Stochastic motions on the 3-sphere governed by wave and heat equations

1987 ◽  
Vol 24 (2) ◽  
pp. 315-327 ◽  
Author(s):  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Maxwell Equations ◽  
Telegraph Equation ◽  
Random Motion ◽  
Heat Equations ◽  
Spherical Coordinates ◽  
Sample Paths

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

Stochastic motions on the 3-sphere governed by wave and heat equations

1987 ◽  
Vol 24 (02) ◽  
pp. 315-327
Author(s):  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Maxwell Equations ◽  
Telegraph Equation ◽  
Random Motion ◽  
Heat Equations ◽  
Spherical Coordinates ◽  
Sample Paths

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

scholarly journals TFR Predictions Based on Brownian Motion Theory

2002 ◽  
pp. 207-219
Author(s):  
Keilman Nico
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Random Walk ◽  
Predictive Distribution ◽  
Major Effect ◽  
Common Time ◽  
Population Forecasts ◽  
Sample Paths ◽  
Long Run ◽  
Future Fertility

In stochastic population forecasts, the predictive distribution of the TFR is of centralconcern. Common time series models can be used to predict the TFR and itsmoments on the short run (up to 10 or 20 years), but on the long run (40-50 years)they result in excessively wide prediction intervals. The aim of this study is toanalyse and apply a time series model for the TFR, which restricts the predictedvalues to a certain pre-specified interval.I will model the time series of log TFR-values as a Brownian motion with absorbingupper barrier. I will give and analyse expressions for the predictive distribution of the log of the TFR assuming itfollows a Brownian motion with absorbing ceiling; expressions for the first and second moments of the predictive distribution ofthe log of the TFR.When the log of the TFR follows a random walk with absorbing ceiling, I find thatthe second moment of the predictive distribution for the long-run TFR in Norwayis insensitive for ceiling levels beyond a threshold of approximately 3.4 childrenper woman. This conclusion holds for a fairly broad range of innovation variances.If the log of the TFR follows a random walk, sample paths that exceed approximately3.4 children per woman may be rejected when simulating future fertility in Westerncountries. This will not have any major effect on the width of the long-termpredictive distribution.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Self-avoiding random walk: A Brownian motion model with local time drift

1987 ◽  
Vol 74 (2) ◽  
pp. 271-287 ◽  
Author(s):  
J. R. Norris ◽  
L. C. G. Rogers ◽  
David Williams
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Local Time ◽  
Motion Model ◽  
Time Drift ◽  
Brownian Motion Model

scholarly journals Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk

2001 ◽  
Vol 186 (2) ◽  
pp. 239-270 ◽  
Author(s):  
Amir Dembo ◽  
Yuval Peres ◽  
Jay Rosen ◽  
Ofer Zeitouni
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Planar Brownian Motion ◽  
Thick Points

scholarly journals A four-dimensional random motion at finite speed

2006 ◽  
Vol 43 (4) ◽  
pp. 1107-1118 ◽  
Author(s):  
Alexander D. Kolesnik
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Transition Density ◽  
Random Motion ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Continuous Component ◽  
Absolutely Continuous ◽  
Finite Speed ◽  
Uniform Law

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

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