All Questions
Tagged with ds.dynamical-systems pr.probability
120 questions
4
votes
0
answers
116
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41
5
votes
1
answer
389
views
Is a random circle rotation weak mixing almost surely?
Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations.
...
- 6,155
2
votes
0
answers
92
views
Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
3
votes
1
answer
194
views
Dynamics of a random stretch map
Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.
Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly ...
- 6,155
1
vote
0
answers
48
views
Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user530766
0
votes
0
answers
107
views
How to show that the map $ R $ here is measure-preserving
Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the ...
- 1,219
2
votes
1
answer
266
views
Ergodicity of linear dynamical systems and convergence of covariance matrices
Let $z(n+1)=Bz(n)+\xi(n+1)$ be an $N$-dimensional linear dynamical system with $\left(\xi(n)\right)_{n\in\mathbb{N}}$ being i.i.d. with $\xi(n)\sim\mathcal{N}(0,\Sigma_{\xi})$.
Assumptions: a) The ...
- 767
20
votes
1
answer
2k
views
Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
- 6,155
3
votes
1
answer
307
views
"Ergodic theorem" for Markov kernels
Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
2
votes
0
answers
313
views
Correlation decay rate
Let $T$ be a continuous transformation of a probability measure space $(X,\mathcal{B}(X),\mu)$ and
$\varphi ,\phi \in L^2(\mu)$ (so-called observable) . The correlation function of $\varphi ,\phi$ (a ...
- 136
1
vote
1
answer
210
views
Shift-ergodic stochastic processes in continuous time
Let $\mathscr{C}:=\{\gamma : \mathbb{R}_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\...
- 463
4
votes
0
answers
200
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Asymptotic behavior of a dynamical system of density functions
On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here:
Link to question on Mathematics Stack Exchange.
I received two up-votes, but no comments or answer. I ...
- 41
4
votes
1
answer
426
views
A process of repeated convolution and conditioning and the resulting sequence of probability distributions
I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$.
Let $D_1$ be the $n$-dimensional Gaussian distribution with ...
- 1,016
0
votes
0
answers
118
views
A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
1
vote
0
answers
177
views
Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
0
answers
96
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Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
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7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,658
3
votes
0
answers
92
views
What dynamical properties should we expect from systems satisfying statistical ones?
Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
2
votes
1
answer
179
views
Union of admissible words are subshift of finite type
Assume that $Q=(q_{ij})$ is a $k\times k$ with $q_{ij}\in \{0, 1\}.$ The two side subshift of finite type associated to the matrix $Q$ is a left shift map $T:\Sigma_{Q}\rightarrow \Sigma_{Q}$, where
...
- 1,043
0
votes
0
answers
83
views
Distortion estimates to control Hausdorff measure of a curve
I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...
3
votes
0
answers
145
views
2-ball billiards in a circle
Consider a 2D circular billiards table with diameter 1m containing two
balls with diameter 0.25m. Let each ball start with a speed of 1m/s.
In general, this speed could change after the balls hit ...
- 1,547
9
votes
1
answer
358
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
4
votes
1
answer
446
views
Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
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10
votes
2
answers
559
views
Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
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1
vote
1
answer
176
views
Invariant distributions for iterated random variables (stochastic dynamical systems)
This is related to discrete dynamical systems, with the initial condition $X_1$ being a random variable with a non singular distribution. The system is driven by the iteration $X_{n+1} = g(X_n)$ for ...
- 3,098
2
votes
0
answers
143
views
inverse of moment-generating function in terms of moments
Let $\{h_i\}$ be decreasing sequence of $n$ positive reals. Define distribution $p(X=h_i)\propto h_i$ and let $g(s)=E_X[e^{sX}]$ be the moment generating function. For instance, for $h=\{1,\frac{1}{4},...
- 2,442
0
votes
1
answer
137
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Do measure-valued dynamical systems correspond to marginals of Markov processes?
