Questions tagged [semigroup-of-operators]
For questions related to theory of semigroups of linear operators and its applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.
623
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Semigroup solving Schrödinger equation weakly is a unitary group
Thank you for reading!
Setup:
Let $(\mathcal{H},\langle \cdot |\cdot\rangle)$ be a separable complex Hilbert space, and $A:D(A)\to \mathcal{H}$ be a densely defined symmetric unbounded operator in $\...
- 43
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Existence of a weak-star limit of a family of operators in $B(\mathcal{H})$
Thank you in advance for reading this question, and your thoughts.
I am working with a family of operators $(A_{s,\alpha})_{s\geq 0,\alpha>0}$ in the space $B(\mathcal{H})$ (the space of bounded ...
- 43
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Existence of an equivalent norm on a Banach space
I am trying to formulated a lemma, because my main result demainds a change of norm.
I have an operator $A\colon D(A)\subset X\to X$, $X$ Banach, and $(0,\infty)\subset \rho(A)$. This operator has the ...
- 505
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15
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limit with integrated semi-group
Let $S(t)_{t\geq 0}$ be an integrated semigroup on a vector space $E$. Let $ 0\leq\alpha< 1$, $ x\in E$.
I haven't been able to calculate this limit, but I expect it to be zero.
$$ \lim_{\epsilon \...
2
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1
answer
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The Spectrum of the derivative operator in a specific Banach space
Consider the Banach space $X=\left\{u\in C^1([0,1]):\, u(0)=0\right\}$ and the subspace $D=\{u\in C^2([0,1]):\, u(0)=u(1)=u'(0)=0\}$, and the operator $A:D\longrightarrow X$ defined by $Au=u'$. I have ...
- 89
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1
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Searching for Functions Exhibiting Semigroup Property for Energy-Efficient Neuronal Modeling
I am working on designing energy-efficient neurons for neuromorphic computing. One of the critical aspects of the dynamics I'm exploring is that they should adhere to the semigroup property. ...
- 33
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1
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Proving that $(G_\lambda)_\lambda$ is a resolvent family
Let $\mathcal{E}$ be a bilinear form on dense subset $\mathcal{D}\subset H$ of a Hilbert space $H$. Assume $\mathcal{E}$ is a closed, symmetric and positive definite, i.e., $\mathcal{E}(u,u)\geq0$. ...
- 23.5k
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The derivative of the Heat semigroup is the convolution by the derivative of the heat kernel?
Let $(T_t)_{t \geq 0}$ be the Heat semigroup of operators acting on the space $L^p(\mathbb{R}^n)$ with kernel $k_t : \mathbb{R}^n \to \mathbb{R}$. I know that that the map $t \mapsto T_t$ is ...
- 325
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Can we find an explicit solution of this Poisson equation?
Let $\kappa$ be a Markov kernel with invariant measure $\pi$ and $$A:=\kappa-\operatorname{id}$$ denote the corresponding (discrete-time) generator of $\kappa$. Let $c>0$ and $$r:=\frac cp$$ where $...
- 13.2k
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How are the variance of a Markov chain ergodic average and the variance of the corresponding continuous time embedding related?
Let
$(E,\mathcal E)$ be a measurable space;
$\kappa$ be a Markov kernrel on $(E,\mathcal E)$;
$\mu$ be a probability measure on $(E,\mathcal E)$ invariant with respect to $\kappa$
$A:=\kappa-\...
- 13.2k
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References for regularity properties of diffusion/Markov semigroups
The question is related to a technical point in Bakry-Emery calculus (page 130 of Bakry, Gentil, and Ledoux's book "Analysis and geometry of Markov diffusion operators"): given a Markov ...
- 387
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Expressing $\int_0^t\operatorname E_\nu\left[f(X_s)g(X_t)\right]\:{\rm d}s$ as an expectation with respect to another measure $\mu$ instead of $\nu$
In the setting below, I want to compute $$\int_0^t\operatorname E_\nu\left[f(X_s)g(X_t)\right]\:{\rm d}s\tag0.$$
Let
$E$ be a $\mathbb R$-Banach space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $...
- 13.2k
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3
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Perturbation of uniformly continuous semigroup
I think I can prove the following claim about perturbing semigroups, but this must be known somehow, and I am just reinventing the wheel. Do you have any suggestions for references that treat this, ...
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What is the importance of Banach space in the theory of semigroups of linear operators?
Pazy's book defines semigroups as follows. Let $X$ be a Banach space. A one parameter family $T(t)$, $0< t < \infty$, of bounded linear operators from $X$ into $X$ is a semigroup of
bounded ...
- 2,847
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1
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Semigroup properties of spectral fractional Laplacian
I need to check if the different fractional Laplacians have the same properties regarding the semigroups they generate. All I know now is in the case of the Restricted Fractional Laplacian:
$(-\Delta)^...
- 99
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36
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Solving "time evolution" partial differential eq using Lagrange shift operator
I was reading about the Weierstrass transform $(W[\cdot])$, and how it's related to the difussion equation in one dimension. It's relation is given by that $W[f]$ is the convolution with the Heat ...
