Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

Tightness of Random Walks in Infinite Dimensional Spaces and Manifolds

created by mennucci on 11 Oct 2022
modified on 23 Dec 2022



Inserted: 11 oct 2022
Last Updated: 23 dec 2022

Year: 2022


In this paper we study random walks \(X^\tau\); these are processes taking values in \(C(R^+;S)\), where \(R^+=[0,\infty)\). These random walks are defined at discrete times \(t\in\tau=\{t_0=0<t_1<t_2\ldots\}\) and then interpolated for \(t\) between \(t_i,t_{i+1}\).

The main objective is to prove tightness for the family of all \(X^\tau\); by Prokhorov's Theorem, this implies that the sequence has limit points that are random functions in \(C(R^+;S)\).

We will provide results in three cases: \(S=H\) a (possibly infinite dimensional) separable Hilbert Space; \(S\) a manifold embedded in \(H\); and then the particular case when \(S\) is the Stiefel Manifold.

These results are motivated by problems in Probability Theory and in Shape Theory, and in particular some models of manifolds of planar immersed curves.