All Questions
Tagged with pr.probability markov-chains
362 questions
1
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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
7
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1
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391
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Idempotent splitting for Markov kernels
Let $X$ be a standard Borel space and $e : X \to X$ a Markov kernel. Suppose that $e$ is idempotent, that is $e \circ e = e$, or written out using the Chapman-Kolmogorov equation,
$$e(A|x) = \int_X e(...
- 6,406
6
votes
1
answer
482
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Average and max. hitting time to a specific vertex
Consider simple random walks that stop when reaching a given node $x$ in an undirected, unweighted and connected graph on $n$ nodes.
Let
$H(i,x)$ denote the (expected) hitting time from $i$ to $x$, ...
- 73
0
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1
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96
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What is the significance of Blumenthal and Getoor's result on the boundedness of paths of a standard Markov process?
In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result:
I don't understand the significance of this result. If I don't misinterpret the assertion, ...
- 167
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0
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72
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If $\kappa$ is a Markov kernel with density $p$, does it generally hold $p(x,z)=\int p(x,y)p(y,z)\:{\rm d}y$?
Let $(E,\mathcal E)$ be a measurable space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$. Assume that $$\kappa(x,B)=\int_Bp(x,y)\:\lambda({\rm d}y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$...
- 167
0
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1
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227
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Constructing Markov chain
Let $(A_1,B_1)$ and $(A_2,B_2)$ be two random variables with the joint distributions $p_{A_1B_1}$ and $p_{A_2B_2}$, respectively. Moreover, we have
$$\mathbb{P}[(A_1,B_1)\neq (A_2,B_2)]=\alpha.$$
Then,...
- 287
0
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1
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262
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Construction of a Markov process with prescribed local behavior and state-dependent jump distribution
Let
$(E,\mathcal E)$ be a measurable space
$\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$
$(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\...
- 167
1
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1
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337
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How can we determine the generator of this Markov process (at least formally)?
Let
$(\Omega,\mathcal A)$ be a measurable space;
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$;
$(Y_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued time-homogeneous Markov process on $(\...
- 167
3
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1
answer
220
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Carne-Varopoulos bound and stationary measure
Let $\Gamma$ denote the Cayley graph for a finitely generated group $G$, and let $p_n(x, y)$ denote the transition probability that a random walk starting at $x$ reaches $y$ at time $n$. A famous &...
- 31
4
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1
answer
186
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Population growth with good and evil children - probability good outnumbers evil
Consider the following discrete-time population model. We start with a single "good" individual who reproduces asexually into $k$ children and dies in the process. At generation $t=2$, those ...
- 524
4
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1
answer
276
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About non-reversible Metropolis Hastings Markov chain
I am reading a paper about constructing a non-reversible Metropolis Hastings Markov chain from a reversible one as described at a high level in paragraph $3$ of page $1$.
But I don't understand how, ...
- 201
3
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1
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474
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Harmonic function and Markov chain
Let $X=(X_k)_{k \in \mathbb{N}}$ be a Markov chain with countable countable state space $S$ and transition matrix $P.$
Let $\mathcal{T}$ be the tail $\sigma$-field of $X:\mathcal{T}=\bigcap_{k \in \...
- 53
1
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1
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88
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Stationary and limiting distributions
Consider a CT Markov Process $X=(X_t)_{t\geq0}$ with state space $E\in\mathbb{R}^N$. Are there any general conditions under which a stationary distribution $\pi$ for $X$ is also a limiting ...
- 203
1
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110
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Birth and death process $M/M/\infty$
I was reading about continuous time Markov chains, when I met for the first time the theory of queue processes. In particular, I considered the following situation which I found on Wikipedia, called M/...
- 445
3
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1
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373
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Concentration of very dependent Markov chains
Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary).
The flip ...
- 161
1
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0
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89
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Understanding the statements of Theorem 5.5 and Lemmas 5.6, 5.7 and 5.8 from a French paper by Yves Guivarc’h and Émile Le Page
I would like to understand the statement and the proof Theorem 5.5 just for the special case when $X$ is a single point from the paper “Simplicité de spectres de Lyapounov et propriété d’isolation ...
