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
questions
0
votes
0
answers
72
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
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 21
1
vote
0
answers
76
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\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\...
- 41
4
votes
1
answer
103
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 ...
- 253
1
vote
0
answers
54
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 \...
- 11
2
votes
1
answer
92
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. ...
- 3,863
3
votes
1
answer
162
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 ...
- 459
1
vote
0
answers
46
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 ...
- 31
2
votes
0
answers
41
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 ...
- 337
0
votes
0
answers
39
views
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,...
- 1,154
2
votes
0
answers
127
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 ...
- 279
0
votes
0
answers
51
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 ...
- 101
2
votes
1
answer
116
views
Periodic solution for linear parabolic equation - existence, regularity
I am interested in proving the existence and regularity of solution to the following problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
- 1,154
4
votes
1
answer
248
views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 1,154
4
votes
1
answer
142
views
Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a $C_0$-semigroup on some ...
- 279
0
votes
0
answers
40
views
On invariant subspaces of a nonautonomous heat semigroup
Consider the one dimensional heat equation with a spacetime dependent smooth conductivity coefficient $\sigma(t,x)>0$ on $[0,T)\times (0,1)$ for some $T>0$ subject to source terms $F$, that is ...
- 3,863
0
votes
0
answers
41
views
The inversion of the Laplacian transform Pazy's Book "semigroups of linear operators and applications to Partial differential equations"
This question has been posted on Math Stack Exchange but no reply, and so I have to put it here. My question is:
In Pazy's Book page 26, the author gives a proof of Lemma 7.1, the lemma 7.1 says that: ...
4
votes
1
answer
111
views
"Open systems" version of Stone's Theorem for one-parameter groups of quantum operations
Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...
- 804
1
vote
1
answer
258
views
What's the name of this semi-group theorem?
I encountered this theorem, that for a bounded linear transform $L$ and a real parameter $t$ and initial data $u_0$, we have
$$\frac{d}{dt} \exp(Lt)[u_0] = L \exp(Lt)[u_0].$$
What is the name of this ...
- 217
1
vote
0
answers
28
views
If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?
Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...
- 131
2
votes
0
answers
106
views
Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?
I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...
0
votes
0
answers
67
views
Determine the adjoint of the generator of a Markov semigroup
Let
$(E,\mathcal E)$ be a measurable space and $$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$$ be equipped with the supremum norm;
$(\kappa_t)_{t\ge0}$ ...
- 131
0
votes
1
answer
135
views
Relationship between elliptic and parabolic problems and their discretizations
Let us consider the fully nonlinear problem
$$
\begin{cases}
F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\
u=0 & \text{ in } \partial \Omega
\end{cases}
$$
Suppose that we know that the ...
0
votes
0
answers
34
views
Properties of the weak generator of a Markov semigroup
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $E$
$(\mathcal D(...
- 131
2
votes
1
answer
163
views
If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?
Let
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ ...
- 131
1
vote
0
answers
411
views
Heat kernel on quaternion Heisenberg group
For the n Heisenberg($\Bbb C^n\times\Bbb R$) it is known that the heat kernel $q_s(z,t)=c_n\int_{\Bbb R} e^{-i\lambda t}\Big( \frac{\lambda}{\sinh(\lambda s)}\Big)^n e^{-\frac{\lambda|z|^2\coth(\...
- 11
1
vote
0
answers
27
views
Reference on (semi)group generated by nonlinear map
Let $X$ be a Banach space and $A : X \rightarrow X$ a map which is not necessarily linear. I am interested in solutions to the problem of finding a family $(y(t))_{t \in \mathbb{R}}$ (or $t \geq 0$) ...
- 493
2
votes
0
answers
61
views
Representation of heat kernel in general domains
I'm looking for results on the representation of the heat kernel in general domains. If $e^{-\Delta_{\Omega} t}$ denotes the heat semigroup in $\Omega$, we have to
$$ (e^{-\Delta_{\Omega} t}f)(x) = \...
- 337
0
votes
0
answers
40
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 ...
- 337
4
votes
0
answers
159
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 ...
- 121
1
vote
0
answers
96
views
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 ...
- 121
1
vote
0
answers
65
views
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
...
- 337
2
votes
0
answers
91
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\...
- 121
1
vote
0
answers
82
views
$H^s$ 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
$$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
- 89
1
vote
0
answers
61
views
Is there any way this property of semigroups can be satisfied?
Suppose you have the heat semigroup $(S(t))_{t>0}$, such that
$$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)....
- 337
2
votes
1
answer
89
views
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-...
- 694
1
vote
1
answer
150
views
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 ...
- 89
-4
votes
1
answer
158
views
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?
- 2,824
2
votes
0
answers
117
views
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.
$$
C(\...
- 21
0
votes
2
answers
109
views
Exponential decay of Fisher information along the OU semigroup
I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$
where $\gamma$ is ...
- 587
1
vote
0
answers
41
views
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 ...
- 165
3
votes
1
answer
222
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....
- 1,154
0
votes
0
answers
82
views
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 ...
- 337
4
votes
1
answer
102
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 ...
- 279
11
votes
0
answers
329
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})$; ...
- 4,883
2
votes
1
answer
193
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 ...
- 395
0
votes
0
answers
84
views
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 ...
- 143
1
vote
2
answers
89
views
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$.
- 99
3
votes
0
answers
76
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 ...
- 921
0
votes
0
answers
140
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,...
6
votes
2
answers
329
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)\...
- 282