Lecture 7. Diffusion semigroups in Lp

In the previous lectures, we have seen that if L is an essentially self-adjoint diffusion operator with respect to a measure, then by using the spectral theorem one can define a self-adjoint strongly continuous contraction semigroup on L2 with generator L. This semigroup is moreover positivity preserving and a contraction on the space of bounded square integrable functions. Our goal, in this lecture, is to define, for 1  \le  p \le  +\infty, the semigroup on \mathbf{L}_{\mu}^p (\mathbb{R}^n,\mathbb{R}). This can be done by using the RieszThorin interpolation theorem that we remind below. In this subsection, in order to simplify the notations we simply denote \mathbf{L}_{\mu}^p (\mathbb{R}^n,\mathbb{R}) by \mathbf{L}_\mu^p. We first start with general comments about semigroups in Banach spaces.

Let (B,\| \cdot \|) be a Banach space (which for us will be \mathbf{L}_{\mu}^p, 1 \le  p \le +\infty).

We first have the following basic definition.

Definition: A family of bounded operators (T_t)_{t \ge 0} on B is called a contraction semigroup if:

  • T_0 =\mathbf{Id} and for s,t \ge 0, T_{s+t}=T_s T_t;
  • For each x \in B and t \ge 0, \| T_t x \| \le \|x \|.

A contraction semigroup (T_t)_{t \ge 0} on B is moreover said to be strongly continuous if for each x \in B, the map t \to T_t x is continuous.

In this Lecture, we will prove the following result:

Theorem: let L be an essentially self-adjoint diffusion operator. Denote by (\mathbf{P}_t)_{t \ge 0} the self-adjoint strongly continuous semigroup associated to L and constructed on \mathbf{L}_{\mu}^2 thanks to the spectral theorem. Let 1 \le p \le +\infty. On \mathbf{L}_\mu^p, there exists a unique contraction semigroup (\mathbf{P}^{(p)}_t)_{t \ge 0} such that for f \in \mathbf{L}_\mu^p \cap \mathbf{L}_\mu^2, \mathbf{P}^{(p)}_t f=\mathbf{P}_t f. The semigroup (\mathbf{P}^{(p)}_t)_{t \ge 0} is moreover strongly continuous for 1 \le p < +\infty.

This theorem can be proved by using two sets of methods: Hille-Yosida theory on one hand and interpolation theory on the other hand. We shall use interpolation theory in L^p space which takes advantage of the fact that \mathbf{P}_t only needs to be constructed on L^1 and L^\infty. The Hille-Yosida method starts from the generator and we sketch it below.

Definition:
Let (T_t)_{t \ge 0} be a strongly continuous contraction semigroup on a Banach space B. There exists a closed and densely defined operator A: \mathcal{D}(A) \subset B \rightarrow B where \mathcal{D}(A)=\left\{ x \in B,\quad  \lim_{t \to 0}  \frac{T_t x -x}{t} \text{ exists} \right\}, such that for x \in  \mathcal{D}(A), \lim_{t \to 0} \left\| \frac{T_t x -x}{t} -Ax \right\|=0. The operator A is called the generator of the semigroup (T_t)_{t \ge 0}. We also say that A generates (T_t)_{t \ge 0}.

The following important theorem is due to Hille and Yosida and provides, through spectral properties, a characterization of closed operators that are generators of contraction semigroups.

Let A:  \mathcal{D}(A) \subset B \rightarrow B be a densely defined closed operator. A constant \lambda \in \mathbb{R} is said to be in the spectrum of A if the the operator \lambda \mathbf{Id}-A is not bijective. In that case, it is a consequence of the closed graph theorem that if \lambda is not in the spectrum of A , then the operator \lambda \mathbf{Id}-A has a bounded inverse. The spectrum of an operator A shall be denoted \rho(A).

Theorem: A necessary and sufficient condition that a densely defined closed operator $A$ generates a strongly continuous contraction semigroup is that:

  • \rho (A) \subset (-\infty,0] ;
  • \| (\lambda \mathbf{Id} -A)^{-1} \| \le \frac{1}{\lambda} for all \lambda > 0.

These two conditions are unfortunately difficult to directly check for diffusion operators.
We can bypass the study of the closure in L^p of a diffusion operator by using interpolation theory.

