# Processos de passeio na reta contínua

Abstract (Summary)

We study a property similar to ergodicity of a class of random processes with localinteraction with continuous space and discrete time. Our process is a sequence of random subsets Ut of a real line, where t = 0, 1, 2, 3, . . . is called time. These sets are of a special kind: their intersections with any limited piece of the real line are linear combinations of a finite list of _-measures, each concentrated in a set consisting of several closed mutually non-intersecting segments, which we call blocks. These sets are generated inductively. Initially, when t = 0, our set U0 is empty. At every time step three operators are applied to Ut to obtain Ut+1. The first operator, W_, includes into our set some of the segments [i, i + 1], where i amp;#8712; Z, chosen at random: each segment is included with a probability _ independently of others. The second operator, WD, includes into our set all small enough gaps between the blocks. The action of the third operator, Wpas, depends on two discrete random variables L and R, each taking only a finite set of values. At one application of Wpas, left ends of all the blocks perform one step of random walk distributed as L independently from each other. The right ends of all the blocks do the same, only using the random variable R instead of L. We say that our process fills the line if for any limited segment the probability that Ut includes this segment tends to one when time tends to infinity. (This is analog of ergodicity.) We show that our process has two types of behavior: If E(L) lt; E(R) (where E means mathematical expectation), our process fills the line for any _ gt; 0. If E(L) gt; E(R), our process does not fill the line if _ is small enough. This contrast has been shown for the discrete line and now we generalize it to the continuous line. Our approach paves the way for a theory of processes with local interaction on a real line, which remains little developed till now
Bibliographical Information:

Advisor:Andrei Toom

School:Universidade Federal de Pernambuco

School Location:Brazil

Source Type:Master's Thesis

Keywords:probabilidade processo estocástico teorema principal preenchimento e não da reta real estatistica

ISBN:

Date of Publication:02/23/2006