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.

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60 views

Commutation of $c_0$-semigroups

Task is: given two $c_0$-semigroups $T(t)$ and $S(t)$ in Banach space X with property $\forall \;t \geq 0 : T(t)S(t) = S(t)T(t)$, show that $\forall \; t, s \geq 0 : T(t)S(s) = S(s)T(t)$. I've tried ...
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0answers
28 views

Determine the generator of a contractive semigroup acting on bounded measurable functions by a solution to a martingale problem

Let $(E,\mathcal E)$ be a measurable space, $B(E,\mathcal E)$ denote the set of bounded $\mathcal E$-measurable functions from $E$ to $\mathbb R$ equipped with the supremum norm and $(\kappa_t)_{t\ge0}...
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0answers
31 views

Generator of compound Poisson process

I'm trying to derive the generator $A$ of a compound Poisson process $(X_t)_{t\ge0}$ with Lévy (aka intensity) measure $\nu$. My guess is $$(Af)(x)=\int f(x+y)-f(x)\:\nu({\rm d}y).$$ The transition ...
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0answers
7 views

Where can i find proof of this Lemma? $S(t)$ is the heat semigroup. If $p \geq 1$, then $[S(t)u_0]^p \leq S(t)u^p_0$

The Lemma below is valid for the heat semigroup Lemma. Assume that $u_0 \in C_0(Ω)$, $u_0 \geq 0$. If $p \geq 1$, then $[S(t)u_0]^p \leq S(t)u^p_0$. If $0 < p < 1$, then $[S(t)u_0]^p \geq S(t)u^...
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0answers
47 views

The semi-group property of Riesz potentials in higher dimensions

I'm trying to wrap my head around the Riesz potential in the sense of a higher dimensional generalization of Riemann-Liouville fractional integrals but some things are coming across as somewhat ...
3
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0answers
23 views

What is a $C^\infty$ regularizing contraction semigroup?

could anyone tell me what a $C^\infty$ regularizing contraction semigroup is? I know the contraction semigroup part, but I don't understand what exactly the $C^\infty$ regularizing part. De ...
2
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1answer
43 views

Time derivative of heat semigroup.

Imagine we have the heat semigroup $\{P_t\}_{0\leq t\leq T}$ and remember that we have $$\frac{d}{dt} (P_t\varphi)(x)=\frac 1 2 (P_t\varphi '')(x).$$ I want to calculate following time derivative $$\...
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0answers
34 views

A question from Reed, Simon, Methods of Modern Mathematical Physics II

In example 2 to Theorem X.70 in Reed-Simon we want to construct a propagator for the heat equation with generator $$A(t) = - \Delta + q(x,t) + M + 1$$ on $C_{\infty}$ where $q$ is a bounded (with ...
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1answer
40 views

What is a semigroup in stochastic analysis?

I just read a paper, in which the author wrote we use the same notation for the Brownian semigroup on $C_b(\mathbb{R}_{+})$ Apart from this the paper made no other reference to semigroup. I'm ...
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32 views

Proving that a $C^\infty$ regularizing contraction semigroup leave a subspace invariant.

In Villani's Hypocoercity, I am faced with proving a statement which is indicated by the title. Consider a $C^\infty$ function $V: \mathbb{R}^n \to \mathbb{R}$, convergeing to $+\infty$ fast enough ...
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1answer
73 views

The square of the field is a quadratic variation

I'm looking for sources which elaborate a little bit on the fact that for Markov process $X_t$ with generator $L$, $\int_{}^{}\Gamma(f,f)(X_s)ds$ is a quadratic variation of $M_t := f(X_t) - f(x) - \...
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0answers
33 views

Infinite matrix to linear map (context: Markov chains)

I am currently working on infinite-state continuous-time Markov chains. Let $S$ be a countably infinite set, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers, equip $S$ with the discrete ...
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1answer
45 views

Can a single point uniquely determine the whole semi-group?

