Questions tagged [semigroups-of-operators]

(Usually one-parameter) semigroups of linear operators and their applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.

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Representation of heat kernel in general domains

I'm looking for results on the representation of the heat kernel in general domains. If $e^{-\Delta_{\Omega} t}$ denotes the heat semigroup in $\Omega$, we have to $$ (e^{-\Delta_{\Omega} t}f)(x) = \...
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How was this heat semigroup estimate made in a paper on reaction–diffusion systems?

In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage $$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$ Here $S(t)$ denotes the heat ...
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3 votes
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Infinitesimal generator of a Markov process acting on a measure

Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...
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Uniformly continuous semigroups are analytic

Reposting from stackexchange. I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous. I want to know where uniform continuity fits in ...
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Is there any class of initial data for which the heat semigroup is increasing in time?

Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is ...
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2 votes
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65 views

Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition

$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$ \begin{cases} \partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
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$H^s$ norm of dispersive semigroup

The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm $$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
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Is there any way this property of semigroups can be satisfied?

Suppose you have the heat semigroup $(S(t))_{t>0}$, such that $$S(t)u(x) = (4\pi t)^{-n/2}\int_{\mathbb{R}^n}e^{-|x-y|^2/4t}u(y)dy.$$ The semigroup has the property that $$S(t)S(s)u(x) = S(t+s)u(x)....
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Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator

Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n  \}_{n \in \mathbb{N}}$ is a non-...
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The semigroup of Laplace-Beltrami operator on 3-flat torus

I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 ...
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$C_{0}$-semigroups with growth bound zero: how do constants $M_{\omega}$ grow as $\omega\to 0^{+}$?

Let $\{T(t)\}_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$ and define the growth bound to be the quantity \begin{equation} \omega_{A}=\inf\{\omega\in\mathbb{R} \mid \exists\text{ } M_{\...
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Is the underlying set of every renormalization group countable and finite? [closed]

Is the underlying set of every renormalization group countable and finite? Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
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2 votes
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Friedrichs Inequality

I'm a little confused with the following proof of Friedrichs inequality in Lawson's & Michelsohn's book Spin geometry, page 194, Theorem 5.4. I don't understand why the last inequality, i.e. $$ C(\...
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Exponential decay of Fisher information along the OU semigroup

I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$ where $\gamma$ is ...
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How to show a space is an invariant core for a strongly continuous semigroup?

This question comes from a paper 2015(Kolokoltsov) Theorem 4.1. In the end of the proof i), “Applying to $T_t$ the procedure applied above to $T_t^h$ shows that $T_t$ defines also a strongly ...
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3 votes
1 answer
128 views

Contractivity of Neumann Laplacean

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on Semigroups of Linear operators I found on many places properties of the Neumann Laplacean. In W....
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Results about Schrödinger equations

Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
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4 votes
1 answer
85 views

Isomorphic generators

Given the two generators $(A,D(A))$ and $(B,D(B))$ of two $C_0$-semigroups on $X$ and $Y$ ( Banach spaces), respectively. We assume that there exists an isomorphism $V:D(A)\longrightarrow D(B)$ such ...
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11 votes
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322 views

Tauberian Theorem for 1-parameter groups of operators

The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
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2 votes
1 answer
183 views

Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$

Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
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Methods to find the spectrum of an operator

Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$? Here is the setting I'm wondering about: consider ...
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1 answer
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Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
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3 votes
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71 views

Ornstein-Ulhenbeck like semigroup for the orthogonal group with a hypercontractive inequality

I am looking for an Ornstein-Uhlenbeck like semigroup $P_t$ and associated generator $\mathcal{L}$ on $G = \operatorname{SO}(n)$ or $\operatorname{O}(n)$ that has a hypercontractive inequality with a ...
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129 views

When is the heat semigroup Gibbs?

Defining the Laplacian on a region $Ω$ of $\mathbb{R}^d$ with Dirichlet boundary conditions, under what conditions on the region (or any other possible assumptions) is the semigroup it generates Gibbs,...
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6 votes
2 answers
301 views

Vacuum region with positive measure for the Schrödinger equation

Let us consider the free Schrödinger equation $(i\partial_t+\Delta_x)\psi=0$ in $\mathbb{R}_t\times\mathbb{R}_x^d$. I'm trying to understand the structure of the vacuum region $$\Omega(\psi):=\{(t,x)\...
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2 votes
1 answer
91 views

On a property of resolvents associated with holomorphic semigroups

This question is about semigroup theory. Let $E$ be a locally compact metric space, and $X=(X_t,t\ge 0;\,P_x,x\in E)$ be a Markov process on $E$. We assume that $X$ is symmetric with respect to $m$, ...
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Identify input maps corresponding to control system ($A,B$), where $B = \delta_0$

The text I'm struggling with comes from "Observation and Control for Operator Semigroups" by Tucsnak and Weiss, page 119. Take $X = L^2(0,\infty)$ and let $X_{-1}$ be the dual of the Sobolev ...
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Independence of variables predicted by the generator

Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
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4 votes
1 answer
337 views

Asymptotic formula for fractional Laplacian

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan ...
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2 votes
0 answers
112 views

Explicit computation of a norm in context of operator-semigroups and differential equations

I am interested in the explicit calculation of the following norm $\vert \cdot \vert$. Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
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2 answers
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Is a linear combination of Markov generator a Markov generator?

Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...
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1 vote
0 answers
47 views

Conservation of the Feller property after passage to the limit

Let $K\subset\mathbb{R}^d$ $(d\geq 1)$ be a compact set, $(D,\|\cdot\|_{\infty)}$ be the set of continuous functions form $[0,1]$ to $K$ endowed with the topology of the supremum norm, and $\mathcal{C}...
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  • 433
2 votes
0 answers
90 views

Boundary terms in integration by parts for the fractional Laplacian

Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$. Is it true that $$ \int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
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1 vote
0 answers
33 views

Existence and uniqueness for fractional parabolic equation with transport term

Let us consider the problem \begin{equation} \begin{cases} u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\ u(x,0)=u_0(x) & \text{in } \...
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1 vote
1 answer
93 views

Integrability of fractional heat kernel

In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
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5 votes
1 answer
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Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
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0 answers
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Eigenvalues of the Laplacian and min-max formula in any space dimension

In which reference book can I find a proof that the eigenvalue of the Laplace operator in a domain $\Omega \subset \mathbb R^d$ with $d \ge 1$ are given by $$ \lambda_1 = \min_{u \in H^1_0(\Omega), \|...
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0 answers
97 views

Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$

How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
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2 votes
0 answers
93 views

Strongly continuous semigroups on weighted $\ell^1$ space

Let $x=(x_i)$ be a sequence in $\ell^1$ such that all $x_i>0.$ Let $T(t):\ell^1 \rightarrow \ell^1$ be a strongly continuous semigroup of, i.e. $t \mapsto T(t)y$ is continuous for every $y \in \ell^...
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2 votes
1 answer
54 views

Link between exact null controllability of two systems

Let $A: D(A) \subset H \rightarrow H$ generate a strongly continuous semigroup $T(t)$ on a Hilbert space $H$ and $B\in \mathcal{B}(H)$. Consider the two control systems: $$(1)\; x'(t)=Ax(t)+ Bu(t) \...
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1 vote
1 answer
112 views

Fractional Laplacian equation on a ball and explicit solutions

Let us consider \begin{align*} (-\Delta)^s u &= 0 && x \in B_r(0) \\ u&=0 && x \in \mathbb R^N \setminus B_r(0), \end{align*} where $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \...
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1 vote
2 answers
174 views

How can I prove that $(I+\lambda G)$ is invertible, where $G$ is the Green function of an elliptic operator?

How can I prove that the operator $$(I+\lambda G)$$ is invertible, where $\lambda >0$ and $G$ is the Green function of an elliptic operator $A$ in a bounded domain $\Omega$? $\Omega$ can be very ...
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3 votes
1 answer
166 views

Neumann/Robin Laplacian semigroup well-known estimate

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on: $$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
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1 vote
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60 views

Show that the Laplacian on these domains is isospectral

Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
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  • 383
2 votes
0 answers
63 views

Weighted translation semigroup

Given a strictly positive function $a:[0,1]\longrightarrow \mathbb{R}$, I have a question about the generation of a $C_0$-semigroup on $L^p([0,1])$ ($1\le p<\infty$) by the following maximal ...
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7 votes
3 answers
560 views

What is dispersive estimate?

Consider free NLS: $i\partial_tu+\Delta u=0, \quad u(0, x)=u_0$ The solution of this IVP, can be written as $$u(x,t)=e^{it\Delta}u_0(x)$$ It is clear to me that how to prove following estimate: $$ \|e^...
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1 vote
1 answer
54 views

Stability of densly defined $C_{0}$-semigroup

Let $(S(t))_{t \geq 0}$ be a $C_{0}$-semigroup on $H$ where $H$ is a Hilbert space. Suppose that $(S(t))_{t \geq 0}$ satisfies the following estimate on a dense subspace on $H$ $$||S(t)x||_H \leq e^{-...
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  • 525
1 vote
1 answer
92 views

Spectrum decomposition of the scaling operator on weighted spaces

Consider the bounded linear operator $M_a$ defined by $M_au(x)=\frac{1}{\sqrt{a}}u\left(\frac{x}{a}\right)$, for $a>1$. On $L^2(\mathbb{R})$, it is easy to see that this is a unitary operator and ...
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1 vote
0 answers
48 views

Is every cyclic right action of a cancellative invertible-free monoid on a set isomorphic to the set of shifts of some homography?

The terms are defined in a related question. [1] Conjecture 1. Let $A$ be a set, $W$ a cancellative invertible-free monoid, and $\cdot\colon A\times W\rightarrow A$ a cyclic right $W$-action generated ...
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4 votes
1 answer
146 views

Is every invertible-free cancellative monoid action represented by "shifting" certain maps?

[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments] Let $W,X$ be ...
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