scholarly journals Brownian motion on stationary random manifolds

2016 ◽  
Vol 16 (02) ◽  
pp. 1660001
Author(s):  
Pablo Lessa
Keyword(s):  
Brownian Motion ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Volume Growth ◽  
Upper And Lower Bounds ◽  
Entropy Theory ◽  
Liouville Property ◽  
Average Volume ◽  
Infinite Dimensional ◽  
Bounded Harmonic Functions

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold satisfies the Liouville property almost surely, and is positive if and only if it admits an infinite dimensional space of bounded harmonic functions almost surely. Upper and lower bounds for the entropy are provided in terms of the linear drift of Brownian motion and average volume growth of the manifold. Other almost sure properties of these random manifolds are also studied.

scholarly journals Lévy’s functional analysis in terms of an infinite dimensional Brownian motion III

1983 ◽  
Vol 90 ◽  
pp. 155-173 ◽  
Author(s):  
Yoshihei Hasegawa
Keyword(s):  
Functional Analysis ◽  
Brownian Motion ◽  
Minimal Surfaces ◽  
Quadratic Forms ◽  
Harmonic Functions ◽  
Dimensional Space ◽  
Probabilistic Methods ◽  
Infinite Dimensional Space ◽  
Infinite Dimensional

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

scholarly journals Folding Stabilizes the Evolution of Catalysts

2004 ◽  
Vol 10 (1) ◽  
pp. 23-38 ◽  
Author(s):  
Stephan Altmeyer ◽  
Rudolf M. Füchslin ◽  
John S. McCaskill
Keyword(s):  
Dimensional Space ◽  
Spatial Dynamics ◽  
Heterogeneous Systems ◽  
Evolutionary Model ◽  
Complete Sequence ◽  
Infinite Dimensional ◽  
Spatially Resolved ◽  
Catalytic Function ◽  
Sequence Recognition ◽  
Intermolecular Recognition

Sequence folding is known to determine the spatial structure and catalytic function of proteins and nucleic acids. We show here that folding also plays a key role in enhancing the evolutionary stability of the intermolecular recognition necessary for the prevalent mode of catalytic action in replication, namely, in trans, one molecule catalyzing the replication of another copy, rather than itself. This points to a novel aspect of why molecular life is structured as it is, in the context of life as it could be: folding allows limited, structurally localized recognition to be strongly sensitive to global sequence changes, facilitating the evolution of cooperative interactions. RNA secondary structure folding, for example is shown to be able to stabilize the evolution of prolonged functional sequences, using only a part of this length extension for intermolecular recognition, beyond the limits of the (cooperative) error threshold. Such folding could facilitate the evolution of polymerases in spatially heterogeneous systems. This facilitation is, in fact, vital because physical limitations prevent complete sequence-dependent discrimination for any significant-size biopolymer substrate. The influence of partial sequence recognition between biopolymer catalysts and complex substrates is investigated within a stochastic, spatially resolved evolutionary model of trans catalysis. We use an analytically tractable nonlinear master equation formulation called PRESS (McCaskill et al., Biol. Chem. 382: 1343–1363), which makes use of an extrapolation of the spatial dynamics down from infinite dimensional space, and compare the results with Monte Carlo simulations.

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