# 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.

197
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### Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms

Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...

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### Regularity up to the boundary of solutions of the heat equation

Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...

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### approximation of a Feller semi-group with the infinitesimal generator

Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...

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### Solution to $u_t = A(t)u + f(t)$ on bounded domain

I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...

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### On the Fractional Laplace-Beltrami operator

I would appreciate it if a reference could be given for the following claim.
Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. ...

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### Looking for an electronic copy of Kato's old paper

I would like to know if anyone has an electronic copy of the following paper:
MR0279626 (43 #5347)
Kato, Tosio
Linear evolution equations of "hyperbolic'' type.
J. Fac. Sci. Univ. Tokyo Sect. I ...

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### Continuity in the uniform operator topology of a map

I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...

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### Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?

I'm thinking about the following situation:0 suppose that
$$
S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy
$$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...

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### Evolution operator pointwise continuity

For any $f\in L^{\infty}(\Omega)$ denote by $U(\tau, t)f$ the evolution operator, i.e. the solution of the following problem (I choose this logistic equation one because it has bounded solution on $(0,...

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### Extrapolated Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...

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### Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator

I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...

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### Periodic solution for linear parabolic equation - existence, regularity

I am interested in proving the existence and regularity of solution to the following problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...

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248
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### Strong positivity of Neumann Laplacian

There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...

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### Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a $C_0$-semigroup on some ...

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40
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### On invariant subspaces of a nonautonomous heat semigroup

Consider the one dimensional heat equation with a spacetime dependent smooth conductivity coefficient $\sigma(t,x)>0$ on $[0,T)\times (0,1)$ for some $T>0$ subject to source terms $F$, that is ...

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### The inversion of the Laplacian transform Pazy's Book "semigroups of linear operators and applications to Partial differential equations"

This question has been posted on Math Stack Exchange but no reply, and so I have to put it here. My question is:
In Pazy's Book page 26, the author gives a proof of Lemma 7.1, the lemma 7.1 says that: ...

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### "Open systems" version of Stone's Theorem for one-parameter groups of quantum operations

Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...

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258
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### What's the name of this semi-group theorem?

I encountered this theorem, that for a bounded linear transform $L$ and a real parameter $t$ and initial data $u_0$, we have
$$\frac{d}{dt} \exp(Lt)[u_0] = L \exp(Lt)[u_0].$$
What is the name of this ...

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### If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?

Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...

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### Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?

I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...

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### Determine the adjoint of the generator of a Markov semigroup

Let
$(E,\mathcal E)$ be a measurable space and $$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$$ be equipped with the supremum norm;
$(\kappa_t)_{t\ge0}$ ...

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135
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### Relationship between elliptic and parabolic problems and their discretizations

Let us consider the fully nonlinear problem
$$
\begin{cases}
F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\
u=0 & \text{ in } \partial \Omega
\end{cases}
$$
Suppose that we know that the ...

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### Properties of the weak generator of a Markov semigroup

Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $E$
$(\mathcal D(...

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1
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163
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### If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?

Let
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ ...

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### Heat kernel on quaternion Heisenberg group

For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...

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### Reference on (semi)group generated by nonlinear map

Let $X$ be a Banach space and $A : X \rightarrow X$ a map which is not necessarily linear. I am interested in solutions to the problem of finding a family $(y(t))_{t \in \mathbb{R}}$ (or $t \geq 0$) ...

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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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40
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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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### 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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91
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### 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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150
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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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### 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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### 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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2
answers

109
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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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41
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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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### 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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82
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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 ...

4
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102
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### 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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329
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### 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 $L^1(\mathbb{R})$; ...

2
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### 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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84
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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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2
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89
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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 $d \ge 3$.

3
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76
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### 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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140
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### 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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2
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329
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### 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)\...