scholarly journals FROM PERSISTENT RANDOM WALK TO THE TELEGRAPH NOISE

2010 ◽  
Vol 10 (02) ◽  
pp. 161-196 ◽  
Author(s):  
S. HERRMANN ◽  
P. VALLOIS
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Random Walks ◽  
Poisson Process ◽  
Counting Process ◽  
Telegraph Equation ◽  
Space Time ◽  
Persistent Random Walks ◽  
Persistent Random Walk ◽  
Telegraph Noise

We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process (Zt) which can be easily expressed in terms of a counting process (Nt). In a particular case the counting process is a Poisson process, and (Zt) permits to represent the solution of the telegraph equation. We study in detail the Markov process ((Zt, Nt); t ≥ 0).

scholarly journals Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

2020 ◽  
Vol 4 (4) ◽  
pp. 51 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Federico Polito ◽  
Alejandro P. Riascos
Keyword(s):  
Cauchy Problem ◽  
Random Walks ◽  
Poisson Process ◽  
Continuous Time ◽  
Laplacian Matrix ◽  
Space Time ◽  
The State ◽  
Diffusion Limit ◽  
Matrix Functions ◽  
Continuous Time Random Walks

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

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.

The time of completion of a linear birth-growth model

2000 ◽  
Vol 32 (3) ◽  
pp. 620-627 ◽  
Author(s):  
S. N. Chiu ◽  
C. C. Yin
Keyword(s):  
Markov Process ◽  
Poisson Process ◽  
Growth Model ◽  
Completion Time ◽  
Space Time ◽  
Constant Speed ◽  
Strong Limit ◽  
Limiting Distributions ◽  
Bounded Interval ◽  
Length Of Interval

Consider the following birth-growth model in ℝ. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.

scholarly journals Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

2020 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Federico Polito ◽  
Alejandro P. Riascos
Keyword(s):  
Cauchy Problem ◽  
Random Walks ◽  
Poisson Process ◽  
Continuous Time ◽  
Laplacian Matrix ◽  
Space Time ◽  
State Density ◽  
Diffusion Limit ◽  
Matrix Functions ◽  
Continuous Time Random Walks

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

The time of completion of a linear birth-growth model

2000 ◽  
Vol 32 (03) ◽  
pp. 620-627
Author(s):  
S. N. Chiu ◽  
C. C. Yin
Keyword(s):  
Markov Process ◽  
Poisson Process ◽  
Growth Model ◽  
Completion Time ◽  
Space Time ◽  
Constant Speed ◽  
Strong Limit ◽  
Limiting Distributions ◽  
Bounded Interval ◽  
Length Of Interval

Consider the following birth-growth model in ℝ. Seeds are born randomly according to an inhomogeneous space-time Poisson process. A newly formed point immediately initiates a bi-directional coverage by sending out a growing branch. Each frontier of a branch moves at a constant speed until it meets an opposing one. New seeds continue to form on the uncovered parts on the line. We are interested in the time until a bounded interval is completely covered. The exact and limiting distributions as the length of interval tends to infinity are obtained for this completion time by considering a related Markov process. Moreover, some strong limit results are also established.

scholarly journals Persistent random walks may have arbitrarily large tails

1989 ◽  
Vol 21 (1) ◽  
pp. 229-230 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Size Distribution ◽  
Symmetric Random Walk ◽  
Jump Size ◽  
Persistent Random Walks ◽  
Probabilistic Proof

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

scholarly journals Persistent random walks may have arbitrarily large tails

1989 ◽  
Vol 21 (01) ◽  
pp. 229-230
Author(s):  
D. R. Grey
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Size Distribution ◽  
Symmetric Random Walk ◽  
Jump Size ◽  
Persistent Random Walks ◽  
Probabilistic Proof

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

Favorite sites of a persistent random walk

2021 ◽  
Vol 501 (2) ◽  
pp. 125180
Author(s):  
Arka Ghosh ◽  
Steven Noren ◽  
Alexander Roitershtein
Keyword(s):  
Random Walk ◽  
Persistent Random Walk

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

scholarly journals Current Trends in Random Walks on Random Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1148
Author(s):  
Jewgeni H. Dshalalow ◽  
Ryan T. White
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Real Time ◽  
Real Space ◽  
Historical Background ◽  
Escape Time ◽  
The Real ◽  
Current Trends ◽  
One Step ◽  
Time Position

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

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