The Heat Kernel — The Classical Heat Equation

1.1 The Classical Heat Equation

In the most classical sense, the heat equation is the following partial differential equation on Rd×R\mathbb {R}^ d \times \mathbb {R}:

(t2xi2)f=0. \left(\frac{\partial }{\partial t} - \sum \frac{\partial ^2}{\partial x_ i^2}\right)f = 0.

This describes the dispersion of heat over time, where f(x,t)f(x, t) is the temperature at position xx at time tt. To simplify notation, we write

Δ=2xi2. \Delta = - \sum \frac{\partial ^2}{\partial x_ i^2}.

Green's strategy to solving such a PDE is to find a solution to the PDE with initial condition

f(x,0)=δ(x). f(x, 0) = \delta (x).

In this case, we can find it very explicitly to be

f(x,t)=1(4πt)d/2exp(x24t). f(x, t) = \frac{1}{(4\pi t)^{d/2}} \exp \left(-\frac{|x|^2}{4t}\right).

We call this Ht(x)H_ t(x). Then given any bounded continuous function f0f_0, it is easy to see that the solution to the initial value problem

f(x,0)=f0(x) f(x, 0) = f_0(x)

is simply given by

f(x,t)=RdHt(xy)f0(y)  dy. f(x, t) = \int _{\mathbb {R}^ d} H_ t(x - y) f_0(y)\; \mathrm{d}y.

The fact that we are allowed to write Ht(xy)H_ t(x - y) at all uses the fact that we are working on Rd\mathbb {R}^ d and our system has translational symmetry. In general, if we work on an arbitrary manifold MM, we would seek a function Ht(x,y)H_ t(x, y) on M×M×(0,)M \times M \times (0, \infty ) such that the solution to the initial value problem f(x,0)=f0(x)f(x, 0) = f_0(x) is

f(x,t)=MHt(x,y)f0(y)  dy. f(x, t) = \int _ M H_ t(x, y) f_0(y)\; \mathrm{d}y.

The function Ht(x,y)H_ t(x, y) then satisfies

(t+Δx)Ht(x,y)=0. \left(\frac{\partial }{\partial t} + \Delta _ x\right) H_ t(x, y) = 0.

This Ht(x,y)H_ t(x, y) is also called the heat kernel, or fundamental solution, and we will mostly use these terms interchangeably. (It is also called a Green's function, but we will not use this name)

The heat kernel also shows up in a closely related problem. Suppose we wanted to solve instead

(tΔ)f=F(x,t) \left(\frac{\partial }{\partial t} - \Delta \right)f = F(x, t)

for some (bounded continuous) forcing term FF. It turns out the solution can also be expressed in terms of HH as

f(x,t)=MHt(x,y)f0(y)  dy+0tdτMHtτ(x,y)F(y,τ)  dy, f(x, t) = \int _ M H_ t(x, y) f_0(y)\; \mathrm{d}y + \int _0^ t \mathrm{d}\tau \int _ M H_{t - \tau }(x, y) F(y, \tau ) \; \mathrm{d}y,

which is again not difficult to check. The interpretation of this is that we can think of the forcing term as adding a new initial condition at each point τ\tau in time. This is known as Duhamel's principle.

Especially from a physical perspective, it is interesting to note that the heat equation propagates information at infinite speed. In other words, for any t>0t > 0, the value of f(x,t)f(x, t) depends on the values of f0f_0 everywhere. This is in contrast with, for example, the wave equation, where information only propagates at finite speed. Nevertheless, in the limit t0t \to 0, the asymptotic behaviour is purely local, as the contribution of the points a finite distance away decays exponentially as t0t \to 0.

On a general Riemannian manifold, we can formulate the same problem, replacing Δ\Delta by the Laplace–Beltrami operator. The Laplace–Beltrami operator (or Laplacian for short) can in fact be defined for all pp-forms. Write Ω=pΩp\Omega ^* = \bigoplus _ p \Omega ^ p. 1 Then the exterior derivative defines a map

d:ΩΩ. \mathrm{d}\colon \Omega ^* \to \Omega ^*.

There is an L2L^2 inner product on pp-forms coming from the metric, and so d\mathrm{d} has a formal adjoint d:ΩΩ\mathrm{d}^*\colon \Omega ^* \to \Omega ^* that lowers degree by 11. The Laplace–Beltrami operator is given by

Δ=(d+d)2=dd+dd:ΩpΩp. \Delta = (\mathrm{d}+ \mathrm{d}^*)^2 = \mathrm{d}^* \mathrm{d}+ \mathrm{d}\mathrm{d}^* \colon \Omega ^ p \to \Omega ^ p.

In the case of p=0p = 0, the Laplace–Beltrami operator is simply given by dd\mathrm{d}^* \mathrm{d}, and in local coordinates with metric gg, we can write this as

Δf=1detgi(detggijjf). \Delta f = \frac{1}{\sqrt{\det g}} \partial _ i (\sqrt{\det g}\, g^{ij} \partial _ j f).