We have two strongly continuous semi-group $\{T(t)\}_{t\geq0}$ and $\{S(t)\}_{t\geq0}$ on a Banach space, and we know for some $t_0>0$, the two operators coincide, i.e., $T(t_0) = S(t_0)$. Can we ...
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3answers
76 views

$e^{itH}$ notation

recently I saw the notation $e^{itH}$, and just wondering how should I interpret it? In my understanding, $u(t,x) = e^{itH} u_0$ is, for example, a solution to Schrodinger-type equation $i\partial_tu =...
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1answer
22 views

What does subspace A-matrix invariance tells me in terms of A Jordan canonical form.

I am asked to show that the semi group $(e^{tA})_{t\geq0}$ for $A \in M_n(\mathbb{C})$ is hyperbolic i.e. there exists direct decomposition $\mathbb(C)^n=X_s \oplus X_u$ in to A-invariant subspaces $...
2
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1answer
30 views

List of diffusion processes with known transition probabilities

Consider a generic multidimensional diffusion SDE \begin{equation} dX_t = b(X_t,t)dt + \sigma(X_t,t)dW_t. \end{equation} By standard theory $X$ is a Markov process, which defines a Markov semigroup of ...
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1answer
67 views

If $\frac{{\rm d}X^{s,\:x}}{{\rm d}t}(t)=v(t,X^{s,\:x}(t))$ and $X^{s,\:x}_s=x$, can we show that $X^{s,\:X^{r,\:x}(s)}(t)=X^{r,\:x}(s+t)$?

Let $\tau>0$, $d\in\mathbb N$, $v:[0,\tau]\times\mathbb R^d\to\mathbb R^d$ and $X^{s,\:x}:[0,\tau]\to\mathbb R^d$ denote the solution of \begin{align}\frac{{\rm d}X}{{\rm d}t}(t)&=v(t,X(t))\;\;\...
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1answer
73 views

Can I show this using a contraction semigroup property?

Let $H$ be a (real) Hilbert space, $L$ be an unbounded operator on $H$ with its domain $D(L)$ and $(e^{-tL})_{t\ge 0}$ be a contraction semigroup on $H$. Then, the following holds from a semigroup ...
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2answers
37 views

$\lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_0^\epsilon e^{-\alpha s}P_s uds = u$ for Feller semigroup

Let $P_t$ be a Feller semigroup, i.e. a contractive, positive preserving, sub-Markovian, strongly continuous semigroup with the Feller property on $C_\infty(\mathbb{R}^d)$. Then I am trying to show ...
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0answers
74 views

Using Cauchy's theorem

In a demonstration of a theorem I have...$A:X\to X $ a bounded operator, $X $ a banach space and $$e^{tA}=\frac{1}{2\pi i}\int_{C_r} e^{\lambda t } R (\lambda ; A) \ d\lambda$$ With uniform norm, ...
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0answers
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In this situation we can shift path of integration?

If $ \dfrac{1}{2\pi i}\int_{C}f(z)dz$ exists for any circle of radius $r>0$ centered at origin, the point $0$ is a singularity for $f$, and always the integral has the same value, say $k = \dfrac{1}...
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0answers
20 views

Eventually continuous semigroup is uniformly continuous on bounded set

A strongly continuous semigroup $T(t)$ is called eventually norm-continuous semigroup if $t\mapsto T(t) $ is norm continuous for $t>t_0$. I want to prove that $T(t)$ is uniform continuous on the ...
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2answers
32 views

Strong continuity of the heat semigroup in $L^2(\mathbb{R}^n)$.

We know that $(K_t * g)(x) = (4\pi t)^{-n/2} \int_{\mathbb{R}^n} e^{-||x-y||^2/4t}g(y)dy$ solves the heat equation $u_t-\Delta u = 0$ in $\mathbb{R}^n \times (0 , \infty)$ with $u=g$ in $\mathbb{R}^n \...
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1answer
35 views

Norm of the resolvent and inequalities

I am considering the equation $$ \lambda q- \frac{\mathrm{d}^{2} q}{\mathrm{d} x^{2}}=g, $$ where $g\in L^p(X)$ and $q\in W^{2,p}(X)$. If, I know that $$ \mid \mid \lambda (\lambda I- \frac{\mathrm{d}^...
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0answers
50 views

How can we extend this Markov semigroup inequality?