1
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1
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193
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Identity for special case of Markov chain
Consider $P(X,Y)$ discrete and $Z = f(Y)$ with $f$ deterministic. The function $f$ identifies a partition of the elements of the alphabet $\mathcal{Y}$ of $Y$. Each outcome $z \in \mathcal{Z}$ is a ...
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8
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1
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691
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Probabilistic proof for derivative of invariant distribution of a Markov chain
Let $P$ be an irreducible Markov matrix, and $\pi$ its stationary distribution. Let $D$ be a perturbation matrix which is zero except for two entries in row $r$:
$$D_{rg}=+1 \qquad D_{r\ell}=-1.$$
Let ...
- 1,068
2
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0
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123
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Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
- 307
0
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2
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804
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Convergence of stationary distributions of a sequence of Markov Chains
I fairly new in the field of Stochastic Processes and Markov Chains so excuse my ignorance.
My question is: If we have a sequence of Markov chains such that each one has a stationary distribution $\pi^...
- 13
1
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1
answer
410
views
Occupation times for two-state Markov processes
Consider a two-state Markov process in continuous time, with states labelled $A$ and $B$. The transition rates for going from state $A$ to $B$, and state $B$ to $A$ are $\alpha$ and $\beta$ ...
1
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1
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233
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Random walks on Galton–Watson trees
I am working on a paper of Elie Aidekon : ‘Speed of the biased random walk on a Galton–Watson tree’ and have a question about one transformation in a proof:
\begin{align}
& 1+\frac{1}{1-\lambda}+\...
- 39
1
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0
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181
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Random walk on 2d lattice with obstacles
Consider a random work on $L=\mathbb Z^2$ endowed with obstacles (i.e each cell $(x,y)$ of $L$ may contain a obstacle, i.e the random walk halts whenever it hits such a cell). Let $P(x,y) = 1$ if cell ...
- 6,853
8
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3
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404
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All two-point correlations equal to $0$, three-point correlation not $0$?
Let $a_1,a_2,a_3,\dotsc \in \{-1,1\}$ be a sequence. Suppose that, for all $j>0$ and all
$\epsilon, \epsilon'\in \{-1,1\}$, the proportion of $n\geq 1$ such that $(a_n,a_{n+j}) = (\epsilon,\epsilon'...
- 20.2k
1
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332
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Markov chains with drift
We consider a Markov process $X$ on a finite set $\mathcal{X} (\neq \emptyset)$. Basically, $X$ is associated with a generator of the following form
\begin{align*}
Af(x)=\lambda(x)\sum_{ y\in \mathcal{...
- 721
2
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1
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101
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Preservation of the Markov Property under Conditioning
Let $(X_t,Z_t)_t$ be an $\mathbb{R}^{n}\times \mathbb{R}^m$-valued time-homogeneous Markov process on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$ with transition ...
- 77
4
votes
2
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683
views
Random walk on $n$-dimensional cube
Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each ...
user171475
1
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0
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75
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Stationary distribution of a Memoryless 2-type priority queue
I have come across the following priority queue, which seems quite natural to me.
A single queue with 2 types of costumers, independent Poisson arrivals and Poisson services.
First class costumers ...
- 61
2
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1
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134
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The reference on Markov chains uncovering the power of the subject in a better way for a working macro-economist
This is by no means a research question. But asking here I hope for the most expert opinion.
A friend of mine, who is a working economist, asked me for advice about a book which uncovers wealth and ...
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1
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2
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228
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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
-2
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1
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181
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Stationary distribution of a weighted directed acyclic graph
Is there any way to calculate the equilibrium (stationary) distribution for a weighted directed acyclic graph?
Some references emphasized adjacency matrix to be symmetric.
https://arxiv.org/abs/1012....
2
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1
answer
176
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Monotonicity of Dirichlet form of Markov chain
Consider a continuous-time, irreducible Markov chain $X_t$ on a finite state space $E$. Assume the jump rates are $R(x,y)$ for $x,y\in E$, the generator is $L$, i.e for any function $f$ on E,
$$Lf(x)=\...
- 59
0
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0
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83
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Constrained MDP
I have a question that is an extension of this one.
