Spherical Brownian Motion Research Articles

An algorithm for simulating Brownian increments on a sphere

This paper presents a novel formula for the transition density of the Brownian motion on a sphere of any dimension and discusses an algorithm for the simulation of the increments of the spherical Brownian motion based on this formula. The formula for the density is derived from an observation that a suitably transformed radial process (with respect to the geodesic distance) can be identified as a Wright–Fisher diffusion process. Such processes satisfy a duality (a kind of symmetry) with a certain coalescent processes and this in turn yields a spectral representation of the transition density, which can be used for exact simulation of their increments using the results of Jenkins and Spanò (2017 Ann. Appl. Probab. 27 1478–09). The symmetry then yields the algorithm for the simulation of the increments of the Brownian motion on a sphere. We analyse the algorithm numerically and show that it remains stable when the time-step parameter is not too small.

Phylogeographic Estimation and Simulation of Global Diffusive Dispersal.

The analysis of time-resolved phylogenies (timetrees) and geographic location data allows estimation of dispersal rates, for example, for invasive species and infectious diseases. Many estimation methods are based on the Brownian Motion model for diffusive dispersal on a 2D plane; however, the accuracy of these methods deteriorates substantially when dispersal occurs at global scales because spherical Brownian motion (SBM) differs from planar Brownian motion. No statistical method exists for estimating SBM diffusion coefficients from a given timetree and tip coordinates, and no method exists for simulating SBM along a given timetree. Here, I present new methods for simulating SBM along a given timetree, and for estimating SBM diffusivity from a given timetree and tip coordinates using a modification of Felsenstein's independent contrasts and maximum likelihood. My simulation and fitting methods can accommodate arbitrary time-dependent diffusivities and scale efficiently to trees with millions of tips, thus enabling new analyses even in cases where planar BM would be a sufficient approximation. I demonstrate these methods using a timetree of marine and terrestrial Cyanobacterial genomes, as well as timetrees of two globally circulating Influenza B clades. My methods are implemented in the R package "castor." [Independent contrasts; phylogenetic; random walk; simulation; spherical 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.

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.

Hypocoercive estimates on foliations and velocity spherical Brownian motion

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose generator is a step-3 generating hypoelliptic Hörmander's type operator. To prove hypocoercivity in that case, the key point is to show the existence of a convenient Riemannian foliation associated to the diffusion. We will then deduce, under suitable geometric conditions, the convergence to equilibrium of the diffusion in $H^1$ and in $L^2$.

Time dependent random fields on spherical non-homogeneous surfaces

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere which is represented by a time-changed spherical Brownian motion of order ν∈(0,1]. We can capture some anisotropies in Cosmology with this model. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotropic random field is indexed. The time dependent random fields we present in this work are therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.

Extremal points of high-dimensional random walks and mixing times of a Brownian motion on the sphere

Nous etudions le comportement asymptotique de la probabilite que l’origine soit un point extremal d’une marche aleatoire dans $\mathbb{R}^{n}$. Nous montrons que cette probabilite est proche de $1/2$ si le nombre de pas de la marche aleatoire est entre $\mathrm{e}^{n/(C\log n)}$ et $\mathrm{e}^{Cn\log n}$ pour une certaine constante $C>0$. Comme corollaire, nous obtenons une borne pour le temps de $\frac{\pi}{2}$-recouvrement d’un mouvement brownien spherique.

Representation of Gaussian isotropic spin random fields

We develop a technique for the construction of random fields on algebraic structures. We deal with two general situations: random fields on homogeneous spaces of a compact group and in the spin line bundles of the 2-sphere. In particular, every complex Gaussian isotropic spin random field can be represented in this way. Our construction extends P. Lévy’s original idea for the spherical Brownian motion.

Convergence of Time Changed Skew Product Diffusion Processes

A limit theorem for the time changed skew product diffusion processes is investigated. Skew product diffusion processes are given by one dimensional diffusion processes and the spherical Brownian motion, and the time change is based on a positive continuous additive functional. It is shown that the limit process is corresponding to Dirichlet form of non-local type according to degeneracy of the limit measure of underlying ones. Some examples of limit processes are given which lead us to Dirichlet forms with diffusion term, jump term and killing term.

Recurrence/transience criteria for skew product diffusion processes

We give recurrence/transience criteria for skew products of one dimensional diffusion process and the spherical Brownian motion with respect to a positive continuous additive functional of the former one dimensional diffusion process. Further we give recurrence/transience criteria for their time changed processes.

Feller property of skew product diffusion processes

Feller property of some diffusion processes and the time changed processes is investigated. Diffusion processes treated here are skew product of one dimensional generalized diffusion processes and the spherical Brownian motion, and the time changed processes are given by additive functional associated with some underlying measure. Concrete expressions of the Dirichlet forms corresponding to time changed processes are also obtained, which may be of non-local type caused by degeneracy of the underlying measures.

Limit theorems for projections of random walk on a hypersphere

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein–Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a “functional” generalisation of Poincaré’s observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli–Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.

Space-time transformations in radial path integrals

Nonlinear space-time transformations in the radial path integral are discussed. A transformation formula is derived, which relates the original path integral to the Green function of a new quantum system with an effective potential containing an observable quantum correction ≈ħ 2. As an example the formula is applied to spherical brownian motion.

The geometry of random drift I. Stochastic distance and diffusion

A stochastic distance measure is defined for a general diffusion process on a parameter space X. This distance is defined by where (gij ) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij ) is the fundamental tensor of the Riemannian space (X, gij ), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift with n alleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher to n alleles. The geometry associated with the diffusion equation for random drift with n alleles is that of a part of an (n − 1)-sphere of radius two. We also show that the diffusion equation for random drift is not spherical Brownian motion, although it is approximated by it near the centroid of frequency space.