scholarly journals Heat Trace Asymptotics of Subordinate Brownian Motion in Euclidean Space

2015 ◽  
Vol 44 (2) ◽  
pp. 331-354 ◽  
Author(s):  
M. A. Fahrenwaldt
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Heat Trace ◽  
Subordinate Brownian Motion

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals “Normal” Orientation Distributions

1992 ◽  
Vol 19 (4) ◽  
pp. 197-202 ◽  
Author(s):  
H. Schaeben
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Euclidean Space ◽  
Normal Distribution ◽  
The Central Limit Theorem ◽  
Bingham Distribution ◽  
Normal Orientation ◽  
Orientation Distributions

Analogues of the normal distribution in Euclidean space for orientations represented by Rodrigues parameters are discussed. It is emphasized that different characterizations of the normal distribution in Euclidean space lead to different distributions in other spaces, none of which is mathematically superior to any other one. Particular analogues of the normal distribution are the Bingham distribution on S+4 for the purposes of mathematical statistics, and the Brownian motion distribution on S+4 in terms of probability theory and stochastic processes. It is reminded of the fact that a simple analogue of the central limit theorem in Euclidean space does not exist for the hyperspheres SP and projective hyperplanes HP−1=S+4.

scholarly journals On the Asymptotic Behaviour of a Diffusion Process with Singular Drift

1975 ◽  
Vol 57 ◽  
pp. 87-106
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Smooth Boundary ◽  
Dimensional Euclidean Space ◽  
Singular Drift ◽  
Strong Markov

We discuss some peculiar features of the diffusion process whose characterization is given below. Let D be a bounded domain in the d-dimensional Euclidean space Ed with a smooth boundary ∂D. The domain D contains open balls (i = 1, · · ·, n) which are mutually disjoint. Our process is a diffusion process on the state space D ∪ ∂D which is locally equivalent to the Brownian motion except on the spheres ∂ and the boundary ∂D. By a diffusion process we mean a continuous strong Markov process. As to the terminology about Markov processes we refer to [2].

Killed Brownian motion and the Brunn–Minkowski inequalities

2012 ◽  
Vol 153 (1) ◽  
pp. 111-121
Author(s):  
HUILING LE
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Poisson Equation ◽  
Harmonic Functions ◽  
Convex Sets ◽  
Compact Convex ◽  
Compact Sets ◽  
Brownian Motions ◽  
The Poisson Equation ◽  
Killed Brownian Motions

AbstractWe construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

scholarly journals PATH INTEGRAL REPRESENTATION FOR SCHRÖDINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

2012 ◽  
Vol 24 (06) ◽  
pp. 1250013 ◽  
Author(s):  
FUMIO HIROSHIMA ◽  
TAKASHI ICHINOSE ◽  
JÓZSEF LŐRINCZI
Keyword(s):  
Brownian Motion ◽  
Path Integral ◽  
Schrödinger Operators ◽  
Integral Representations ◽  
Schrodinger Operators ◽  
Kato Class ◽  
Path Integral Representation ◽  
Bernstein Functions ◽  
Subordinate Brownian Motion ◽  
The One

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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