Skew-product Decomposition Research Articles

Skew-product decomposition of Brownian motion on an ellipsoid

In this article we obtain a skew-product decomposition of a Brownian motion on an ellipsoid of dimension n in a Euclidean space of dimension n+1. We only consider such ellipsoid whose restriction to first n dimensions is a sphere and its last coordinate depends on a variable parameter. We prove that the projection of this Brownian motion on to the last coordinate is, after a suitable transformation, a Wright–Fisher diffusion process with atypical selection coefficient.

Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.

A note on the exact simulation of spherical Brownian motion

We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright–Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017). The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.

Invariance principle for non-homogeneous random walks

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R} ^{n+\ell }$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R} ^{n+\ell }$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

When do skew-products exist?

The classical skew-product decomposition of planar Brownian motionrepresents the process in polar coordinates as an autonomously Markovian part and an part that is an independent Brownian motion on the unit circle time-changed according to the part. Theorem 4 of L09 gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action ofa Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian radial part that lives on the space of orbits of $K$ and an angular part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the part. We present two apparent counterexamples to Theorem 4 of L09. In the first counterexample the part is not a time-change of any Brownian motionon $K/M$, whereas in the second counterexample the part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the part. In both of these examples $K/M$ has dimension $1$. The statement and proof of Theorem 4 in L09 remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the part is a time-changed Brownian motion but this Brownian motion is not independent of the part.

Penalisations of multidimensional Brownian motion, VI

As in preceding papers in which we studied the limits of penalized one-dimensional Wiener measures with certain functionals Γt, we obtain here the existence of the limit, as t → ∞, of d-dimensional Wiener measures penalized by a function of the maximum up to time t of the Brownian winding process (for d = 2), or in d ≥ 2 dimensions for Brownian motion prevented to exit a cone before time t. Various extensions of these multidimensional penalisations are studied, and the limit laws are described. Throughout this paper, the skew-product decomposition of d-dimensional Brownian motion plays an important role.

P-Adic spherical coordinates and their applications

On the space ℚ , where p ≠ 2 and p does not divide n, we construct a p-adic counterpart of spherical coordinates. As applications, a description of homogeneous distributions on ℚ and a skew product decomposition of p-adic Lévy processes are given.

REDUCTION, RECONSTRUCTION, AND SKEW-PRODUCT DECOMPOSITION OF SYMMETRIC STOCHASTIC DIFFERENTIAL EQUATIONS

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. In addition, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.

Skew-product representations of multidimensional Dunkl Markov processes

In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. S\'{e}minaire de Probabilit\'{e}s XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.

Notes on the two-dimensional fractional Brownian motion

We study the two-dimensional fractional Brownian motion with Hurst parameter H>½. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion.

Random Holonomy for Hopf Fibrations

In this paper we study the heat flow on sections of the complex and quaternion line bundles associated to the Hopf fibration over complex and quaternionic projective space. These bundles are equipped with natural metrics and connections. We represent solutions to the heat-equation by the expectation of an appropriate stochastic functional on path space. This uses the stochastic parallel transport. We obtain the first eigenvalue of the horizontal Laplacian by calculating conditional expectations of the stochastic parallel transport. We use a stochastic argument similar to a skew-product decomposition to reduce the problem to an ordinary differential equation of Sturm–Liouville type. We find all eigenvalues and eigenfunctions of this equation.

Brownian motion in cones

We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an infinite series representation of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition of Brownian motion is derived and used to show properties of the transition density.

Brownian excursions from hyperplanes and smooth surfaces

A skew-product decomposition of the $n$-dimensional $(n \geq 2)$ Brownian excursion law from a hyperplane is obtained. This is related to a Kolmogorov-type test for excursions from hyperplanes. Several results concerning existence, uniqueness and form of Brownian excursion laws from sufficiently "flat" surfaces are given. Some of these theorems are potential-theoretic in spirit. An extension of the results concerning excursion laws to an exit system in a Lipschitz domain is supplied.