Brownian motion, negative curvature, and harmonic maps

1981 ◽  
pp. 479-491 ◽  
Author(s):  
W. S. Kendall
Keyword(s):  
Brownian Motion ◽  
Negative Curvature ◽  
Harmonic Maps

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.

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