All Questions
Tagged with markov-chains semigroups-of-operators
14 questions
3
votes
0
answers
58
views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 31
0
votes
0
answers
56
views
Generator of sub-Markov semigroup induces generator of Markov semigroup
I have to show that for the generator $A:L^1 \rightarrow L^1$ of a sub-Markov semigroup and a non-negative $f_* \in L^1$ (with $L^1$ Set of Lebesgue-integrable functions) with $\int_{-\infty}^\infty ...
- 21
0
votes
0
answers
49
views
Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...
- 90
3
votes
0
answers
241
views
A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post.
There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
- 620
1
vote
0
answers
37
views
If $(\kappa_t)_{t\ge0}$ is a Markov semigroup with invariant measure $μ$, under which assumption is $t\mapsto\kappa_tf$ measurable for $f\in L^p(μ)$?
Let
$(E,\mathcal E)$ be a measurable space;
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$;
$\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\...
- 167
2
votes
1
answer
181
views
If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?
Let
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ ...
- 167
0
votes
0
answers
122
views
Methods to find the spectrum of an operator
Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$?
Here is the setting I'm wondering about: consider ...
- 153
1
vote
2
answers
228
views
Is a linear combination of Markov generator a Markov generator?
Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...
- 449
3
votes
0
answers
90
views
How does one define the gradient of a Markov semigroup?
In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:
...
- 167
0
votes
1
answer
76
views
Friedrich's extension of the generator of a continuous time markov chaoin
Consider the infinitesimal generator $G$ of a Markov chain with state space $\mathbb{Z}$ such that it is symmetric with respect to a measure $\mu$ on $\mathbb{Z}$. Then, the operator $(G,C_c(\mathbb{Z}...
- 407
0
votes
0
answers
224
views
Show convergence of a sequence of resolvent operators
Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...
- 167
2
votes
0
answers
416
views
How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?
In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...
- 167
1
vote
1
answer
370
views
Markov chain with Feller property
Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller property?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $...
- 13
3
votes
1
answer
199
views
Markov-semigroup Sobolev inequality
I have a question about the following definition:
A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality ...
- 259