A Diffusion on a Fractal State Space

Abstract

A fractal is defined in the plane known as the Vicsek Snowflake by constructing a skeletal lattice graph and then rescaling spatial dimensions to give a sequence of lattices that converges to a fractal. By defining a simple random walk on the skeletal lattice and then rescaling both time and space, we define a sequence of random walks on the approximating lattices that converge weakly to a limiting process on the snowflake. This limit has continuous sample paths and the strong Markov property, and that it is the unique diffusion limit of random walk on the snowflake in a natural sense. This diffusion has a scaling property reminiscent of Brownian motion, and we introduce a coupling argument to show that the diffusion has transition densities with respect to Hausdorff measure on the snowflake. Keywords: Diffusion; Fractals.

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Document Details

Document Type
Technical Report
Publication Date
Sep 01, 1989
Accession Number
ADA213555

Entities

People

  • William B. Krebs

Organizations

  • Florida State University

Tags

DTIC Thesaurus Topics

  • Brownian Motion
  • Classification
  • Convergence
  • Equations
  • Markov Chains
  • Markov Processes
  • Mathematics
  • Probability
  • Random Variables
  • Random Walk
  • Scaling Laws
  • Security
  • Sequences
  • Statistics
  • Stochastic Processes
  • Transitions
  • Weak Convergence

Fields of Study

  • Mathematics

Readers

  • Aerospace Propulsion Engineering.
  • Calculus or Mathematical Analysis
  • Statistical inference.

Technology Areas

  • Space
  • Space - Space Objects