All Questions
13 questions with no upvoted or accepted answers
2
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155
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Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
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2
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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
2
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0
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70
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If $X^n$ is a sequence of càdlàg processes whose FDDs converge to a continous process $X$, does $X^n$ converge to $X$ in the Skorohod topology?
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $E$ be a complete locally compact separable metric space, $(X^n_t)_{t\ge0}$ be an $E$-valued càdlàg process on $(\Omega,\mathcal A,\...
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1
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115
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Concatenation of Markov processes and independence
In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post.
It is rather ...
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1
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37
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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
1
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0
answers
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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0
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106
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Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
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1
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0
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56
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Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter
Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.
I want to ...
- 167
1
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0
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81
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If $\text P\left[X_2\in B_2\mid X_1\right]=\kappa(X_1,B_2)$ a.s. for all $B_2$, can we select a common null-set over all $B_2$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E_i,\mathcal E_i)$ be a measurable space
$X_1:\Omega\to E_1$
$X_2:\Omega\to E_2$ be $(\mathcal A,\mathcal E_2)$-measurable
$\kappa$ ...
- 167
1
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0
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120
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Existence of Time-Reversed Markov Kernels
Suppose I have a probability measure $\pi$ and a Markov kernel $q$ which leaves $\pi$ invariant, in the sense that
\begin{align}
\int_x \pi(dx) q(x \to dy) = \pi(dy)
\end{align}
Then, a (the) time-...
- 801
0
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161
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Markov process with time varying transition kernels
I cross post this question from StackExchange as it may be more appropriate.
I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
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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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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,\...
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