The Iwasawa decomposition and the limiting behaviour of Brownian motion on a symmetric space of noncompact type

1988 ◽  
pp. 303-332 ◽  
Author(s):  
J. C. Taylor
Keyword(s):  
Brownian Motion ◽  
Symmetric Space ◽  
Iwasawa Decomposition ◽  
Noncompact Type ◽  
Limiting Behaviour

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

Brownian Motion on a Symmetric Space of Non-Compact Type: Asymptotic Behaviour in Polar Coordinates

1991 ◽  
Vol 43 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
J. C. Taylor
Keyword(s):  
Brownian Motion ◽  
Symmetric Space ◽  
Asymptotic Behaviour ◽  
Polar Coordinates ◽  
Skew Product ◽  
Compact Type ◽  
Product Representation

AbstractThe results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.

On an Upper Bound for the Heat Kernel on SU*(2n)/ Sp(n)

1994 ◽  
Vol 37 (3) ◽  
pp. 408-418 ◽  
Author(s):  
P. Sawyer
Keyword(s):  
Symmetric Space ◽  
Heat Kernel ◽  
Upper Bound ◽  
Symmetric Spaces ◽  
Legendre Function ◽  
Noncompact Type ◽  
Image Position

AbstractJean-Philippe Anker made an interesting conjecture in [2] about the growth of the heat kernel on symmetric spaces of noncompact type. For any symmetric space of noncompact type, we can writewhere ϕ0 is the Legendre function and q, "the dimension at infinity", is chosen such that limt—>∞Vt(x) = 1 for all x. Anker's conjecture can be stated as follows: there exists a constant C > 0 such thatwhere is the set of positive indivisible roots. The behaviour of the function ϕ0 is well known (see [1]).The main goal of this paper is to establish the conjecture for the spaces SU*(2n)/ Sp(n).

LIMITING PROPERTIES OF LÉVY PROCESSES IN SYMMETRIC SPACES OF NONCOMPACT TYPE

2012 ◽  
Vol 12 (04) ◽  
pp. 1250001 ◽  
Author(s):  
MING LIAO ◽  
LONGMIN WANG
Keyword(s):  
Symmetric Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Symmetric Spaces ◽  
Large Time ◽  
Levy Processes ◽  
Levy Process ◽  
Noncompact Type

We study the large time limiting properties of a Lévy process in a symmetric space of noncompact type, both pathwise and in terms of distribution.

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