scholarly journals Construction of Liouville Brownian motion via Dirichlet form theory

2019 ◽  
Vol 148 ◽  
pp. 128-132 ◽  
Author(s):  
Jiyong Shin
Keyword(s):  
Brownian Motion ◽  
Dirichlet Form ◽  
Form Theory

Dirichlet Form Theory and its Applications

2014 ◽  
Vol 11 (4) ◽  
pp. 2667-2756
Author(s):  
Sergio Albeverio ◽  
Zhen-Qing Chen ◽  
Masatoshi Fukushima ◽  
Michael Röckner
Keyword(s):  
Dirichlet Form ◽  
Form Theory

A REMARK ON THE GENERATOR OF A RIGHT-CONTINUOUS MARKOV PROCESS

2007 ◽  
Vol 10 (04) ◽  
pp. 633-640 ◽  
Author(s):  
MICHAEL RÖCKNER ◽  
GERALD TRUTNAU
Keyword(s):  
Markov Process ◽  
Dirichlet Form ◽  
Borel Measure ◽  
Initial Condition ◽  
Full Support ◽  
Finite Borel Measure ◽  
Continuous Markov Process ◽  
Form Theory ◽  
Densely Defined ◽  
Generalized Dirichlet

Given a right-continuous Markov process (Xt)t ≥ 0 on a second countable metrizable space E with transition semigroup (pt)t ≥ 0, we prove that there exists a σ-finite Borel measure μ with full support on E, and a closed and densely defined linear operator [Formula: see text] generating (pt)t ≥ 0 on Lp (E; μ). In particular, we solve the corresponding Cauchy problem in Lp (E; μ) for any initial condition [Formula: see text]. Furthermore, for any real β > 0 we show that there exists a generalized Dirichlet form which is associated to (e-βt pt)t ≥ 0. If the β-subprocess of (Xt)t ≥ 0 corresponding to (e-βt pt)t ≥ 0, β > 0, is μ-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for [Formula: see text]. If (Xt)t ≥ 0 is transient, then β can be chosen to be zero.

scholarly journals BV functions in a Gelfand triple for differentiable measure and its applications

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Michael Röckner ◽  
Rongchan Zhu ◽  
Xiangchan Zhu
Keyword(s):  
Dirichlet Form ◽  
Adjoint Operator ◽  
Stochastic Quantization ◽  
Bv Functions ◽  
Form Theory ◽  
Differentiable Measure ◽  
Definition Of ◽  
Non Gaussian ◽  
Gelfand Triple ◽  
Quantization Problem

AbstractIn this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [Ann. Probab. 40 (2012), 1759–1794], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator

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.

scholarly journals A localization formula in Dirichlet form theory

2012 ◽  
Vol 140 (5) ◽  
pp. 1815-1822
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Localization Formula ◽  
Form Theory

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 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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