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Register a new userNegative Entropy, Zero temperature and stationary Markov Chains on the interval
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We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^mathbb{N}to mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $mu_infty$ on $ [0,1]^mathbb{N}$ that maximize the value $ int A d mu,$ among all stationary Markov probabilities $mu$ on $[0,1]^mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $mu_beta$ which weakly converges to $mu_infty$. The probabilities $mu_beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $frac{partial^2 A}{partial_x partial_y} (x,y) eq 0$, for all $(x,y) in [0,1]^2$, we show the graph property.
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We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $tgeq 0$, taking values on S. We denote by $Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($Omega, {cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${cal L}^infty (P)$. From the probability $P$ we define a Ruelle operator ${cal L}^t, tgeq 0$, acting on functions $f:Omega to mathbb{R}$ of ${cal L}^infty (P)$. Given $V:Omega to mathbb{R}$, such that is constant in sets of the form ${X_0=c}$, we define a modified Ruelle operator $tilde{{cal L}}_V^t, tgeq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ u_V$ on $Omega$ associated to $tilde{{cal L}}^t_V, tgeq 0$. We also show the following property for the probability $ u_V$: for any integrable $gin {cal L}^infty (P)$ and any real and positive $t$ $$ int e^{-int_0^t (V circ Theta_s)(.) ds} [ (tilde{{cal L}}^t_V (g)) circ theta_t ] d u_V = int g d u_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).
We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.
In this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative including additive white noises. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak$^*$-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation is provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.
In probability theory, the independence is a very fundamental concept, but with a little mystery. People can always easily manipulate it logistically but not geometrically, especially when it comes to the independence relationships among more that two variables, which may also involve conditional independence. Here I am particularly interested in visualizing Markov chains which have the well known memoryless property. I am not talking about drawing the transition graph, instead, I will draw all events of the Markov process in a single plot. Here, to simplify the question, this work will only consider dichotomous variables, but all the methods actually can be generalized to arbitrary set of discrete variables.
Parametric Interval Markov Chains (pIMCs) are a specification formalism that extend Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into account imprecision in the transition probability values: transitions in pIMCs are labeled with parametric intervals of probabilities. In this work, we study the difference between pIMCs and other Markov Chain abstractions models and investigate the two usual semantics for IMCs: once-and-for-all and at-every-step. In particular, we prove that both semantics agree on the maximal/minimal reachability probabilities of a given IMC. We then investigate solutions to several parameter synthesis problems in the context of pIMCs -- consistency, qualitative reachability and quantitative reachability -- that rely on constraint encodings. Finally, we propose a prototype implementation of our constraint encodings with promising results.
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