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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.

3
votes
1answer
696 views

If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute. The converse of this statement can be an easy application of the Cauchy ...
1
vote
2answers
262 views

Why does the semigroup commute with integration?

I have a question about Theorem 7.4.2 in Evan's PDEs book. If $S(t)$ is a contraction semigroup on a Banach space $X$. He uses $$S(r)\int_0^t S(s)u\,ds = \int_0^t S(r+s)u\,ds$$ and I don't understand ...
0
votes
1answer
355 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
5
votes
2answers
2k views

How to find the infinitesimal generator of this semigroup?

Definition 1: Let $X$ be a Banach space. A semigroup is a family $\{T(t)\}_{t\geq 0}$ of continuous linear operators $T(t):X\to X$ such that $(i)\;\;T(0)=I$, where $I$ is the identity operator; $(...
4
votes
1answer
346 views

dissipative operator meaning

Can someone explain to me the "meaning" of the dissipative operator ? https://en.wikipedia.org/wiki/Dissipative_operator I am a bit confused. Thanks in advance.
13
votes
1answer
2k views

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 ...
7
votes
1answer
738 views

Weak generator of Feller semigroup

Let $(T_t)_{t \geq 0}$ a Feller semigroup and define a linear operator $(A,\mathcal{D}(A))$ by $$\mathcal{D}(A) := \left\{u \in C_{\infty}(\mathbb{R}^d); \exists f \in C_{\infty} \forall x \in \mathbb{...
4
votes
2answers
221 views

Generating a $C_0$-semigroup on $L^2$

Consider the linear operator $$A : H^4(\mathbb{R}; \mathbb{R}) \to L^2(\mathbb{R};\mathbb{R})$$ defined by $u\mapsto -(1-\partial_{xx}^2)^2$. Show that $A$ generates a $C_0$-semigroup on $L^2$. I ...
4
votes
2answers
146 views

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 ...
3
votes
1answer
507 views

Proof of a property of the uniformly continuous semigroups.

Let $X$ be a Banach space and $T:=\{T(t)\}_{t\geq 0}$ an uniformly continuous semigroup of bounded linear operators on $X$. So, we know that (i) $T(t)$ is linear and bounded for all $t\geq 0$; (...
2
votes
1answer
102 views

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:...
1
vote
1answer
67 views

How can we prove that a (locally bounded) semigroup is strongly continuous on the closure of its generator?

Let $E$ be a $\mathbb R$-Banach space and $(T(t))_{t\ge0}$ be a semigroup on $E$, i.e. $T(t)$ is a bounded linear operator on $E$ for all $t\ge0$, $T(0)=\operatorname{id}_E$ and $$T(s+t)=T(s)T(t)\;\;\;...
1
vote
0answers
76 views

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 ...
0
votes
1answer
312 views

Prove that $\{S(t)\}_{t \ge 0}$ is a contration semigroup on $L^2(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the heat ...
7
votes
1answer
165 views
+100

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 ...
1
vote
1answer
76 views

If $S$ is a strongly continuous semigroup on a Hilbert space $H$ and $u\in C^1((0,T),H)$, show that $\lim_{h\to0+}S(h)\frac{u(s)-u(s-h)}h=u'(s)$

Let $T>0$ $H$ be a $\mathbb R$-Hilbert space $S:[0,\infty)\to H$ be a strongly continuous semigroup on $H$ $A$ be the infinitesimal generator of $A$ $u\in C^1((0,T),H)$ and $$u(t)\in\mathcal D(A)\;...
1
vote
0answers
72 views

Regularity of the mild solution of a semilinear PDE

Let $H$ be a separable $\mathbb R$-Hilbert space $(\mathcal D(A),A)$ be a linear operator on $H$ $(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ be an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;...
1
vote
0answers
30 views

How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
1
vote
0answers
34 views

Generation theorem for Feller semigroups

Let $E$ be a locally compact Hausdorff space. I want to show that a linear operator $(\mathcal D(A),A)$ on $C_0(E)$$^1$ is closable and the closure $(\mathcal D(\overline A),\overline A)$ is the ...
0
votes
1answer
150 views

analytic semigroups and norm continuous semigroups

Are every analytic semigroups norm continuous? Are there counterexamples otherwise? What would make analytic semigroup norm continuous as well? (My apology, frankly, I am not sure if norm continuous ...
0
votes
0answers
47 views

An exemple of a strongly-continuous contraction semigroup

I try to prove that $P_t := e^{\lambda t (P-I)}$ (where $Pf:= \int f(y) P(\cdot , dy)\in \mathcal{C}_0(\mathbb{R}^d)$, for $f\in \mathcal{C}_0(\mathbb{R}^d)$), is a strongly-continuous contraction ...