# 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.

337 questions
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### Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very similar ...
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### Semi-group theory and Poisson equation on the upper half plane

We first look at the 2D Laplace equation , say on the upper half plane: $$\Delta u=0,\quad -\infty<x<\infty, y>0$$ $$u(x,0)=g(x),$$ where $g\in L^p(\mathbb{R})$ for some $1\leq p<\infty$. ...
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### What does Trotter Product Formula mean?

For some reason, I have to work with Trotter product formula recently, but I do not have a strong background in functional analysis. The following is the statement of the formula from MathWorld ...
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### Does Continuity in Weak Operator Topology imply Continuity in Strong Operator Topology?

This homework problem has puzzled me for almost a year. As nobody in the class has figured it out, I would like to seek a proof or a disproof here. Problem (Prove or Disprove) Let $X$ be a Banach ...
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### Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
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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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### Show that the semigroup S(t) here described is a contraction semigroup

Definition 1: Let $H$ be a Hilbert space. A strongly continuous semigroup is a family $\{S(t)\}_{t \ge 0}$ of continuous linear operators $S(t): H \rightarrow H$ such that $S(0)=I$, where $I$ is the ...
Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...