My question is: Can we say that for every policy, there exists a deterministic policy in case of a finite-state, finite-action infinite-horizon ...
- 154
1
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0
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152
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Harnack inequalities for Markov chains
We consider a (continuous time) Markov chain $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in V})$ on a finite set $V$. We assume moreover that $V$ is embedded into $\mathbb{R}^d$. The generator $\mathcal{L}$ of $...
- 721
0
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1
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109
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Sets of invariant measures of Markov operators
A family of Markov operators $P_i \colon C \to C, i \in I$ is given. Let $V_i$ be the set of the $P_i$-invariant measures. Is there any result in the literature about a necessary and sufficient ...
0
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0
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85
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If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
- 167
1
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1
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154
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If $L_t=\sum_{i=1}^{N_t}Y_i$ is a compound Poisson process, then $\left|\left\{s\in[0,t]:\Delta L_s\in B\right\}\right|=\sum_{i=1}^{N_t}1_B(Y_i)$
Let $H$ be a $\mathbb R$-Hilbert space, $\mu$ be a finite measure on $\mathcal B(H)$ with $\mu(\{0\})=0$ and $(L_t)_{t\ge0}$ be a $H$-valued càdlàg Lévy process on a probability space $(\Omega,\...
- 167
1
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0
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276
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Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
1
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1
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183
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finiteness of moments of the stationary distribution of a Markov chain
I have a Markov chain $\{X_k\}_{k\geq 0}$ on $\mathbb{R}$. The corresponding probability density functions satisfy
$$
f_{k+1}(t) = \int_{-\infty}^\infty \Psi(t,\tau)f_k(\tau)\,d\tau,\qquad k=0,1,2,\...
- 427
4
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0
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61
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Mixing times for the exclusion process with rejection
Consider the following Markov chain on $k$-subsets of $\{1,\ldots, L\}$, equivalently, sequences $x\in \{0,1\}^L$ with $k$ 1's.
Let $p_1,\ldots, p_L\in (0,1)$ and $q_i=1-p_i$.
At each step, choose an ...
- 371
2
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1
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114
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Strong Data Processing Inequality for capped channels
Let $X$ and $Y$ be two $\rho$ correlated Gaussian vectors, such that $X,Y\sim N(0,1)^n$ and $E[X_iY_i]=\rho$.
Let $M_X = f(X)$ and $M_Y = f(Y)$ be $k$-bit functions of $X$ and $Y$, that is $H(X)=H(Y)=...
1
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0
answers
61
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Convergence of empirical measure to Mc-Kean Vlasov equation for mean-field model with jumps
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
2
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0
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175
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Representing a continuous time-inhomogeneous Markov chain by a stochastic integral
I am interested in the following mean-field model introduced in the reference below:
There are $N$ particles. At each instant of time, a particle's state is a particular value taken from the finite ...
- 31
1
vote
1
answer
189
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If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?
Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
- 167
1
vote
1
answer
173
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Spectral gap of a Markov chain on the nonnegative integers
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers such that $\sum_{k=1}^\infty k\lambda_k<\infty,$ let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb ...
- 276
1
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1
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306
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English translation of a Russian paper by Gordin and Lifšic
Unfortunately I can't read Russian, I was wondering if there is an English translation of this paper
“The central limit theorem for stationary Markov processes”, Dokl. Akad. Nauk SSSR, 239:4 (1978), ...
- 757
1
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2
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242
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Extension of spectral gap inequality in Wasserstein distance
Let $E$ be a separable $\mathbb R$-Banach space, $\rho_r$ be a metric on $E$ for $r\in(0,1]$ with $\rho_r\le\rho_s$ for all $0<r\le s\le1$, $\rho:=\rho_1$, $$d_{r,\:\delta,\:\beta}:=1\wedge\frac{\...
- 167
3
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0
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90
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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
1
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1
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175
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Existence of Markov chain on nonnegative integers with specified rates
Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers, let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers and ...
- 276
2
votes
1
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205
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Eigenspace of Gaussian Markov operator
Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator
\begin{equation*}
\begin{array}{rccc}
R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
- 123