# 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.

338 questions
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### If $X^{(n)},X$ are càdlàg and $X^{(n)}\to X$ in distribution, do the corresponding transition semigroups strongly converge?

Let $\left(\kappa^{(n)}_t\right)_{t\ge0}$ and $(\kappa_t)_{t\ge0}$ be Markov semigroups on $(\mathbb R,\mathcal B(\mathbb R))$ for $n\in\mathbb N$ $(T_n(t))_{t\ge0}$ and $(T(t))_{t\ge0}$ be strongly ...
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### I don't understand the proof of Corollary 4.8.7 in the book of Ethier and Kurtz

I'm trying to understand the proof of Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas G. Kurtz. Here is the theorem and its proof:...
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### Question about $e^{\frac{-itA}{\hbar}}(\hat{Q}+ \hat{P})e^{\frac{itA}{\hbar}}$

This arises in the context of trying to rigorously understand quantum dynamics but it's a functional analysis issue. For simplicity suppose we are in dimension $1$. Let $\hat{Q}$ be the operator ...
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### Meaning of Compactness

Let $\Omega \subset\mathbb{R}$ be a bounded domain (interval) and observe the following problem : \begin{align*} (P) \begin{cases} u_{t} = \Delta u + |u|^{p-1}u\, \quad x\in\Omega, t>0\\ u(0,x) = ...
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### Use of L'Hosptial Rule in generalized setting related to heat equation

Let $X$ be a space of continuous functions with compact support in a bounded domain $\Omega \subset \mathbb{R}^{N}$ with Lipschitz continuous boundary, $F : X \to X$ be a Lipschitz continuous function,...
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### Can norm of resolvent of closed operator grow away from its spectrum(but near to its numerical range)

I hope my question is simple to answer but I could not find anything up to now. Given a closed densely defined operator $A \colon H \supseteq D(A) \to H$ in a Hilbert space $H$. For many types of ...
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### Heat semigroup representation

It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in ...
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### Exponential stability of a second order ordinary linear operator

Let's consider an unbounded second order linear differential operator $A := k(x)\frac{d^{2}}{dx^{2}}+\frac{d}{dx}$ defined over $L^{2}(0,1)$ whose domain is $H^{2}(0,1) \cap H_{0}^{1}(0,1)$. $k(x)$ is ...
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### Extension of domain of contactive semigroups, and why contractivity is important?

I am reading a paper, where I think the authors use this fact that the contractivity of a semigroup on a space of initial values, imeadiatly implies that we can extend the domain of semigroup to this ...
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### Showing that $\lambda - (A + B)$ has dense range

Let $A$ be the generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ of contractions on a Banach space $X$ and $B \in \mathcal L(X)$ a bounded operator. To apply some approximation formula I want to show ...
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### Do contraction semigroups admit exponential representation?

Given a Banach space $\mathcal N$, as contraction semigroup is defined as a set of bounded operators $P^t$, $0\le t\le+\infty$ defined everywhere in $\mathcal N$, such that \begin{equation*} P^0=1, \...
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### Do self adjoint operators on a Hilbert space generates an analytic semigroup?

Is this generally true that, a densely defined, closed and self adjoint operator on a Hilbert space generates an analytic semigroup?
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### Extending the domain of the Dirichlet form associated with a symmetric Markov semigroup

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(P_t)_{t\ge0}$ be a Markov semigroup ...
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### Generator of semigroup?

I have seen such expression: the generator of the semigroup is defined by $$[Uf] = \lim_{t\rightarrow 0} \frac{U^t f - f}{t}.$$ However, I don't understand, isn't the $[Uf] = df/dt$ that simple?
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### Show that $C_c^∞(\mathbb R)$ is a core of the generator of the Feller semigroup induced by the strong solution of an SDE with Lipschitz coefficients

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $b,\sigma:\mathbb R\to\mathbb R$ be Lipschitz continuous and $$Lf:=bf'+\frac12\sigma^2f''\;\;\;\text{for }f\in C^2(\mathbb R)$$ $W$ be ...
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### Show that the carré du champ operator is nonnegative

Let $(E,\mathcal E)$ be a measurable space $\mathcal M_b(E,\mathcal E):=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov ...
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### Show that a semigroup is strongly continuous on the domain of its generator

Let $(\kappa_t)_{t\ge0}$ be a semigroup of linear, conservative, contractive and nonnegative operators on $C_0(E)$ with $$(\kappa_tf)(x)\xrightarrow{t\to0}f(x)\tag1\;\;\;\text{for all }x\in\mathbb R$$ ...