Theorem: (Riesz-Thorin interpolation theorem) Let 1 \le p_0, p_1,q_0,q_1 \le \infty, and \theta \in (0,1). Define 1 \le p,q \le \infty by \frac{1}{p}=\frac{1-\theta}{p_0} + \frac{\theta}{p_1}, \quad  \frac{1}{q}=\frac{1-\theta}{q_0} + \frac{\theta}{q_1}. If T is a linear map such that T:\mathbf{L}_\mu^{p_0} \rightarrow \mathbf{L}_\mu^{q_0}, \quad \| T \|_{ \mathbf{L}_\mu^{p_0} \rightarrow \mathbf{L}_\mu^{q_0} } =M_0 and T:\mathbf{L}_\mu^{p_1} \rightarrow \mathbf{L}_\mu^{q_1}, \quad \| T \|_{ \mathbf{L}_\mu^{p_1} \rightarrow \mathbf{L}_\mu^{q_1} } =M_1, then, for every f \in \mathbf{L}_\mu^{p_0} \cap \mathbf{L}_\mu^{p_1}, \| T f \|_q \le M_0^{1-\theta} M_1^{\theta} \| f \|_p. Hence T extends uniquely as a bounded map from \mathbf{L}_\mu^{p} to \mathbf{L}_\mu^{q} with \| T \|_{ \mathbf{L}_\mu^{p} \rightarrow \mathbf{L}_\mu^{q} }  \le M_0^{1-\theta} M_1^{\theta} .

The statement that T is a linear map such that T:\mathbf{L}_\mu^{p_0} \rightarrow \mathbf{L}_\mu^{q_0}, \quad \| T \|_{ \mathbf{L}_\mu^{p_0} \rightarrow \mathbf{L}_\mu^{q_0} } =M_0 and T:\mathbf{L}_\mu^{p_1} \rightarrow \mathbf{L}_\mu^{q_1}, \quad \| T \|_{ \mathbf{L}_\mu^{p_1} \rightarrow \mathbf{L}_\mu^{q_1} } =M_1 means that there exists a map T:  \mathbf{L}_\mu^{p_0} \cap \mathbf{L}_\mu^{p_1}\rightarrow \mathbf{L}_\mu^{q_0} \cap \mathbf{L}_\mu^{q_1} with \sup_{ f \in  \mathbf{L}_\mu^{p_0} \cap \mathbf{L}_\mu^{p_1} , \| f \|_{p_0} \le 1 } \| Tf \|_{q_0} =M_0 and \sup_{ f \in  \mathbf{L}_\mu^{p_0} \cap \mathbf{L}_\mu^{p_1} , \| f \|_{p_1} \le 1 } \| Tf \|_{q_1} =M_1. In such a case, T can be uniquely extended to bounded linear maps T_0: \mathbf{L}_\mu^{p_0} \rightarrow \mathbf{L}_\mu^{q_0} , T_1: \mathbf{L}_\mu^{p_1} \rightarrow \mathbf{L}_\mu^{q_1}. With a slight abuse of notation, these two maps are both denoted by T in the theorem.

The proof of the theorem can be found in this post by Tao.

One of the (numerous) beautiful applications of the Riesz-Thorin theorem is to construct diffusion semigroups on L^p by interpolation. More precisely, let L be an essentially self-adjoint diffusion operator. We denote by (\mathbf{P}_t)_{t \ge 0} the self-adjoint strongly continuous semigroup associated to L constructed on \mathbf{L}_{\mu}^2 thanks to the spectral theorem. We recall that (\mathbf{P}_t)_{t \ge 0} satisfies the submarkov property: That is, if 0 \le f \le 1 is a function in \mathbf{L}_{\mu}^2 , then 0 \le \mathbf{P}_t f \le 1.

