scholarly journals Generating stochastic trajectories with global dynamical constraints

2021 ◽  
Vol 2021 (12) ◽  
pp. 123204
Author(s):  
Benjamin De Bruyne ◽  
Satya N Majumdar ◽  
Henri Orland ◽  
Grégory Schehr
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Langevin Equation ◽  
Occupation Time ◽  
Future Time ◽  
Lévy Flights ◽  
Effective Force ◽  
Fixed Area ◽  
Dynamical Constraints ◽  
Positive Axis

Abstract We propose a method to exactly generate Brownian paths x c (t) that are constrained to return to the origin at some future time t f , with a given fixed area A f = ∫ 0 t f d t x c ( t ) under their trajectory. We derive an exact effective Langevin equation with an effective force that accounts for the constraint. In addition, we develop the corresponding approach for discrete-time random walks, with arbitrary jump distributions including Lévy flights, for which we obtain an effective jump distribution that encodes the constraint. Finally, we generalise our method to other types of dynamical constraints such as a fixed occupation time on the positive axis T f = ∫ 0 t f d t Θ x c ( t ) or a fixed generalised quadratic area A f = ∫ 0 t f d t x c 2 ( t ) .

scholarly journals Generating discrete-time constrained random walks and Lévy flights

2021 ◽  
Vol 104 (2) ◽  
Author(s):  
Benjamin De Bruyne ◽  
Satya N. Majumdar ◽  
Grégory Schehr
Keyword(s):  
Random Walks ◽  
Discrete Time ◽  
Lévy Flights ◽  
Levy Flights

scholarly journals Survival probability of random walks and Lévy flights on a semi-infinite line

2017 ◽  
Vol 50 (46) ◽  
pp. 465002 ◽  
Author(s):  
Satya N Majumdar ◽  
Philippe Mounaix ◽  
Grégory Schehr
Keyword(s):  
Random Walks ◽  
Survival Probability ◽  
Lévy Flights ◽  
Levy Flights ◽  
Infinite Line

scholarly journals A limit theorem for a splitting distribution of a quantum walk

2018 ◽  
Vol 16 (03) ◽  
pp. 1850023
Author(s):  
Takuya Machida
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Random Walks ◽  
Discrete Time ◽  
Limit Distribution ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Unitary Evolution ◽  
Time Quantum ◽  
Long Time

Discrete-time quantum walks are considered a counterpart of random walks and their study has been getting attention since around 2000. In this paper, we focus on a quantum walk which generates a probability distribution splitting to two parts. The quantum walker with two coin states spreads at points, represented by integers, and we analyze the chance of finding the walker at each position after it carries out a unitary evolution a lot of times. The result is reported as a long-time limit distribution from which one can see an approximation to the finding probability.

Simple random walk statistics. Part I: Discrete time results

1996 ◽  
Vol 33 (2) ◽  
pp. 311-330 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Order Statistics ◽  
Discrete Time ◽  
Rank Order ◽  
Simple Random Walk ◽  
Alternative Method ◽  
Starting Point ◽  
Famous Paper ◽  
Rank Order Statistics

In a famous paper, Dwass (1967) proposed a method to deal with rank order statistics, which constitutes a unifying framework to derive various distributional results. In the present paper an alternative method is presented, which allows us to extend Dwass's results in several ways, namely arbitrary endpoints, horizontal steps and arbitrary probabilities for the three step types. Regarding these extensions the pertaining rank order statistics are extended as well to simple random walk statistics. This method has proved appropriate to generalize all results given by Dwass. Moreover, these discrete time results can be taken as a starting point to derive the corresponding results for randomized random walks by means of a limiting process.

On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

1998 ◽  
Vol 7 (4) ◽  
pp. 397-401 ◽  
Author(s):  
OLLE HÄGGSTRÖM
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Complete Graph ◽  
Continuous Time ◽  
Product Graph ◽  
Product Graphs ◽  
Continuous Time Random Walks ◽  
Discrete Time Case

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

scholarly journals The connection between evolution algebras, random walks and graphs

2019 ◽  
Vol 19 (02) ◽  
pp. 2050023 ◽  
Author(s):  
Paula Cadavid ◽  
Mary Luz Rodiño Montoya ◽  
Pablo M. Rodriguez
Keyword(s):  
Probability Theory ◽  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Associative Algebras ◽  
Biological Phenomena ◽  
Evolution Algebra ◽  
New Type ◽  
Markov Evolution

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper, we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph, we believe that our results may add a new landscape in the study of Markov evolution algebras.

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