A Wiener Process with Reflection and Harmonic Functions with Finite Energy Integral

1989 ◽  
Vol 33 (3) ◽  
pp. 547-550
Author(s):  
A. D. Bendikov ◽  
I. V. Pavlov
Keyword(s):  
Wiener Process ◽  
Harmonic Functions ◽  
Finite Energy ◽  
Energy Integral

scholarly journals Infinite weighted graphs with bounded resistance metric

2018 ◽  
Vol 123 (1) ◽  
pp. 5-38
Author(s):  
Palle Jorgensen ◽  
Feng Tian
Keyword(s):  
Harmonic Functions ◽  
Voltage Drop ◽  
Interpolation Formula ◽  
Finite Energy ◽  
Boundary Representation ◽  
Special Importance ◽  
Weighted Graphs ◽  
Resistance Distance ◽  
Lipschitz Continuous ◽  
Vertex Set

We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countably infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on $V$, voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on $G$ of finite energy.We compute a resistance metric $d$ from a given conductance function. (The resistance distance $d(x,y)$ between two vertices $x$ and $y$ is the voltage drop from $x$ to $y$, which is induced by the given assignment of resistors when $1$ amp is inserted at the vertex $x$, and then extracted again at $y$.)We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of ${1}/{2}$-Lipschitz-continuous and bounded functions on $V$, relative to the metric $d$. We further show that, in this case, the metric completion $M$ of $(V,d)$ is automatically compact, and that the vertex-set $V$ is open in $M$. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on $V$ of finite energy. We further compare $M$ to other compactifications; e.g., to certain path-space models.

Tracking a wiener process with a process of finite energy

1989 ◽  
Vol 5 (2) ◽  
pp. 327-334
Author(s):  
Peter G. Doyle ◽  
Larry A. Shepp
Keyword(s):  
Wiener Process ◽  
Finite Energy

Reflected Dirichlet Spaces

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Dirichlet Form ◽  
Harmonic Functions ◽  
Dirichlet Space ◽  
Random Variables ◽  
Finite Energy ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Hunt Process ◽  
Equivalent Definition ◽  
Definition Of

This chapter turns to reflected Dirichlet spaces. It first introduces the notion of terminal random variables and harmonic functions of finite energy for a Hunt process associated with a transient regular Dirichlet form. The chapter next establishes several equivalent notions of reflected Dirichlet space (ℰ ref,ℱ ref) for a regular transient Dirichlet form (E,F). One of these equivalent notions is then used to define reflected Dirichlet space for a regular recurrent Dirichlet form. Moreover, the chapter gives yet another equivalent definition of reflected Dirichlet space that is invariant under quasi-homeomorphism of Dirichlet forms. Various concrete examples of reflected Dirichlet spaces are also exhibited for regular Dirichlet forms. Finally, the chapter defines a Silverstein extension of a quasi-regular Dirichlet form (E,F) on L²(E; m) and investigates the equivalence of analytic and probabilistic concepts of harmonicity.

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