Let $(\kappa_t)_{t\ge0}$ be a Markov semigroup on a measurable space $(E,\mathcal E)$, $$\iota:[0,\infty)\to[0,1)\;,\;\;\;x\mapsto x-\lfloor x\rfloor,$$ $\xi:[0,1]\to[0,1)$ be nonincreasing and $$\...
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0answers
41 views

Extend a bound for a Markov semigroup using Jensen's inequality

Let $E$ be a $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$. Assume there are $(C,R,M)\in[0,\infty)\times[0,1)\times[...
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0answers
61 views

If $X:\Omega\times[0,\infty)\times E\to E$ is a stochatic flow, is $\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]$ a semigroup?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space and $X:\Omega\times[0,\infty)\times E\to E$ be a stochastic flow, i.e. $X$ is $(\mathcal A\...
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2answers
95 views

On the resolvent set of an unbounded operator

Suppose $A$ is the infinitesimal generator of a $C_0$ semigroup $S(t)$ on an Hilbert space $X$. If $$\langle Ax, x\rangle \leq \omega \|x \|^2 \ \ \ \forall x \in \mathfrak{D}(A)$$ then $$\|S(t)\...
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1answer
24 views

For $T(t)$ strongly continuous, check that $T(t)x - x = tAx + \int_0^t (t-s) T(s)A^2 x ds$.

I am reading Lemma 2.8 of "Semigroups of Linear Operators and Applications to Partial Differential Equations" by Pazy: Let $A$ be the infinitesimal generator of a strongly continuous semigroup $T(t)...
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1answer
41 views

Question on Semigroup theory for evolution equations: Strong continuity of analytic semigroup $e^{-tA}$

I'm studying about Semigroups in Parabolic equations this semester and I'm having a really hard time understanding how these complex line integrals behave from times to times (my complex analysis ...
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0answers
23 views

Continuity of a map involving strongly continuous semigroups

Could someone help me with the following exercise about strongly continuous semigroups ? Let $Y$ be a Banach space, $\{T(s)\}_{s\geq 0}$ a strongly continuous semigroup and $a>0$. If $w:[0,a[\...
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1answer
37 views

What can we infer from a growth bound What can we infer from the condition $\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|}n\le0$?

Let $A_n$ be a compact linear operator on a $\mathbb R$-Hilbert space $H$. What can we infer from the condition $$\limsup_{n\to\infty}\frac{\ln\left\|A_n\right\|_{\mathfrak L(H)}}n\le0\tag1$$ and in ...
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2answers
68 views

When are spectral values a pole of the resolvent?

Let $(T_t)_{\geq 0}$ be a $C_0$-semigroup on a Banach space $X$ with the generator $A.$ What are some conditions sufficient for $\lambda \in \sigma(A)$ to be a pole of resolvent? I'm looking for ...
4
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1answer
75 views

Limit of resolvent in terms of limit of semigroup

Let $(T_t)_{t\geq 0}$ be a $C_0$-semigroup on a Banach space $X$ with generator $A$ such that the spectral bound $s(A)=0.$ Suppose there exists an operator $P$ on $X$ such that $$T(t) \stackrel{t\to \...
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1answer
31 views

Show that the initial value problem for the transport problem defines a contraction semigroup

I'm working through some problems with contraction semigroup right now. Show that the initial value problem for the transport equation: $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\...
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1answer
76 views

When does strong convergence imply convergence in operator norm?

I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show $\lim_{t \to \infty} T_t =0 $ with respect to the operator norm. After some effort, I was able to prove $\lim_{t \to \infty} T_t =0 $ ...
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1answer
31 views

Range of strong limit of a semigroup belongs to the fixed space?