- 453
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1
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25
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Smoothness of heat kernel on Lipschitz and polygon (cornered) domain
I'm wondering about the spatial smoothness of the heat kernel $K(t,x,x_0)$ on Lipschitz and polygon domains (or cornered domains). It's well known that $K(t,x,x_0)$ is smooth in $t$ for very general ...
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Polynomial decay of any order implies exponential decay for operators norm
Context and Motivation
Consider a Banach space $X$ and a linear bounded operator $T \in \mathcal{L}(X)$. We have established that for each non-negative integer $k$, there exists a constant $C_k$ ...
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1
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Is a dissipative closed operator maximal dissipative?
This is a follow up to the following questions about m-dissipativity:
Are m-dissipative operators closed?
Are m-dissipative operators closed (II)?
Is a dissipative closed operator $A:D(A)\subset X\...
- 147
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Heat semigroup in Heisenberg groups properties
I'm trying to find the construction of the heat semigroup in Heisenberg groups, but I haven't found anything. In particular, I'm trying to find out if the semigroup in the Heisenberg group has the ...
- 2,847
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Semigroup of heat equation: $\|P^\kappa_t \|_{L^p \to L^{p'}} \le c t^{-\frac{d(p'-p)}{2pp'}}$
For any $\kappa>0$, we consider the Gaussian heat kernel
$$
p^\kappa_t (x) := (\kappa \pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{\kappa t}},
\quad t>0, x \in {\mathbb R}^d.
$$
Let $L^0 := L^0 (\...
- 16.7k
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1
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Express $e^{A+B}$ where $B$ is time-dependent unbounded operator as $e^A + \cdots$
Consider the ordered time derivative operator:
$$
D_t := \overset{\longleftarrow}{\frac{\partial }{\partial t}}
$$
Here the arrow indicates differentiation of operators appearing to the left of the ...
- 1,130
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1
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58
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How to get generator of this Gaussian contraction semigroup?
$X=C_0(\mathbb R^n)$ is the closure of Schwartz function space $\mathcal S(\mathbb R^n)$ under the $L^{\infty}(\mathbb R^n)$ norm. Define
$$
T_tu=\left\{
\begin{aligned}
...
- 31
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27
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Semigroup not strongly continuous in 0
The solution of homogenous heat equation in bounded regular domain $\omega$ of $R^{n}$ is
$$u(t,x)=\sum_{n\geq 1}a_{n}(0)\exp(-\lambda_{n}t)e_{n}=S(t) u_{0}$$
where $e_{n}$ is Hilbert basis of $L^{2}...
- 21
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Analytic semigroup whose are uniformly bounded with 0 in resolvent of A
Are analytic semigroups $T(t)$ defined on a Banach space $X$, that is uniformly bounded with $0 \in \rho (A)$ the resolvent set of the infinitesimal generator A, have this property:
$$ \lim_{t \to 0 } ...
- 1
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How can we calculate the variance of this stopping time?
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$ with generator$^1$ $A$;
$\mu,\pi$ be probability measures on $(E,\...
- 13.2k
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45
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Derivitive of operation
I study semigroups and their application in PDEs, and I'm stuck with an idea that I can't understand.
Let $T(t)$ and $S(t)$ be $C_0$ semigroup of bounded linear operator with the same infinitesimal ...
- 1
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Maximal $L^p$-regularity of Laplace-Beltrami operator $\Delta$ on closed manifold
It's well-known that the Dirichlet Laplacian $\Delta$ on flat domain is R-sectorial on $\Sigma_{\pi}$ in $L^p$ space for all $p\in (1,\infty)$.
I'm wondering if the Laplace-Beltrami operator $\Delta$ ...
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1
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How is Hölder's inequality and Parseval's identity applied here?
I am reading this lemma from Pazy's semigroup of linear operators
I am having a hard time understanding
1- Why does the highlighted part hold?
2- How is the Hölder's inequality applied and how the ...
- 147
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Brezis' theorem 7.5: a simplified proof
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $A: D(A) \subset H \to H$ be a maximal monotone (unbounded linear) operator. We define by induction ...
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Commutator of self-adjoint operator with semigroup generated by another self-adjoint operator
Let $\mathcal{H}$ be a (complex) Hilbert space.
Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
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Non local Initial conditions
I am having a really hard time to try to understand this problem. what is non local initial conditions and how it work (or how we can use it) in practical applications like, in physics and engineering ...
- 1
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0
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30
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Space decomposition from the infinitesimal generator of a contraction $C_0$-semigroup
On this week I'm trying to show the subspace $\mathcal{N
(A)} \bigoplus \overline{Im(A)}$ is closed and the equation
$$ \lim_{t \to \infty} \frac{1}{t} \int_b^t T(s)u ds = \lim_{t \to \infty} \frac{1}{...
- 31
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Is there a version of Floquet-Bloch theory for semigroups?