Theorem: The space \mathbf{L}_{\mu}^1 \cap \mathbf{L}_{\mu}^{\infty} is invariant under \mathbf{P}_t and \mathbf{P}_t may be extended from \mathbf{L}_{\mu}^1 \cap \mathbf{L}_{\mu}^{\infty} to a contraction semigroup (\mathbf{P}^{(p)}_t)_{t \ge 0} on \mathbf{L}_{\mu}^{p} for all 1 \le p \le \infty: For f \in  \mathbf{L}_{\mu}^p, \| \mathbf{P}_t f \|_{ \mathbf{L}_{\mu}^p} \le \| f \|_{ \mathbf{L}_{\mu}^p}. These semigroups are consistent in the sense that for f \in \mathbf{L}_{\mu}^p \cap \mathbf{L}_{\mu}^{q}, \mathbf{P}^{(p)}_t f=\mathbf{P}^{(q)}_t f.

Proof: If f,g \in \mathbf{L}_{\mu}^1 \cap \mathbf{L}_{\mu}^{\infty} which is a subset of \mathbf{L}_{\mu}^1 \cap \mathbf{L}_{\mu}^{\infty}, then \left| \int_{\mathbb{R}^n} (\mathbf{P}_t f) g d\mu \right| = \left| \int_{\mathbb{R}^n} f(\mathbf{P}_t g)  d\mu \right | \le \| f \|_{ \mathbf{L}_{\mu}^1} \| \mathbf{P}_t g \|_{ \mathbf{L}_{\mu}^\infty}  \le \| f \|_{ \mathbf{L}_{\mu}^1} \|  g \|_{ \mathbf{L}_{\mu}^\infty}. This implies \| \mathbf{P}_t f \|_{ \mathbf{L}_{\mu}^1} \le \|  f \|_{ \mathbf{L}_{1}^\infty}. The conclusion follows then from the Riesz-Thorin interpolation theorem \square

Exercise: Show that if f \in \mathbf{L}_{\mu}^{p} and g \in \mathbf{L}_{\mu}^{q} with \frac{1}{p}+\frac{1}{q}=1 then, \int_{\mathbb{R}^n} f \mathbf{P}^{(q)}_t g d\mu=\int_{\mathbb{R}^n} g \mathbf{P}^{(p)}_t f d\mu.

Exercise:

  • Show that for each f \in  \mathbf{L}_{\mu}^{1}, the \mathbf{L}_{\mu}^{1}-valued map t \rightarrow \mathbf{P}^{(1)}_t f is continuous.
  • Show that for each f \in  \mathbf{L}_{\mu}^{p}, 1 < p < 2, the \mathbf{L}_{\mu}^{p}-valued map t \rightarrow \mathbf{P}^{(p)}_t f is continuous.
  • Finally, by using the reflexivity of \mathbf{L}_{\mu}^{p}, show that for each f \in  \mathbf{L}_{\mu}^{p} and every p \ge 1, the \mathbf{L}_{\mu}^{p}-valued map t \rightarrow \mathbf{P}^{(p)}_t f is continuous.

We mention, that in general, the \mathbf{L}_{\mu}^{\infty} valued map t \rightarrow \mathbf{P}^{(\infty)}_t f is not continuous.

Due to the consistency property, we always remove the subscript p from \mathbf{P}^{(p)}_t and only use the notation \mathbf{P}_t.

To finish this Lecture, we finally connect the heat semigroup in L^p to L^p solutions of the heat equation.

Proposition: Let f \in L^p_\mu(\mathbb{R}^n), 1 \le p \le \infty, and let u (t,x)= P_t f (x), \quad t \ge 0, x\in \mathbb{R}^n. Then, if L is elliptic with smooth coefficients, u is smooth on (0,+\infty)\times \mathbb{R}^n and is a strong solution of the Cauchy problem \frac{\partial u}{\partial t}= L u,\quad u (0,x)=f(x).

Proof: The proof is identical to the L^2 case. For \phi \in C^\infty_0 ((0,+\infty) \times \mathbb{R}^n), we have
\int_{\mathbb{R}^n \times \mathbb{R}} \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right) u(t,x) d\mu(x) dt
=\int_{\mathbb{R}} \int_{\mathbb{R}^n} \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right)  P_t f (x) dx dt
= \int_{\mathbb{R}} \int_{\mathbb{R}^n}   P_t \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right)  f (x) dx dt
= \int_{\mathbb{R}} \int_{\mathbb{R}^n}   -\frac{\partial}{\partial t} \left(  P_t \phi (t,x) f(x) \right) dx dt =0.
Therefore u is a weak solution of the equation \frac{\partial u}{\partial t}= L u. Since u is smooth it is also a strong solution \square.

