Lecture 4. The heat kernel of a diffusion semigroup

The goal of this lecture is to prove that if a diffusion operator L is elliptic, then the semigroup it generates admits a smooth kernel. As a consequence, the semigroup generated by an elliptic diffusion operator is regularizing in the sense that it transforms any function into a smooth function. The key point is the following estimate that can be proved by using the theory of Sobolev spaces.

Proposition: Let L be an elliptic diffusion operator with smooth coefficients on \mathbb{R}^n which is symmetric with respect to a Borel measure \mu. Let u \in \mathbf{L}_\mu^2 (\mathbb{R}^n,\mathbb{R}) such that Lu,L^2u,\cdots, L^ku \in \mathbf{L}_\mu^2 (\mathbb{R}^n,\mathbb{R}), for some positive integer k. If k > \frac{n}{4}, then u is a continuous function, moreover, for any bounded open set \Omega \subset \mathbb{R}^n, and any compact set K \subset \Omega, there exists a positive constant C (independent of u) such that \left(\sup_{x \in K} | u(x) | \right)^2 \le C \left( \sum_{j=0}^k \|L^j u \|^2_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})} \right). More generally, if k > \frac{m}{2}+\frac{n}{4} for some non negative integer m, then u \in \mathcal{C}^m(\mathbb{R}^n,\mathbb{R}) and for any bounded open set \Omega \subset \mathbb{R}^n, and any compact set K \subset \Omega, there exists a positive constant C (independent of u) such that \left(\sup_{|\alpha| \le m}  \sup_{x \in K} |\partial^\alpha u(x) | \right)^2 \le C \left( \sum_{j=0}^k \|L^j u \|^2_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})} \right).

We can explain by simple computations on the Laplace operator \Delta how the \frac{m}{2}+\frac{n}{4} comes into the play in the above proposition. Let u be a smooth and rapidly decreasing function. By the inverse Fourier transform formula \| \Delta^j u \|^2_{ \mathbf{L}^2 (\Omega,\mathbb{R})} =(2\pi)^{4j} \int_{\mathbb{R}^n} \| \lambda \|^{4j} | \hat{u} (\lambda) |^2 d\lambda, so that by Cauchy-Schwarz inequality we may bound
\partial^\alpha u(x) = \int_{\mathbb{R}^n} (2i\pi\lambda)^{\alpha} e^{2i\pi \langle x, \lambda\rangle} \hat{u} (\lambda) d\lambda = \int_{\mathbb{R}^n}  (2i\pi\lambda)^{\alpha} e^{2i\pi \langle x, \lambda\rangle} \sqrt{ \sum_{j=0}^k (2\pi)^{4j}  \| \lambda \|^{4j} } \frac{\hat{u} (\lambda)}{\sqrt{ \sum_{j=0}^k (2\pi)^{4j}  \| \lambda \|^{4j} } } d\lambda,
by \sum_{j=0}^k \|\Delta^j u \|^2_{ \mathbf{L}^2 (\mathbb{R}^n,\mathbb{R})} only when \int_{\mathbb{R}^n} \frac{\|\lambda\|^{2\alpha} d\lambda}{ \sum_{j=0}^k (2\pi)^{4j}  \| \lambda \|^{4j}} < \infty that is if k  > \frac{m}{2}+\frac{n}{4}. We are now in position to prove the following regularization estimate for the semigroup associated with an elliptic operator.

Proposition: Let L be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure \mu. Denote by (\mathbf{P}_t)_{t \ge 0} the corresponding semigroup on \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}).

  • If K is a compact set of \mathbb{R}^n, there exists a positive constant C such that for f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), \sup_{x \in K} |\mathbf{P}_t f(x)| \le C \left( 1 +\frac{1}{t^{\kappa}} \right) \| f \|_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})}, where \kappa is the smallest integer larger than \frac{n}{4}.
  • For f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), the function (t,x)\rightarrow \mathbf{P}_tf (x) is smooth on (0,+\infty)\times \mathbb{R}^n.

Proof: Let us first observe that from the spectral theorem that if f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) then for every k \ge 0, L^k \mathbf{P}_t f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) and \|L^k \mathbf{P}_t f \|_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})} \le \left(\sup_{\lambda \ge 0} \lambda^k e^{-\lambda t}\right) \|f \|_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})}. Now, let K be a compact set of \mathbb{R}^n. From the previous proposition, there exists therefore a positive constant C such that \left(\sup_{x \in K} | \mathbf{P}_t f(x) | \right)^2 \le C \left( \sum_{k=0}^{\kappa} \|L^k \mathbf{P}_t f \|^2_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})} \right). Since it is immediately checked that \sup_{\lambda \ge 0} \lambda^k e^{-\lambda t}=\left( \frac{k}{t}\right)^k e^{-k}, the bound \sup_{x \in K} |\mathbf{P}_t f(x)| \le C \left( 1 +\frac{1}{t^{\kappa}} \right) \| f \|_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})} easily follows. We now turn to the second part. Let f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}). First, we fix t > 0. As above, from the spectral theorem, for every k \ge 0, L^k \mathbf{P}_t f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), for any bounded open set \Omega. By hypoellipticity of L, we deduce therefore that \mathbf{P}_t f is a smooth function.

