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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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 $t \geq 0$) ...
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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
$$ (e^{-\Delta_{\Omega} t}f)(x) = \...
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How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
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Infinitesimal generator of a Markov process acting on a measure
Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...
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Uniformly continuous semigroups are analytic
Reposting from stackexchange.
I know that every analytic $C_0$-semigroup is differentiable and then every differentiable semigroup is norm continuous.
I want to know where uniform continuity fits in ...
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Is there any class of initial data for which the heat semigroup is increasing in time?
Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is
...
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Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
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$H^s$ norm of dispersive semigroup
The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm
$$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
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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
$$S(t)S(s)u(x) = S(t+s)u(x)....
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Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator
Let $\{X_i \}_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X_1 = 0$ and $\mathbb{E} X_1 ^2 < \infty$. Assume that $\{S_n \}_{n \in \mathbb{N}}$ is a non-...
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The semigroup of Laplace-Beltrami operator on 3-flat torus
I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 ...
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Is the underlying set of every renormalization group countable and finite？ [closed]
Is the underlying set of every renormalization group countable and finite？
Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
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Friedrichs Inequality
I'm a little confused with the following proof of Friedrichs inequality in Lawson's & Michelsohn's book Spin geometry, page 194, Theorem 5.4.
I don't understand why the last inequality, i.e.
$$
C(\...
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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 ...
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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 ...
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