A dirichlet form on the wiener space and properties on Brownian motion

1984 ◽  
pp. 290-300 ◽  
Author(s):  
Masatoshi Fukushima
Keyword(s):  
Brownian Motion ◽  
Dirichlet Form ◽  
Wiener Space

On homogeneous chaos

1991 ◽  
Vol 110 (2) ◽  
pp. 353-363 ◽  
Author(s):  
Nigel Cutland ◽  
Siu-Ah Ng
Keyword(s):  
Brownian Motion ◽  
Wiener Space ◽  
Chaos Decomposition ◽  
Image Position

AbstractThis paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decompositionFirst these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.

scholarly journals Resistance scaling on 4N-carpets

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Claire Canner ◽  
Christopher Hayes ◽  
Robin Huang ◽  
Michael Orwin ◽  
Luke G. Rogers
Keyword(s):  
Brownian Motion ◽  
Harmonic Function ◽  
Boundary Value Problem ◽  
Dirichlet Form ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Scaling Properties ◽  
Constant Potential ◽  
Self Similar ◽  
Group Of Symmetries

Abstract The 4 ⁢ N {4N} -carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a 4 ⁢ N {4N} -carpet F, let { F n } n ≥ 0 {\{F_{n}\}_{n\geq 0}} be the natural decreasing sequence of compact pre-fractal approximations with ⋂ n F n = F {\bigcap_{n}F_{n}=F} . On each F n {F_{n}} , let ℰ ⁢ ( u , v ) = ∫ F N ∇ ⁡ u ⋅ ∇ ⁡ v ⁢ d ⁢ x {\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx} be the classical Dirichlet form and u n {u_{n}} be the unique harmonic function on F n {F_{n}} satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. Lond. Ser. A 431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is ρ = ρ ⁢ ( N ) > 1 {\rho=\rho(N)>1} such that ℰ ⁢ ( u n , u n ) ⁢ ρ n {\mathcal{E}(u_{n},u_{n})\rho^{n}} is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.

Convergence of Brownian motion with a scaled Dirac delta potential

2012 ◽  
Vol 55 (2) ◽  
pp. 403-427 ◽  
Author(s):  
Florian Conrad ◽  
Martin Grothaus ◽  
Janna Lierl ◽  
Olaf Wittich
Keyword(s):  
Brownian Motion ◽  
Dirichlet Form ◽  
Scaling Limits ◽  
Brownian Motions ◽  
Dirac Delta ◽  
Infinite Dimensional ◽  
Delta Potential ◽  
Dirac Delta Potential

AbstractThe method of deriving scaling limits using Dirichlet-form techniques has already been successfully applied to a number of infinite-dimensional problems. However, extracting the key tools from these papers is a rather difficult task for non-experts. This paper meets the need for a simple presentation of the method by applying it to a basic example, namely the convergence of Brownian motions with potentials given by n multiplied by the Dirac delta at 0 to Brownian motion with absorption at 0.

scholarly journals On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin

2019 ◽  
Vol 62 (8) ◽  
pp. 1477-1492
Author(s):  
Patrick J. Fitzsimmons ◽  
Liping Li
Keyword(s):  
Brownian Motion ◽  
Dirichlet Form ◽  
Three Dimensional

scholarly journals On the heat kernel and the Dirichlet form of Liouville Brownian motion

2014 ◽  
Vol 19 (0) ◽  
Author(s):  
Rémi Rhodes ◽  
Christophe Garban ◽  
Vincent Vargas
Keyword(s):  
Brownian Motion ◽  
Heat Kernel ◽  
Dirichlet Form

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.

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