Abstract:
The bulk of modern studies of locally interacting particle processes is based on the assumption that the set of sites, called the space, does not change in the process of interaction. Elements of this space, called components, may be in different states, e.g. 0 and 1, often interpreted as absence vs. presence of a particle, and may go from one state to another, which may be interpreted as birth or death of a particle, but the sites themselves do not appear or disappear in the process of functioning. Operators and processes which do not create or eliminate sites will be called constant-length ones. However, in various areas of knowledge we deal with long sequences of components, which are subject to some local random transformations, which may change their lengths. The simplest and the most well-known of such transformations are often called "insertion" and "deletion" and are widely discussed in informatics and molecular biology. In such cases we use the phrase variable-lenght processes and our goal is to provide a rigorous definition of some class of variable-length processes with infinite space and study their properties. Thus we study a new kind of random processes with discrete time. In fact we study a certain class of operators called substitution operators whose iterations generate a process. Our operators act on probability measures on a configuration space A^Z, where A is a finite set called alphabet, elements of which are called letters. Elements of A^Z are bi-infinite sequences of letters. Let us call any finite sequence of letters a word. Length of a word is the number of letters in it. Informally speaking, action of our operator consists in the following: any occurence of a word G in a configuration may be substituted by another word H with a certain probability rho. Many well-known operators fit into this description with G and H having equal lenghts. Our main novelty is that the lengths of G and H may be different. This makes our operators non-linear and causes our main difficulties. Our main task is to define such operators rigorously (which is not trivial since our space is Z rather than its finite segment), prove some of their properties and study invariant measures. (In collaboration with A.D.Ramos, A.V.Rocha and A.B.Simas.)
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