We now address the uniqueness of solutions. As in the L^2 case, we assume that L is elliptic with smooth coefficients and that there is a sequence h_n\in C_0^\infty(\mathbb{R}^n), 0 \le h_n \le 1, such that h_n\nearrow 1 on \mathbb{R}^n, and ||\Gamma (h_n,h_n)||_{\infty} \to 0, as n\to \infty.

Proposition: Let v(x,t) be a non negative function such that \frac{\partial v}{\partial t} \le L v,\quad v(x,0)=0, and such that for every t > 0, \| v ( \cdot,t) \|_{L^p_\mu(\mathbb{R}^n)} <+\infty, where 1 < p  < +\infty. Then v(x,t)=0.

Proof: Let x_0 \in \mathbb{R}^n and h \in C_0^\infty(\mathbb{R}^n). Since u is a subsolution with the zero initial data, for any \tau\in (0,T),
\int_0^\tau \int_{\mathbb{R}^n}  h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt
\geq  \int_0^\tau \int_{\mathbb{R}^n}  h^2(x) v^{p-1} \frac{\partial v}{\partial t} d\mu(x) dt
=  \frac{1}{p} \int_0^\tau \frac{\partial }{\partial t} \left( \int_{\mathbb{R}^n} h^2(x) v^{p} d\mu(x)\right) dt
=  \frac{1}{p}\int_{\mathbb{R}^n}  h^2(x) v^{p}(x,\tau) d\mu(x).
On the other hand, integrating by parts yields
\int_0^\tau \int_{\mathbb{R}^n}  h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt =  - \int_0^\tau \int_{\mathbb{R}^n} 2h v^{p-1} \Gamma(h,v)  d\mu dt - \int_0^\tau \int_{\mathbb{R}^n}  h^2 (p-1) v^{p-2} \Gamma(v)  d\mu dt .

Observing that
0\leq  \left(\sqrt{\frac{2}{p-1}\Gamma(h)}v - \sqrt{\frac{p-1}{2}\Gamma(v)}h \right)^2  \leq \frac{2}{p-1}\Gamma(h)v^2 + 2 \Gamma(h,v) h v +\frac{p-1}{2}\Gamma(v)h^2 ,
we obtain the following estimate.
\int_0^\tau \int_{\mathbb{R}^n}  h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt
\leq  \int_0^\tau \int_{\mathbb{R}^n}  \frac{2}{p-1} \Gamma(h) v^p  d\mu dt - \int_0^\tau \int_{\mathbb{R}^n}  \frac{p-1}{2}h^2 v^{p-2} \Gamma(v)  d\mu dt
=  \int_0^\tau \int_{\mathbb{R}^n}  \frac{2}{p-1} \Gamma(h) v^p  d\mu dt - \frac{2(p-1)}{p^2} \int_0^\tau  \int_{\mathbb{R}^n}  h^2 \Gamma(v^{p/2})  d\mu dt

Combining with the previous conclusion we obtain ,
\int_{\mathbb{R}^n} h^2(x) v^{p}(x,\tau) d\mu(x) + \frac{2(p-1)}{p} \int_0^\tau \int_{\mathbb{R}^n}  h^2 \Gamma(v^{p/2})  d\mu dt \leq \frac{2 p}{(p-1) } \| \Gamma(h) \|_\infty^2 \int_0^\tau \int_{\mathbb{R}^n}  v^p  d\mu dt.
By using the previous inequality with an increasing sequence h_n\in C_0^\infty(\mathbb{R}^n), 0 \le h_n \le 1, such that h_n\nearrow 1 on \mathbb{R}^n, and ||\Gamma (h_n,h_n)||_{\infty} \to 0, as n\to \infty, and letting n \to +\infty, we obtain
\int_{\mathbb{R}^n}  v^{p}(x,\tau) d\mu(x)=0 thus v=0 \square.

As a consequence of this result, any solution in L^p_\mu(\mathbb{R}^n), 1 < p < +\infty of the heat equation \frac{\partial u}{\partial t}= L u is uniquely determined by its initial condition, and is therefore of the form u(t,x)=P_tf(x). We stress that without further conditions, this result fails when p=1 or p=+\infty.

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