Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in probability theory as well as stochastic processes. Nonstandard analysis allows construction of a single object---a hyperfinite probability space---which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings.
In this talk, for every general-state-space discrete-time Markov process satisfying appropriate conditions, we construct a hyperfinite Markov process which has all the basic order logical properties of a finite Markov process to represent it. We show that the mixing time and the hitting time agree with each other up to some multiplicative constants for discrete-time general-state-space reversible Markov processes satisfying certain condition. Finally, we show that our result is applicable to a large class of Gibbs samplers and Metropolis-Hasting algorithms.
The logic of metric structures was introduced by Ben Yaacov, Berenstein , Henson and Usvyatsov. It is a version of continuous logic which allows fruitful model theory for many kinds of metric structures .There are many aspects of this logic which make it similar to first order logic, like compactness, a complete proof system, an omitting types theorem for complete types etc. But when one tries to generalize the omitting type criteria to general (non-complete) types the problem turns out to be essentially more difficult than the first order situation.For instance one can have two types (in a complete theory) that each one can be omitted , but they can not be omitted simultaneously.
The precise structural understanding of uncountably categorical theories given by the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an ω-stable metric theory. Finally we will examine the extent to which we recover the Baldwin-Lachlan theorem in the presence of strongly minimal sets.
Metric structures are like first-order structures except that the formulas take truth values in the unit interval, and instead of equality there is a distance predicate with respect to which every function and predicate is uniformly continuous. Pre-metric structures are similar the distance predicate is only a pseudo-metric. In recent years the model theory of metric and pre-metric structures has been successfully developed in a way that is closely parallel to first order model theory, with many applications to analysis.
We consider general structures, where formulas still have truth values in the unit interval, but the predicates and functions need not be continuous with respect to a distance predicate. It is shown that every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, any two such expansions have the same notion of uniform convergence. This can be used to extend almost all of the model theory of metric structures to general structures in a precise way. For instance, the notion of a stable theory extends in a natural way to general structures, and the main results carry over.