Most Probable Paths for Anisotropic Brownian Motions on Manifolds

Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths

We use anisotropic diffusion processes to generalize normal distributions to manifolds and to construct a framework for likelihood estimation of template and covariance structure from manifold valued data. The procedure avoids the linearization that arise when first estimating a mean or template before performing PCA in the tangent space of the mean. We derive flow equations for the most probab...

Radiation Reaction on Brownian Motions

Tracking the “real” trajectory of a quantum particle is one of the interpretation problem and it is expressed by the Brownian (stochastic) motion suggested by E. Nelson. Especially the dynamics of a radiating electron, namely, “radiation reaction” which requires us to track its trajectory becomes important in the high-intensity field physics by PW-class lasers at present. It has been normally t...

On the Sample Paths of Brownian Motions on Compact Infinite Dimensional Groups by Alexander Bendikov

Cut Points on Brownian Paths

Let X be a standard 2-dimensional Brownian motion. There exists a.s. t ∈ (0, 1) such that X([0, t)) ∩ X((t, 1]) = ∅. It follows that X([0, 1]) is not homeomorphic to the Sierpiński carpet a.s.

Gibbs Measures on Brownian Paths

This is a summary of results based upon 1, 4] outlining how Gibbs measures can be deened on Brownian paths and what are their most important properties. Such Gibbs measures have a number of applications in Euclidean quantum eld theory, statistical mechanics, stochastic (partial) diierential equations and other areas.