Probability Density of Random Walks on Random Fractals: Stretched Gaussians and Multifractal Features

1990 ◽  
pp. 305-307
Author(s):  
H. Eduardo Roman ◽  
Armin Bunde ◽  
Shlomo Havlin
Keyword(s):  
Random Walks ◽  
Probability Density ◽  
Random Fractals

Biased movement at a boundary and conditional occupancy times for diffusion processes

2003 ◽  
Vol 40 (3) ◽  
pp. 557-580 ◽  
Author(s):  
Otso Ovaskainen ◽  
Stephen J. Cornell
Keyword(s):  
Random Walks ◽  
Probability Density ◽  
Diffusion Processes ◽  
Diffusion Problem ◽  
Movement Behaviour ◽  
Habitat Patches ◽  
Hitting Probabilities ◽  
Interior Boundary ◽  
Biological Organisms

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

Biased movement at a boundary and conditional occupancy times for diffusion processes

2003 ◽  
Vol 40 (03) ◽  
pp. 557-580 ◽  
Author(s):  
Otso Ovaskainen ◽  
Stephen J. Cornell
Keyword(s):  
Random Walks ◽  
Probability Density ◽  
Diffusion Processes ◽  
Diffusion Problem ◽  
Movement Behaviour ◽  
Habitat Patches ◽  
Hitting Probabilities ◽  
Interior Boundary ◽  
Biological Organisms

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

A CLOSED FORM FOR THE DENSITY FUNCTIONS OF RANDOM WALKS IN ODD DIMENSIONS

2015 ◽  
Vol 93 (2) ◽  
pp. 330-339 ◽  
Author(s):  
JONATHAN M. BORWEIN ◽  
CORWIN W. SINNAMON
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Random Walks ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Dimensional Space ◽  
Density Functions ◽  
Piecewise Polynomial ◽  
Odd Dimensions

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

Diffusion on Self-Similar Structures

Fractals ◽  
1997 ◽  
Vol 05 (03) ◽  
pp. 379-393 ◽  
Author(s):  
H. Eduardo Roman
Keyword(s):  
Random Walks ◽  
Probability Density ◽  
Analytical Solutions ◽  
Asymptotic Form ◽  
Theoretical Framework ◽  
Percolation Clusters ◽  
Self Similar

Diffusion on self-similar structures is reviewed within a unified theoretical framework. Much attention is devoted to the asymptotic form of the probability density of random walks on fractals, for which analytical solutions are discussed. New predictions for the structure of percolation clusters at criticality are presented.

scholarly journals From diffusion in compartmentalized media to non-Gaussian random walks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jakub Ślęzak ◽  
Stanislav Burov
Keyword(s):  
Random Walks ◽  
Probability Density ◽  
Exponential Decay ◽  
Diffusion Constant ◽  
Mean Square Displacement ◽  
Cellular Membrane ◽  
Long Time ◽  
The Media ◽  
The One ◽  
Non Gaussian

AbstractIn this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

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