Conditioned Weiner Processes as Nonlinearities: A Rigorous Probabilistic Analysis of Dynamics

TL;DR

This paper rigorously analyzes the dynamics of conditioned Wiener processes, providing a probabilistic framework to understand invariant sets and their structures, which aids in the statistical analysis of dynamical systems from finite samples.

Contribution

It introduces a topological and probabilistic approach to characterize the global dynamics of conditioned Wiener processes, establishing a foundation for analyzing systems from limited data.

Findings

Characterizes invariant sets in conditioned Wiener processes

Provides probability estimates for the correctness of the dynamic characterization

Offers a theoretical basis for statistical analysis of dynamical systems

Abstract

We study a Weiner process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, we compute the probability that this characterization is correct. This work is probabilistic in nature and intended to provide a theoretical foundation for the statistical analysis of dynamical systems which can only be queried via finite samples.