From the Big Bang Theory to the Theory of a Stationary Universe

We consider chaotic inflation in the theories with the effective potentials phi^n and e^{\alpha\phi}. In such theories inflationary domains containing sufficiently large and homogeneous scalar field \phi permanently produce new inflationary domains of a similar type. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time, both at the stage of inflation and after it. This represents a strong deviation of inflationary cosmology from the standard Big Bang paradigm. We compare our approach with other approaches to quantum cosmology, and illustrate some of the general conclusions mentioned above with the results of a computer simulation of stochastic processes in the inflationary Universe.Comment: No changes to the file, but original figures are included. They substantially help to understand this paper, as well as eternal inflation in general, and what is now called the "multiverse" and the "string theory landscape." High quality figures can be found at http://www.stanford.edu/~alinde/LLMbigfigs

Recursive Markov chains, stochastic grammars, and monotone systems of non-linear equations

We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, an

Analysis of a recurrence related to critical non-homogeneous Branching processes

Some classes of controlled branching processes (with nonhomogeneous migration or with nonhomogeneous state-dependent immigration) lead in the critical case to a recurrence for the extinction probabilities. Under some additional conditions it is known that this recurrence depends on some parameter β and converges for 0 < β < 1. Now we show that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes. We also give a generalization of the recurrence and an asymptotic analysis of its behavior.SCOPUS: ar.jinfo:eu-repo/semantics/publishe