scholarly journals Brownian motion on a surface of negative curvature

1984 ◽  
pp. 70-76
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Brownian Motion ◽  
Negative Curvature

scholarly journals A diffusion model for Bookstein triangle shape

1996 ◽  
Vol 28 (02) ◽  
pp. 334-335
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Hyperbolic Plane ◽  
Negative Curvature ◽  
General Linear Group ◽  
Shape Space ◽  
Shape Theory ◽  
Independent Brownian Motion ◽  
Invertible Matrices

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3 } is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi ; determines a triple {X1 X2, X3 } whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

scholarly journals Brownian Motion and Negative Curvature

2011 ◽  
pp. 143-161 ◽  
Author(s):  
Marc Arnaudon ◽  
Anton Thalmaier
Keyword(s):  
Brownian Motion ◽  
Negative Curvature

scholarly journals A diffusion model for Bookstein triangle shape

1996 ◽  
Vol 28 (2) ◽  
pp. 334-335
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Hyperbolic Plane ◽  
Negative Curvature ◽  
General Linear Group ◽  
Shape Space ◽  
Shape Theory ◽  
Independent Brownian Motion ◽  
Invertible Matrices

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3} is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi; determines a triple {X1 X2, X3} whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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