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

Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$; see this for instance. It seems that we can deduce the time derivative of the heat ...
tianJ's user avatar
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Semigroup property in SPDEs

In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$ However, in various literatures, I ...
Y. Li's user avatar
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1 vote
0 answers
63 views

The derivative of semigroup in the weak sense imply strong sense

Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
Richard's user avatar
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2 votes
0 answers
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interchange of integrals and semigroup without the semigroup being an integral operator

In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears The formula (1.5.2) is Duhamel formula: $$u(t) = T(t)u(...
Ilovemath's user avatar
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2 votes
1 answer
119 views

Domain of the infinitesimal generator of a composition $C_0$-semigroup

In the paper [1] the following $C_0$-group is presented, $$ T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E $$ where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
Scottish Questions's user avatar
1 vote
0 answers
22 views

Analyticity of the semigroup generated by the sublaplacian on unimodular Lie group

Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates ...
Ilovemath's user avatar
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6 votes
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79 views

Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)

Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
ABIM's user avatar
  • 5,079
2 votes
1 answer
155 views

Koopman operators on $L^p(X)$

On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
Scottish Questions's user avatar
1 vote
1 answer
118 views

Is the heat kernel of a manifold $p$-integrable?

If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
Alex M.'s user avatar
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0 answers
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Generator of sub-Markov semigroup induces generator of Markov semigroup

I have to show that for the generator $A:L^1 \rightarrow L^1$ of a sub-Markov semigroup and a non-negative $f_* \in L^1$ (with $L^1$ Set of Lebesgue-integrable functions) with $\int_{-\infty}^\infty ...
Mathhead123's user avatar
1 vote
0 answers
66 views

Parabolic regularity for weak solution with $L^2$ data

I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions: $$\begin{cases}\...
Bogdan's user avatar
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3 votes
1 answer
229 views

$L^{\infty}$ estimate for heat equation with $L^2$ initial data

Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{...
Bogdan's user avatar
  • 1,506
0 votes
0 answers
50 views

Rescaling of cosine families

First of all, the best wishes for 2024. Recently, I got aware of cosine operator families (in the framework of evolution equations). It is well-known, that operator semigroups can be rescaled (see for ...
Alondra Isla Stablum's user avatar
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0 answers
48 views

Reference needed for powers of semi-group generators

Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$. For example, if the ...
matilda's user avatar
  • 90
3 votes
1 answer
151 views

Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard

In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
monotone operator's user avatar
4 votes
1 answer
140 views

Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup

I am looking for a reference of the following result: Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let $$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
Peter Wacken's user avatar
1 vote
1 answer
84 views

Bounded $C_0$-semigroups on barrelled spaces are equicontinuous

I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, ...
Sonam Idowu's user avatar
3 votes
0 answers
201 views

A few questions on Feller processes

Update. Most of my questions have been answered in the comments. I am adding these answers to the post. There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
tsnao's user avatar
  • 600
1 vote
0 answers
64 views

A question about semigroups in a Heisenberg group

I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
Ilovemath's user avatar
  • 645
3 votes
0 answers
74 views

Algebra core for generator of Dirichlet form

This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
Curious's user avatar
  • 143
4 votes
0 answers
76 views

Reference/Help request for formula $[A,e^{-itB}]$ found in physics thread

I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are ...
DerGalaxy's user avatar
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0 answers
32 views

Concerning the conversion of an essential supremum to a pointwise estimate

In the following paper : Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian, Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3747-...
Sarvesh Ravichandran Iyer's user avatar
1 vote
0 answers
36 views

Existence for a nonlinear evolution equation with a monotone operator that is not maximal

We consider the nonlinear evolution equation $$ \dot{u}(t) + Bu(t) = 0, \quad u(0)=0 $$ with $$ A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
ChocolateRain's user avatar
1 vote
0 answers
77 views

Commutator of self-adjoint operators and $C^1$-type formula

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 ...
DerGalaxy's user avatar
3 votes
2 answers
134 views

Lumer-Phillips-type theorem for non-autonomous evolutions

The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
Peter Wacken's user avatar
0 votes
0 answers
140 views

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(...
elmas's user avatar
  • 55
1 vote
0 answers
217 views

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\...
joaquindt's user avatar
4 votes
1 answer
144 views

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 ...
Marco's user avatar
  • 293
1 vote
0 answers
59 views

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 \...
Maschadi's user avatar
3 votes
1 answer
147 views

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. ...
Ali's user avatar
  • 4,113
3 votes
1 answer
176 views

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 ...
Math's user avatar
  • 491
1 vote
0 answers
66 views

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 ...
Mathlover's user avatar
2 votes
0 answers
54 views

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 ...
Ilovemath's user avatar
  • 645
3 votes
0 answers
143 views

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 ...
Malik Amine's user avatar
0 votes
0 answers
87 views

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 ...
ahdahmani's user avatar
  • 101
2 votes
1 answer
265 views

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)...
Bogdan's user avatar
  • 1,506
4 votes
1 answer
341 views

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 ...
Bogdan's user avatar
  • 1,506
4 votes
1 answer
237 views

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 ...
Malik Amine's user avatar
0 votes
0 answers
52 views

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: ...
monotone operator's user avatar
4 votes
1 answer
138 views

"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 ...
Yonah Borns-Weil's user avatar
1 vote
1 answer
298 views

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 ...
askquestions2's user avatar
1 vote
0 answers
36 views

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 $(\...
0xbadf00d's user avatar
  • 157
2 votes
0 answers
112 views

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 ...
Uriel Herrera's user avatar
0 votes
1 answer
152 views

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 ...
user485442's user avatar
2 votes
1 answer
178 views

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 $x\in E$ $\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$ $(\kappa_t)_{t\ge0}$ ...
0xbadf00d's user avatar
  • 157
1 vote
0 answers
441 views

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(\...
user484672's user avatar
1 vote
0 answers
34 views

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 $t \geq 0$) ...
Seven9's user avatar
  • 525
2 votes
0 answers
136 views

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) = \...
Ilovemath's user avatar
  • 645
0 votes
0 answers
49 views

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 ...
Ilovemath's user avatar
  • 645
4 votes
0 answers
263 views

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 ...
Ziv's user avatar
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