Search ResultsShowing 1-11 of 11
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arXiv:2409.19246 (Published 2024-09-28)
Quasi-stationary distributions of non-absorbing Markov chains
Comments: 20 pages, 7 figuresWe consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the ``hitting time of the stationary distribution''. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the \emph{Yaglom limit}. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.
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arXiv:1410.4814 (Published 2014-10-17)
Conditioned, quasi-stationary, restricted measures and escape from metastable states
Comments: 39 pagesCategories: math.PRWe study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a trap defined by very general properties. We give an explicit description of the law of $X^{(n)}$ conditioned to stay within the trap, and from this we deduce the exponential distribution of $\tau^{(n)}$. Our approach is very broad ---it does not require reversibility, the target $G$ does not need to be a rare event, and the traps and the limit on $n$ can be of very general nature--- and leads to explicit bounds on the deviations of $\tau^{(n)}$ from exponentially. We provide two non trivial examples to which our techniques directly apply.
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arXiv:0704.3156 (Published 2007-04-24)
How to clean a dirty floor: Probabilistic potential theory and the Dobrushin uniqueness theorem
Comments: LaTex2e, 80 pages including 4 figuresJournal: Markov Processes and Related Fields 14, 1--78 (2008)Keywords: dobrushin uniqueness theorem, probabilistic potential theory, dirty floor, countably infinite index set, formal power seriesTags: journal articleMotivated by the Dobrushin uniqueness theorem in statistical mechanics, we consider the following situation: Let \alpha be a nonnegative matrix over a finite or countably infinite index set X, and define the "cleaning operators" \beta_h = I_{1-h} + I_h \alpha for h: X \to [0,1] (here I_f denotes the diagonal matrix with entries f). We ask: For which "cleaning sequences" h_1, h_2, ... do we have c \beta_{h_1} ... \beta_{h_n} \to 0 for a suitable class of "dirt vectors" c? We show, under a modest condition on \alpha, that this occurs whenever \sum_i h_i = \infty everywhere on X. More generally, we analyze the cleaning of subsets \Lambda \subseteq X and the final distribution of dirt on the complement of \Lambda. We show that when supp(h_i) \subseteq \Lambda with \sum_i h_i = \infty everywhere on \Lambda, the operators \beta_{h_1} ... \beta_{h_n} converge as n \to \infty to the "balayage operator" \Pi_\Lambda = \sum_{k=0}^\infty (I_\Lambda \alpha)^k I_{\Lambda^c). These results are obtained in two ways: by a fairly simple matrix formalism, and by a more powerful tree formalism that corresponds to working with formal power series in which the matrix elements of \alpha are treated as noncommuting indeterminates.
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arXiv:math/0410191 (Published 2004-10-07)
Spatial birth-and-death processes in random environment
Comments: 48 pagesWe consider birth-and-death processes of objects (animals) defined in ${\bf Z}^d$ having unit death rates and random birth rates. For animals with uniformly bounded diameter we establish conditions on the rate distribution under which the following holds for almost all realizations of the birth rates: (i) the process is ergodic with at worst power-law time mixing; (ii) the unique invariant measure has exponential decay of (spatial) correlations; (iii) there exists a perfect-simulation algorithm for the invariant measure. The results are obtained by first dominating the process by a backwards oriented percolation model, and then using a multiscale analysis due to Klein to establish conditions for the absence of percolation.
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Construction of a specification from its singleton part
Comments: 19 pages. Version to appear in ALEA. We added Proposition 4.3 and made small corrections with respect to version 1Categories: math.PRSubjects: 82B05We state a construction theorem for specifications starting from single-site conditional probabilities (singleton part). We consider general single-site spaces and kernels that are absolutely continuous with respect to a chosen product measure (free measure). Under a natural order-consistency assumption and weak non-nullness requirements we show existence and uniqueness of the specification extending the given singleton part. We determine conditions granting the continuity of the specification. In addition, we show that, within a class of measures with suitable support properties, consistency with singletons implies consistency with the full specification.
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Chains with complete connections: General theory, uniqueness, loss of memory and mixing properties
Comments: 35 pagesCategories: math.PRWe introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail $\sigma$-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. These results are complementary of those existing in the literature, and generalize the Markovian results based on the Dobrushin ergodic coefficient.
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Chains with complete connections and one-dimensional Gibbs measures
Comments: 31 pagesWe discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily Gibbsian).
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arXiv:math/0107081 (Published 2001-07-11)
Variational principle and almost quasilocality for some renormalized measures
Comments: 19 pages, LaTeXWe restore part of the thermodynamic formalism for some renormalized measures that are known to be non-Gibbsian. We first point out that a recent theory due to Pfister implies that for block-transformed measures free energies and relative entropy densities exist and are conjugate convex functionals. We then determine a necessary and sufficient condition for consistency with a specification that is quasilocal in a fixed direction. As corollaries we obtain consistency results for models with FKG monotonicity and for models with appropriate "continuity rates". For (noisy) decimations or projections of the Ising model, these results imply almost quasilocality of the decimated "+" and "-" measures.
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arXiv:math-ph/0101014 (Published 2001-01-15)
Non-Gibbsianness of the invariant measures of non-reversible cellular automata with totally asymmetric noise
Comments: 29 pages (double space)We present a class of random cellular automata with multiple invariant measures which are all non-Gibbsian. The automata have configuration space {0,1}^{Z^d}, with d > 1, and they are noisy versions of automata with the "eroder property". The noise is totally asymmetric in the sense that it allows random flippings of "0" into "1" but not the converse. We prove that all invariant measures assign to the event "a sphere with a large radius L is filled with ones" a probability \mu_L that is too large for the measure to be Gibbsian. For example, for the NEC automaton -ln(\mu_L) ~ L while for any Gibbs measure the corresponding value is ~ L^2.
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Processes with Long Memory: Regenerative Construction and Perfect Simulation
Comments: 27 pages, one figure. Version accepted by Annals of Applied Probability. Small changes with respect to version 2Journal: Ann. Appl. Probab. Volume 12, Number 3 (2002), 921-943Keywords: regenerative construction, long memory, perfect simulation algorithm, unit interval, stationary processesTags: journal articleWe present a perfect simulation algorithm for stationary processes indexed by Z, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Mart{\'\i}nez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval.
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Perfect simulation for interacting point processes, loss networks and Ising models
Comments: Revised version after referee of SPA: 39 pagesJournal: Stochastic Process. Appl. 102 (2002), no. 1, 63--88Subjects: 60K35Keywords: interacting point processes, ising models, invariant measures, target process, perfect simulation algorithmTags: journal articleWe present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birth-and-death processes. Examples include area- and perimeter-interacting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve couplings of the process with different initial conditions and it is not tied up to monotonicity requirements. Furthermore, it directly provides perfect samples of finite windows of the infinite-volume measure, subjected to time and space ``user-impatience bias''. The algorithm is based on a two-step procedure: (i) a perfect-simulation scheme for a (finite and random) relevant portion of a (space-time) marked Poisson processes (free birth-and-death process, free loss networks), and (ii) a ``cleaning'' algorithm that trims out this process according to the interaction rules of the target process. The first step involves the perfect generation of ``ancestors'' of a given object, that is of predecessors that may have an influence on the birth-rate under the target process. The second step, and hence the whole procedure, is feasible if these ``ancestors'' form a finite set with probability one. We present a sufficiency criteria for this condition, based on the absence of infinite clusters for an associated (backwards) oriented percolation model.