Let $(\mu_n)_{n=1}^{\infty}$ be a sequence in $\mathcal{P}_1(X)$ for some compact metric space $(X,d)$. Suppose that there is a weakly-continuous function $F:\mathcal{P}_1(X)\rightarrow \mathcal{P}_1(...
- 5,405
7
votes
1
answer
274
views
Uniqueness of stationary measures for $(G,\mu)$ boundaries
Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ is a $\...
- 481
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
1
vote
0
answers
193
views
Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map
Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
- 3,098
2
votes
1
answer
126
views
Values appearing with density in an ergodic system
Values appearing with density in an ergodic system
Let $(X,\mu)$ be a probability space with invertible, measure preserving, totally-ergodic map $T:X \to X$. ($(X,\mu,T)$ is a $\mathbb{Z}$ dynamical ...
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10
votes
2
answers
488
views
A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
- 128k
1
vote
0
answers
66
views
When are all average trajectories of $w_{k+1}=Aw_k+b$ bounded?
Below is an open-problem in my field, and I'm wondering if someone has insights I'm missing. (cross-posted on math.se)
Suppose observation $x$ is drawn from some distribution $\mathcal{D}$, $w_0\in \...
- 2,442
14
votes
0
answers
358
views
What is the asymptotic dynamics of the winning position in this game?
$n$ players indexed $1,2,...,n$ play a game of mock duel. The rules are simple: starting from player $1$, each player takes turns to act in the order $1,2,...,n,1,2,...$. In his turn, a player ...
- 2,619
5
votes
0
answers
183
views
Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
3
votes
1
answer
233
views
A subadditive maximal ergodic theorem
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
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6
votes
1
answer
611
views
The "Chaos Game" as a particular series of i.i.d. random variables
Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X_0,X_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable
$X_\infty = (1-\alpha)\sum_{k=0}^\infty ...
- 646
1
vote
0
answers
72
views
Distance between value function of deterministic and stochastic control problems
Suppose that one wants to control a diffusion process
$$
dX_t^u = \mu(X_t^u,u)dt + \sigma dW_t; \qquad X_0^u=x
$$
in order to optimize a stochastic control problem with value function
$$
V_T(u)=\...
- 5,405
1
vote
1
answer
170
views
Stationary distribution of Markov Chain with departure
I have a Markov Chain of $N$ states. Such states represent the energy levels in a molecule.
The states' connectivity is as follows:
States $j\in\{0,\ldots,N\}$ transition to $k\in\{\max(j-M,0),...,\...
- 151
2
votes
1
answer
117
views
Size of the orbit of a dense set
This question is a follow-up to: this post.
Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
- 63
2
votes
1
answer
409
views
Existence and uniqueness of a stationary measure
This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure.
Recently I have posted the following question on MO ...
- 333
3
votes
1
answer
372
views
Attractors in random dynamics
Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
- 333
11
votes
1
answer
393
views
Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
- 151k
8
votes
0
answers
157
views
Pursuit-evasion with many slow pursuers
Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...
- 7,545
1
vote
0
answers
179
views
Two mixing rates of random dynamical system
Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define ...
- 553
1
vote
0
answers
59
views
Regularity of the pdf of partial Birkhoff sums
Suppose that $T: X \to X$ is some measurable map on a Riemannian manifold $X$ (possibly with boundary). Let $\mu$ denote the Riemannian measure on $X$. For measurable, real-valued $g$ we may consider ...
- 403
0
votes
0
answers
48
views
Characterization of Time-homogeneous flows for conditional expectation
Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
- 5,405
1
vote
0
answers
86
views
Coboundary in the slow mixing systems
Given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
- 553
1
vote
1
answer
94
views
Does the following percolation model have a name?
Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and ...
- 403
0
votes
1
answer
174
views
Probabilistic approach for cellular automata
Few months ago my scientific adviser asked me to use probabilistic ideas in such problem :
Consider a matrix NxN. Each element of matrix is a number 1 or 0. We may change all elements of this matrix ...
- 137