Let $\left(T(t)\right)_{t\geq0}$ be a $C_0$-semigroup on a Banach lattice $E$ such that $T(t)$ converges strongly to a positive operator $S$ as $t \to \infty.$ Then $$T(t)S=S \text{ for all }t\geq0.$$ ...
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1answer
32 views

Characterization of space $D(A^2)$ in semigroup theory for laplacian operator

Let us assume $\Omega$ is sufficient smooth. Let $H=L^2(\Omega)$ and define $A:D(A)\subset H\to H$ by $Au=\Delta u$, with $D(A)=H_0^1(\Omega)\cap H^2(\Omega).$ Brezis's book, Brezis, Functional ...
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18 views

Analytic Semigroups and Regularity

Suppose $L$ is a second order elliptic differential operator with domain $D(L) = W^{2, p}(\mathbb{R}^d)$ generating an analytic semigroup $(T_t)_{t \geq 0}$ in $L^p(\mathbb{R}^d)$ as in "Lundari - ...
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0answers
22 views

Analytic semigroup on L^p

I am learning about analytic semigroups, then appear phrases like: "...is well know that the operator generates an analytic semigroup...", hence appear the following question: How can I prove that ...
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1answer
22 views

An estimate in the proof of $Q(t)x = \lim_{\epsilon \rightarrow 0} \exp(tA_\epsilon)x$ for semigroup of operators.

I am studying Rudin's Functional Analysis and come across a step in proving Theorem 13.35 (e): $Q(t)$ be a semigroup of operators that is strongly continuous (i.e. $\lim_{t\rightarrow 0}||Q(t)x-x|| = ...
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1answer
35 views

Left translation on $C_b (\mathbb{R})$ is not a strongly continous semigroup

I'm studying strongly continuous semigroups right now with the book One-Parameter Semigroups for Linear Evolution Equations by Engel, Nagel. In this book there is a small section about the left ...
2
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1answer
50 views

Feynman-Kac formula on $L^2(\mathbb{R}^d)$ (the right hand side)

Let $B_t$ be a $d$ dimensional Brownian motion and $V\in C^\infty_0(\mathbb{R}^d)$. Define the operator $T_t$ on $L^2(\mathbb{R}^d)$ by \begin{equation} (T_tf)(x)=\mathrm{E}_x[ e^{-\int_0^t V(B_s) ...
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0answers
30 views

Where can I find a proof of the fact that the Gaussian semigroup is generated by the Dirichlet Laplacian?

Let $H_0$ denote the Dirichlet Laplacian on $\mathbb{R}^d$ given by $H_0u=\Delta u$ for $u \in \text{dom}(H_0):= \{f \in H^1(\mathbb{R}^d): \Delta f \in L_2(\mathbb{R}^d) \}$. Further, for $t>0$ ...
2
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0answers
59 views

Existence and uniqueness for a PDE by classical semigroup theory

I am currently reading the following paper: Chen, X. The Hele-Shaw problem and area-preserving curve-shortening motions. Arch. Rational Mech. Anal. 123, 117–151 (1993). https://doi.org/10.1007/...
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1answer
114 views

Exercise 4.7 Kesavan's book Topics in Functional Analysis and Applications

Below I present Exercise 4.7 from Kesavan's book Topics in Fucntional Analysis and Applications chapter 4. I could solve item (a) but I don't know how to get (b) from (a). I would like some hint or ...
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1answer
61 views

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

In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the following definitions and result: Consider the following semilinear ...
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0answers
21 views

Are there linear delay systems which are strongly stable (=attractive) but not exponentially stable?

Let $T$ be a strongly continuous semigroup of linear operators over a Banach space $X$. $T$ is called strongly stable (= attractive), if for any $x \in X$ it holds that $T(t)x \to 0$ as $t\to\infty$....
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0answers
33 views

can an integral operator be the generator of a semigroup?

Lets assume operators/semigroups on a function space $X$ over $\mathbb{R}$. For $u\in X$ set $(Au)(x)=\int_{\mathbb{R}}K(x,y)u(y)dy$. Is $A$ the generator of a semigroup? Is there a good reference for ...
3
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1answer
94 views

Integration in Banach Spaces - Bochner Integral and Rieman Integral.

Im at the beginning of my studies of operator semigroup theory and I have some trouble understanding the integration of operators in Banach Spaces. Let $(T(t))_{t\geq0}$ be a $C_0$-semigroup on a ...

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