Lecture 8. The heat semigroup on the circle

We now turn to a new part in this course. The first few lectures were devoted to the study of diffusion operators and the construction of associated semigroups. The goal of this new part of the course will be to construct the heat semigroup on a Riemannian manifold. We shall see that given a Riemannian structure on a differentiable manifold, it is possible to canonically associate to it a diffusion operator which is called the Laplace-Beltrami operator of the Riemannian structure.

As an appetizer, we first study the heat semigroup on the simplest (non Euclidean) Riemannian manifold: the circle \mathbb{S}^1= \left\{ e^{i\theta}, \theta \in \mathbb{R} \right\}. The Laplace operator on \mathbb{R}^n, \Delta=\sum_{i=1}^n \frac{\partial^2}{\partial x_i^2} is the canonical diffusion operator on \mathbb{R}^n. A natural question to be asked is: in the same way, is there a canonical diffusion operator on \mathbb{S}^1. A first step, of course, is to understand what is a diffusion operator on \mathbb{S}^1. We characterized diffusion operators as linear operators on the space of smooth functions that satisfy the maximum principle. Once a notion of smooth functions on \mathbb{S}^1 is understood, this maximum principle property can be taken as a definition. The circle \mathbb{S}^1 may be identified with the quotient space \mathbb{R} / 2\pi \mathbb{Z} . More precisely, it is easily shown that a smooth function, f :\mathbb{R} \rightarrow  \mathbb{C} which is 2\pi periodic, i.e. f(\theta+2\pi)=f(\theta), can be written as f(\theta)=g\left( e^{i\theta} \right), for some function g: \mathbb{S}^1 \rightarrow \mathbb{C}. Conversely, any function g: \mathbb{S}^1 \rightarrow \mathbb{C} defines a 2\pi periodic function on \mathbb{R} by setting f(\theta)=g\left( e^{i\theta} \right). So, with this in mind, we simply say that g: \mathbb{S}^1 \rightarrow \mathbb{C} is a smooth function if f is. With this identification between the set of smooth 2\pi periodic functions on \mathbb{R} and the set of smooth functions on \mathbb{S}^1, it then immediate that the canonical diffusion operator \Delta_{\mathbb{S}^1} on \mathbb{S}^1 should write, \Delta_{\mathbb{S}^1} g ( e^{i\theta})=f''(\theta). The corresponding diffusion semigroup is also easily computed from the heat semigroup on \mathbb{R}. Indeed, a natural computation leads to
\left( e^{t \Delta_{\mathbb{S}^1} }g \right) ( e^{i\theta}) =\left( e^{t \frac{d^2}{d\theta^2} }f \right) ( \theta)
= \int_{-\infty}^{\infty} f(y+\theta) \frac{e^{-\frac{y^2}{4t} } }{\sqrt{4\pi t}} dy
=\sum_{k \in \mathbb{Z}} \int_{2k \pi }^{2k\pi +2\pi} f(y+\theta) \frac{e^{-\frac{y^2}{4t} } }{\sqrt{4\pi t}} dy
=\sum_{k \in \mathbb{Z}} \int_{0 }^{2\pi} f(y-2k\pi+\theta) \frac{e^{-\frac{(y-2k\pi)^2}{4t} } }{\sqrt{4\pi t}} dy
= \int_{0 }^{2\pi} f(y+\theta) p(t,y) dy,
where p(t,y)=\frac{1}{\sqrt{4\pi t}} \sum_{k \in \mathbb{Z}} e^{-\frac{(y-2k\pi)^2}{4t} }. This allows to define the heat semigroup on \mathbb{S}^1 as the family of operators defined by \mathbf{P}_t g (e^{i\theta} )= \int_{0 }^{2\pi} g(e^{i\nu }) p(t,\theta-\nu) d\nu. The natural domain of this operator is \mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R}) where \mu is the measure on \mathbb{S}^1 which is defined through the property \int_{\mathbb{S}^1} g d\mu = \int_0^{2\pi} f(\theta) d\theta. The reader may then check the following properties for this semigroup of operators:

  • (Semigroup property) \mathbf{P}_{t+s} =\mathbf{P}_t \mathbf{P}_s;
  • (Strong continuity) The map t \rightarrow \mathbf{P}_t is continuous for the operator norm on \mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R});
  • (Contraction property) \|\mathbf{P}_t g \|_{\mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R}) } \le \| g \|_{\mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R}) };
  • (Self-adjointness) For g_1,g_2 \in \mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R}), \int_{\mathbb{S}^1} (\mathbf{P}_t g_1)  g_2 d\mu=\int_{\mathbb{S}^1} g_1(\mathbf{P}_t g_2)  d\mu
  • (Markov property) If g \in \mathbf{L}^2_\mu(\mathbb{S}^1,\mathbb{R}) is such that 0 \le g \le 1, then 0 \le \mathbf{P}_t g \le 1.

Exercise.

  • Prove the Poisson summation formula: If f:\mathbb{R} \rightarrow \mathbb{R} is a smooth and rapidly decreasing function, then \sum_{m \in \mathbb{Z}} f(m) e^{im \theta}=\sum_{k \in \mathbb{Z}} \hat{f} (\theta -2k\pi).
  • Deduce that the heat kernel on \mathbb{S}^1 may also be written p(t,y)=\frac{1}{2\pi}\sum_{m \in \mathbb{Z}} e^{-m^2 t} e^{im y}.

Exercise. From the previous exercise, the heat kernel on \mathbb{S}^1 is given by p(t,y) =\frac{1}{2\pi}\sum_{m \in \mathbb{Z}} e^{-m^2 t} e^{im y} =\frac{1}{\sqrt{4\pi t}} \sum_{k \in \mathbb{Z}} e^{-\frac{(y 2k\pi)^2}{4t} }.

  • By using the subordination identity e^{-\tau | \alpha | } =\frac{\tau}{2\sqrt{\pi}} \int_0^{+\infty} \frac{e^{-\frac{\tau^2}{4t}-t \alpha^2}}{t^{3/2}} dt, \quad \tau  \neq 0, \alpha \in \mathbb{R}, show that for \tau > 0, \frac{1+e^{-2\pi \tau}}{1-e^{-2\pi \tau}} =\frac{1}{2\pi} \sum_{k \in \mathbb{Z}} \frac{2\tau}{\tau^2+n^2}
  • The Bernoulli numbers B_k are defined via the series expansion \frac{x}{e^x -1}=\sum_{k=0}^{+\infty} B_k \frac{x^k}{k!}. By using the previous identity show that for k \in \mathbb{N}, k \neq 0, \sum_{n=1}^{+\infty} \frac{1}{n^{2k}} =(-1)^{k-1} \frac{(2\pi)^{2k} B_{2k} }{2(2k)!}.

Exercise. Show that the heat kernel on the torus \mathbb{T}^n=\mathbb{R}^n / (2 \pi \mathbb{Z})^n is given by p(t,y) =  \frac{1}{(4\pi t)^{n/2}}   \sum_{k \in \mathbb{Z}^n} e^{-\frac{\|y+2k\pi\|^2}{4t} }=\frac{1}{(2\pi)^n} \sum_{l\in \mathbb{Z}^n} e^{i l \cdot y -\| l \|^2 t}.

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