Next, we prove joint continuity in the variables (t,x)\in (0,+\infty)\times \mathbb{R}^n. It is enough to prove that if t_0 >0 and if K is a compact set in \mathbb{R}^n, \sup_{x \in K} | \mathbf{P}_{t} f(x) - \mathbf{P}_{t_0} f(x) | \rightarrow_{t \to t_0} 0. From the previous proposition, there exists a positive constant C such that \sup_{x \in K} | \mathbf{P}_{t} f(x) - \mathbf{P}_{t_0} f(x) |  \le C \left( \sum_{k=0}^{\kappa} \|L^k \mathbf{P}_t f-L^k \mathbf{P}_{t_0} f \|^2_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})} \right). Now, again from the spectral theorem, it is checked that \lim_{t \to t_0} \sum_{k=0}^{\kappa} \|L^k \mathbf{P}_t f-L^k \mathbf{P}_{t_0} f \|^2_{ \mathbf{L}_\mu^2 (\Omega,\mathbb{R})}=0. This gives the expected joint continuity in (t,x). The joint smoothness in (t,x) is a consequence of the second part of the previous proposition and the details are let to the reader \square

Remark: If the bound \sup_{x \in K} |\mathbf{P}_t f(x)| \le C(t) \| f \|_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})} uniformly holds on \mathbb{R}^n, that is if \| \mathbf{P}_t \|_{ \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) \rightarrow \mathbf{L}_{\mu}^\infty (\mathbb{R}^n,\mathbb{R})} < \infty, then the semigroup (\mathbf{P}_t)_{t \ge 0} is said to be ultracontractive.

Exercise: Let L be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure \mu . Let \alpha be a multi-index. If K is a compact set of \mathbb{R}^n, show that there exists a positive constant C such that for f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), \sup_{x \in K} |\partial^{\alpha} \mathbf{P}_t f(x)| \le C \left( 1 +\frac{1}{t^{|\alpha|+\kappa}} \right) \| f \|_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R})}, where \kappa is the smallest integer larger than \frac{n}{4}.

We are now in position to prove the following fundamental theorem:

Theorem: Let L be an elliptic and essentially self-adjoint diffusion operator. Denote by (\mathbf{P}_t)_{t \ge 0} the corresponding semigroup on \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}). There is a smooth function p(t,x,y), t \in (0,+\infty), x,y \in \mathbb{R}^n, such that for every f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) and x \in \mathbb{R}^n , \mathbf{P}_t f (x)=\int_{\mathbb{R}^n} p(t,x,y) f(y) d\mu (y). The function p(t,x,y) is called the heat kernel associated to (\mathbf{P}_t)_{t \ge 0}. It satisfies furthermore:

  • (Symmetry) p(t,x,y)=p(t,y,x);
  • (Chapman-Kolmogorov relation) p(t+s,x,y)=\int_{\mathbb{R}^n} p(t,x,z)p(s,z,y)d\mu(z).


Proof: Let x\in \mathbb{R}^n and t > 0. From the previous proposition, the linear form f \rightarrow \mathbf{P}_t f (x) is continuous on \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), therefore from the Riesz representation theorem, there is a function p(t,x,\cdot)\in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), such that for f \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), \mathbf{P}_t f (x)=\int_{\mathbb{R}^n} p(t,x,y) f(y) d\mu (y). From the fact that \mathbf{P}_t is self-adjoint on \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}), \int_{\mathbb{R}^n} (\mathbf{P}_t f) g d\mu=\int_{\mathbb{R}^n} f(\mathbf{P}_t g)  d\mu, we easily deduce the symmetry property p(t,x,y)=p(t,y,x). And the Chapman-Kolmogorov relation p(t+s,x,y)=\int_{\mathbb{R}^n} p(t,x,z)p(s,z,y)d\mu(z) stems from the semigroup property \mathbf{P}_{t+s} =\mathbf{P}_t \mathbf{P}_s. Finally, from the previous proposition the map (t,x) \rightarrow p(t,x,\cdot) \in \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) is smooth on (0,+\infty) \times \mathbb{R}^n for the weak topology of \mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}). This implies that it is also smooth on (0,+\infty) \times \mathbb{R}^n for the norm topology. Since, from the Chapman-Kolmogorov relation p(t,x,y)=\langle p(t/2,x,\cdot), p(t/2,y.\cdot) \rangle_{\mathbf{L}_{\mu}^2 (\mathbb{R}^n,\mathbb{R}) }, we conclude that (t,x,y)\rightarrow p(t,x,y) is smooth on (0,+\infty) \times \mathbb{R}^n \times \mathbb{R}^n \square

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