A Symbolic Dynamics Approach to Random Walk on Koch Fractal

2014 ◽  
Vol 610 ◽  
pp. 17-22
Author(s):  
Hong Luo ◽  
Ying Tan ◽  
Shou Li Peng
Keyword(s):  
Random Walk ◽  
Hausdorff Measure ◽  
Symbolic Sequence ◽  
Koch Curve ◽  
One Dimensional ◽  
Underlying Space ◽  
Chemical Distance ◽  
Koch Fractal ◽  
The One ◽  
Partition Of Integer

The paper presents a new symbolic dynamic approach to the research of the random walk andBrownianmotion(BM)on Koch fractal. From the symbolic sequence of Koch automaton, on the one hand, we obtained the geometric description of the Koch curve completely, and constructed the state space of the random walk with the symbolic sequence. And the precise arithmetic representation of Koch curve is provided by the deterministicRademachersequence. On the other hand, the arithmetic feature of the Koch automaton, the position numbers, forms a partition of integer , which is naturally a one-dimensional lattice, it will be underlying space of theBMdirectly. When the chemical distance is introduced to measure the distance between two states, analytic results of the model for random walk on Koch fractal are obtained, particularly the relation between the chemical distance and the Hausdorff measure is discussed, and the Wiener Process in terms of Hausdorff measure is constructed parallel.

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

scholarly journals Zero-one-only process: A correlated random walk with a stochastic ratchet

2014 ◽  
Vol 28 (29) ◽  
pp. 1450201
Author(s):  
Seung Ki Baek ◽  
Hawoong Jeong ◽  
Seung-Woo Son ◽  
Beom Jun Kim
Keyword(s):  
Transport Phenomenon ◽  
Random Walk ◽  
Random Walks ◽  
Stochastic Processes ◽  
Function Method ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Persistent Random Walk ◽  
The One ◽  
Previous Step

The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (03) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

Phase transition in one-dimensional random walk with partially reflecting boundaries

1985 ◽  
Vol 17 (3) ◽  
pp. 594-606 ◽  
Author(s):  
Ora E. Percus
Keyword(s):  
Random Walk ◽  
Transition Probability ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Boundary Case ◽  
Reflecting Boundaries ◽  
The Mean ◽  
The One ◽  
The Relationship ◽  
Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

scholarly journals Improved Bounds for Restricted Families of Projections to Planes in ℝ3

2018 ◽  
Vol 2020 (19) ◽  
pp. 5797-5813 ◽  
Author(s):  
Tuomas Orponen ◽  
Laura Venieri
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Unit Sphere ◽  
Orthogonal Projection ◽  
Partial Result ◽  
Borel Set ◽  
Borel Sets ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
The One

Abstract For $e \in S^{2}$, the unit sphere in $\mathbb{R}^3$, let $\pi _{e}$ be the orthogonal projection to $e^{\perp } \subset \mathbb{R}^{3}$, and let $W \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K \subset \mathbb{R}^{3}$ is a Borel set with $\dim _{\textrm{H}} K \leq \tfrac{3}{2}$, then $\dim _{\textrm{H}} \pi _{e}(K) = \dim _{\textrm{H}} K$ for $\mathcal{H}^{1}$ almost every $e \in S^{2} \cap W$, where $\mathcal{H}^{1}$ denotes the one-dimensional Hausdorff measure and $\dim _{\textrm{H}}$ the Hausdorff dimension. This was known earlier, due to Järvenpää, Järvenpää, Ledrappier, and Leikas, for Borel sets $K$ with $\dim _{\textrm{H}} K \leq 1$. We also prove a partial result for sets with dimension exceeding $3/2$, improving earlier bounds by D. Oberlin and R. Oberlin.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (01) ◽  
pp. 169-175
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b &gt; 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 &lt;m &lt;a) under different conditions.

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