Floquet-Bloch theory provides "nice" spectral decompositions for operators which commute with group actions (usually abelian groups). The simplest example is that of periodic Schrodinger ...
- 4,158
2
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1
answer
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Non-existence of Banach-Tarski in the plane from non-existence on the line
The following theorem is well known:
There exists a isometry invariant finitely additve measure, measuring all subsets of $\mathbb{R}^d$ that extends the Lebesgue measure if and only if $d\le 2$
...
- 5,403
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Generator of a transformed semigroup
Let
$(E,\mathcal E)$ be a measurable space;
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ be equipped with the supremum norm;
$(\kappa_t)_{t\...
- 13.2k
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1
answer
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Interpretation of condition probability of $X_0 = x$ for a Markov process and semigroup $(P_t)$ [closed]
When studying Markov processes, I have seen a lot of authors define the semigroup as $P_tf(x) = \mathbb E_x(f(X_t))$ (with the assumption that $X_t$ is homogeneous) and the call $\mathbb E_x$ as the &...
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Verifying that the Laplacian is the infinitesimal generator of a semigroup.
Let
\begin{align}
\frac{\partial}{\partial_t}u(t,x)&=\Delta u(t,x),\quad t>0,\, x\in\mathbb{R}^n\\
u(0,x)&=f(x)
\end{align}
where $f\in L^2(\mathbb{R}^n)$ (Heat's equation)
The heat kernel ...
- 3,336
3
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1
answer
53
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$L^p$ implies continuity in some way
I want to prove the following,
$$
\lim_{t->0} \|T_p(t)f-f\|_p = 0,
$$
where $T_p(t)f(s) = \chi_{[0,e^{-t}]}(s)e^{t/p}f(se^t)$, with $f\in L^p([0,1])$ and $t>0$ and $s \in [0,1]$.
In fact, we ...
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0
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100
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Proving the existence of a solution of the fractional heat equation using semigroup methods
I am trying to solve the following problem:
$$u_t + (-\Delta)^su = 0$$
in $\Omega \subset \mathbb{R}^N$ with $N > 2s$, where $s \in (0,1)$ and Dirichlet Boundary conditions.
Let my operator $A = (-\...
- 245
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Positivity of semigroup $(P_t)_{t\geq0}$ implies its contractivity, $\| P_tf\| \leq \|f\|$
Let $\Omega$ be a Polish space, denote $E = (C_b(\Omega),\|\cdot\|)$ be the Banach space of continuous and bounded functions with the usual supremum norm. A family $(P_t)_{t\geq 0} $ of linear ...
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Using Semigroup Theory of Linear Operators to show that the operator $(-\Delta)^s$ is closed.
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 245
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1
answer
29
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Reference of duality in Markov Semigroup
I heard that there is a duality of Markov semigroups acting on functionals or measures, does anyone of a reference of this materials?
- 1,209
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0
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35
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On subadditive functions everywhere finite bounded on compact sets
In the book "Functional analysis and Semi-Groups" by E Hille and R S Phillips, theorem 7.4.1 states that subadditive functions defined on some interval $I$ and finite everywhere are bounded ...
- 103
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1
answer
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Are there good bounds for the norm of a uniformly continuous semigorup $e^{tA}$ in terms of its generator $A$?
Are there good bounds for the norm of a uniformly continuous semigorup $e^{tA}$ in terms of its generator $A$? I'm only looking for references. In the standard literature, I wasn't able to find ...
- 13.2k
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0
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75
views
Intuitively, why does the spectral gap control the speed of convergence to equilibrium.
Is there an intuitive way to understand the following principle :
Given a Markov Process $X_t$ with generator $L$, why does the spectral gap of $L$ control the speed of convergence to equilibrium for $...
- 2,140
3
votes
0
answers
72
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Equivalent definitions of a dissipative operators in Banach Space
In the proof of proposition 3.23 (bellow in the picture) in Engel-Nagel book's (One-Parameter Semigroups for Linear Evolution Equations) they claim that $z'_\lambda$ (in the sphere of $X^\ast$) has a ...
2
votes
1
answer
87
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How can I prove $L^p$ continuity of the left-shift operator?
I've been self-studying $C_0$-semigroups through Innsbruck's online lectures and one of the exercises is to show that the left-shift operator
$$(S(t)f)(s)=f(t+s)$$
forms a strongly continuous ...
- 3,084
1
vote
0
answers
32
views
An inequality related to a semigroup of operators.
I would like to prove that
$$4CLh\int_0^t {\frac{{ds}}{{\left( {t - s + h} \right){{\left( {t - s} \right)}^{1 - \theta }}}}} \leq {C_1}{h^\theta }$$
where $\theta \in \left( {0,1} \right)$. $C, L, ...
- 11
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votes
0
answers
97
views
Does the exponential operator $e^{tA}$ defined via functional calculus and via semi group coincide?
Given $X$ is a Hilbert space and consider the operator $A$ on $X$.
Functional calculus:
given an unbounded operator $A$ (densely defined, closed and self-adjoint), we can define $e^{tA}$ by the ...
- 81