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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62 views
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Asymptotics for fractional Laplacian

This question is motivated by Asymptotic formula for fractional Laplacian For the equation $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\...
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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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39 views

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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200 views
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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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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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121 views

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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41 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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45 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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28 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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1answer
71 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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66 views

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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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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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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86 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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1answer
48 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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54 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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153 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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107 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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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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58 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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312 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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50 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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68 views

Spectrum of generalized Ornstein-Uhlenbeck generator?

At some point, I stumbled over the work by Giorgio Metafune who basically solved, with collaborators, the problem of understanding the spectrum of the Ornstein-Uhlenbeck operator on $L^p$ with respect ...
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1answer
65 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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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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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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Convergence of steady states for Lindblad systems in infinite volume

In the physics of open quantum systems it is common to consider the Lindblad form. Which for a (super)-operator $\mathcal{L} \in B(B(\mathbb{C}^n ))$ means that \begin{align*} \mathcal{L}(\rho) = - i \...
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1answer
93 views

Holomorphic semigroups on $L^1$ spaces

Let $E$ be a locally compact metric space and $\mu$ a non-negative Radon measure on $E$ (we also assume that the support is $E$). I am concerned with holomorphic semigroups on $L^1(E,\mu)$. In ...
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Reference on infinitesimal generators for functional SDEs

I am trying to solve a problem using convergence of infinitesimal generators of functional SDEs. I havt yet found good material on this. The setting is that of Wanatbe Ikeda -88 page 167. It looks ...
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1answer
47 views

Jensen’s inequality for Heat semigroup is valid for Schrödinger semigroup?

The following result can be found in this article (Jensen’s inequality) Let $v = v(x, t)$ be any nonnegative function. Then it holds that, for all $t > 0$, $$[S(t)v(s)]^q \leq S(t)v^q(s)$$ if $q \...
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1answer
62 views

Characterization of the generator of a Lévy process using martingale problems

Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
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45 views

Time dependent reaction-diffusion semigroup

I'm interested in the following linear reaction-diffusion equation \begin{align*} &\partial_tu(t,x) = \sigma(t)\Delta u(t,x),\\ & u(0)=u_0\in X \end{align*} where $X$ is a Banach space and $\...
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27 views

Generator of a Hilbert space valued Wiener process from the solution of a martingale problem

Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
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Uniform stability of linear operators - reference request

Let $T$ be a bounded linear operator on a complex Banach space $X$. I am looking for a reference for the following result: Theorem 1. Let $p \in [1,\infty]$. The following assertions are equivalent: (...
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247 views

The contractivity of the heat semigroup in $L^p$ spaces

Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
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1answer
147 views

Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme)

In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$ $$R_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1_{x\geq y},$$ which is the Poisson distribution pdf. (This is ...
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43 views

Semigroup theory for non-symmetric Markov processes / complex-valued potentials

Let $X$ be a continuous-time Markov process on a countable state space $E$, and let $V:E\mapsto\mathbb C$ be some complex function. $X$ can be characterized by its transition rates $(\lambda_{xy})_{x,...
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Continuity of operator exponential on subspace

Let $H$ be a separable Hilbert space with overcomplete basis $(e_t)_{t \in (0,\infty)}$ such that $\langle e_t, e_s \rangle = e^{-\vert t-s \vert}.$ We can then define the projection $P_t$ which is ...
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66 views

What types of semigroups have a Laplacean type operator as infinitesimal generator?

Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded connected set having Lipschitz uniform boundary. Moreover let $d\in L^{\infty}(\Omega,\mathbb{R}^M),\ d_1(x),d_2(x),\dots, d_M(x)>d>0,\ \...
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Literature request: Schatten class difference of semigroups

Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
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$\|(A_n-z)^{-1} - (A-z)^{-1}\|\to 0\;\Rightarrow\; \|e^{-tA_n}-e^{-tA}\|\to 0$ for general $C_0$ semigroups?

In short, the question is whether norm-resolvent convergence implies operator-norm convergence of the assocoated semigroups. More specifically, assume the following: The $A_n$ generate contraction ...
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A strange convergence for a semigroup of operators

I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows: Let $A,B$ ...
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59 views

The derivative of a $C_0$-semigroup with respect to a perturbation parameter

Let $H$ be a Hilbert space, and $A : H \to H$ be the (semi-bounded) generator of the $1$-parameter $C_0$-semigroup $[0, \infty) \ni t \mapsto \mathrm e ^{-t A}$. Let $B : H \to H$ be a bounded ...
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How does one define the gradient of a Markov semigroup?

In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper: ...
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140 views

Reference on noncommutative PDE

I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
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1answer
120 views

Equality in spectral inclusion theorem

I asked this question on Math SE but didn't receive any response. Let $(T_t)$ be a $C_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda_0\in \mathbb{C}$ is such that $e^{\lambda_0 t}$...
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1answer
85 views

Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...
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1answer
118 views

Analyticity of the semigroup generated by a time-changed Brownian motion

Let $d$ be an integer. We denote by $m$ the Lebesgue measure on $\mathbb{R}^d$. We define $BL(\mathbb{R}^d)$ by \begin{align*} BL(\mathbb{R}^d)=\{f \in L^2_{\rm loc}(\mathbb{R}^d,m) \mid |\nabla f|\in ...
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1answer
45 views

Friedrich's extension of the generator of a continuous time markov chaoin

Consider the infinitesimal generator $G$ of a Markov chain with state space $\mathbb{Z}$ such that it is symmetric with respect to a measure $\mu$ on $\mathbb{Z}$. Then, the operator $(G,C_c(\mathbb{Z}...
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3answers
208 views

$u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup

I asked this question on Mathematics Stackexchange, but got no answer. In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the ...