any , where is the degree of the regular graph and sum is over the vertices that are adjacent to . This is a contraction of unit speed, and the probability that the random walk at lands at at time is
where are the Dirac deltas at . Using (5), we can rewrite this as
where we are now using the energy form (4). We can write
where is the simple random walk of length on the integers, that is to say where are independent uniform Bernoulli signs. Thus we wish to show that
By finite speed of propagation, the inner product here vanishes if . For we can use Cauchy-Schwarz and the unitary nature of to bound the inner product by . Thus the left-hand side may be upper bounded by
and the claim now follows from the Chernoff inequality.
This inequality has many applications, particularly with regards to relating the entropy, mixing time, and concentration of random walks with volume growth of balls; see this text of Lyons and Peres for some examples.
For sake of comparison, here is a continuous counterpart to the Varopoulos-Carne inequality:
Theorem 2 (Continuous Varopoulos-Carne inequality) Let , and let be supported on compact sets respectively. Then
where is the Euclidean distance between and .
Proof: By Fourier inversion one has
for any real , and thus
By finite speed of propagation, the inner product vanishes when ; otherwise, we can use Cauchy-Schwarz and the contractive nature of to bound this inner product by . Thus
Bounding by , we obtain the claim.
Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than .