Finiteness of hitting times under taboo

We consider a continuous-time Markov chain with a finite or countable state space. For a site y and subset H of the state space, the hitting time of y under taboo H is defined to be infinite if the process trajectory hits H before y, and the first hitting time of y otherwise. We investigate the probability that such times are finite. In particular, if the taboo set is finite, an efficient iterative scheme reduces the study to the known case of a singleton taboo. A similar procedure applies in the case of finite complement of the taboo set. The study is motivated by classification of branching processes with finitely many catalysts. Keywords and phrases: Markov chain, hitting time, taboo probabilities, catalytic branching process

Strong and weak convergence of population size in supercritical catalytic branching process

A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for phase of pure global survival and strong local survival. Under wide conditions the limit theorems for the normalized total and local particles numbers in supercritical CBP are proved in the sense of almost surely convergence as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well. In the proofs the main role is played by recent results by the author devoted to classification of CBP and the moment analysis of the total and local particles numbers in CBP. Keywords and phrases: catalytic branching process, extinction probability, strong local survival, pure global phase, limit theorems, multi-type Bellman-Harris process

Catalytic Branching Random Walk with Semi-exponential Increments

A catalytic branching random walk on a multidimensional lattice, with arbitrary finite number of catalysts, is studied in supercritical regime. The dynamics of spatial spread of the particles population is examined, upon normalization. The components of the vector random walk jump are assumed independent (or close to independent) and have semi-exponential distributions with, possibly, different parameters. A limit theorem on the almost sure normalized positions of the particles at the population ``front'' is established. Contrary to the case of the random walk increments with ``light'' distribution tails, studied by Carmona and Hu (2014) in one-dimensional setting and Bulinskaya (2018) in multidimensional setting, the normalizing factor has a power rate and grows faster than linear in time function. The limiting shape of the front in the case of semi-exponential tails is non-convex in contrast to a convex one in the case of light tails.Comment: 1 figur

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables. Keywords and phrases: catalytic branching random walk, heavy tails, regular varying tails, spread of population, L log L condition

Correlation Functions of Local Operators in 2D Gravity Coupled to Minimal Matter

Recent advances are being discussed on the calculation, within the conformal field theory approach, of the correlation functions for local operators in the theory of 2D gravity coupled to the minimal models of matter.Comment: 12 page

Naive model from 1970th applied to CMR manganites: it seems to work

Existing experimental data for various colossal magnetoresistance manganites have been examined employing an ovesimplified model that roots in 1970th. This model considers a classical semiconductor where conducting bands are affected by the strong Weiss exchange field that arises from the magnetic order in the substance. The field--caused shifts of the conducting bands results in the change in the number of thermally activated carriers, and this change is presumed to be responsible for the resistivity dependences on temperature and magnetic field and for the CMR itself. Employing this model we calculate this hypothetical Weiss field from the experimental data for various CMR manganites employing minimal set of the adjustable parameters, namely two. The obtained Weiss field behaves with temperature and external field similarly to the local magnetization, its supposed source, hence supporting the model.Comment: 14 pages, 7 figure

Classification of catalytic branching processes and structure of the criticality set

We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may only occur at the points of catalysis. The results obtained generalize and strengthen those known in cases of CBP with a single catalyst and of branching random walk on d-dimensional integer lattice with a finite number of sources of particles generation. We propose to classify CBP with N catalysts as supercritical, critical or subcritical according to the value of the Perron root of a specified NxN matrix. Such classification agrees with the moment analysis performed here for local and total particles numbers. By introducing the criticality set C we also consider the influence of catalysts parameters on the process behavior. The proof is based on construction of auxiliary multi-type Bellman-Harris processes with the help of hitting times under taboo and on application of multidimensional renewal theorems. Keywords and phrases: catalytic branching process, classification, hitting times under taboo, moment analysis, multi-type Bellman-Harris process.Comment: 1 figur

Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice

For a supercritical catalytic branching random walk on Z^d (d is positive integer) with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. Namely, we divide by t the position coordinates of each particle existing at time t and then let t tend to infinity. It is shown that in the limit there are a.s. no particles outside the closed convex surface in R^d which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation. Recent strong limit theorems for total and local particles numbers established by the author play an essential role. The results obtained develop ones by Ph.Carmona and Y.Hu (2014) devoted to the spread of catalytic branching random walk on Z. Keywords and phrases: branching random walk, supercritical regime, spread of population, propagation front, many-to-one lemma.Comment: 2 figure

Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Subcritical catalytic branching random walk on d-dimensional lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particles numbers are established. To prove the results different approaches are used including the connection between fractional moments of random variables and fractional derivatives of their Laplace transforms. In the previous papers on this subject only supercritical and critical regimes were investigated assuming finiteness of the first moment of offspring number and finiteness of the variance of offspring number, respectively. In the present paper for the offspring number in subcritical regime finiteness of the moment of order 1+\delta is required where \delta is some positive number. Keywords and phrases: branching random walk, subcritical regime, finite variance of offspring number, infinite variance of offspring number, local particles numbers, limit theorems, fractional moments, fractional derivatives

Online Handwritten Devanagari Stroke Recognition Using Extended Directional Features

This paper describes a new feature set, called the extended directional features (EDF) for use in the recognition of online handwritten strokes. We use EDF specifically to recognize strokes that form a basis for producing Devanagari script, which is the most widely used Indian language script. It should be noted that stroke recognition in handwritten script is equivalent to phoneme recognition in speech signals and is generally very poor and of the order of 20% for singing voice. Experiments are conducted for the automatic recognition of isolated handwritten strokes. Initially we describe the proposed feature set, namely EDF and then show how this feature can be effectively utilized for writer independent script recognition through stroke recognition. Experimental results show that the extended directional feature set performs well with about 65+% stroke level recognition accuracy for writer independent data set.Comment: 8th International Conference on Signal Processing and Communication Systems 15 - 17 December 2014, Gold Coast, Australi