scholarly journals The flashing Brownian ratchet and Parrondo’s paradox

2018 ◽  
Vol 5 (1) ◽  
pp. 171685 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Diffusion Process ◽  
Directed Motion ◽  
Time And Space ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Games Of Chance ◽  
Parrondo’S Paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

scholarly journals The Tilted Flashing Brownian Ratchet

2019 ◽  
Vol 18 (01) ◽  
pp. 1950005 ◽  
Author(s):  
S. N. Ethier ◽  
Jiyeon Lee
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Planck Equation ◽  
Directed Motion ◽  
Brownian Ratchet ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Motion Effect ◽  
Viable Method ◽  
Direction Opposite

The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous force that acts in the direction opposite to that of the directed motion, there is a reduction (or even a reversal) of the directed motion effect. Such a process may be called a tilted flashing Brownian ratchet. We show how one can study this process numerically, using a random walk approximation or, equivalently, using numerical solution of the Fokker–Planck equation. Stochastic simulation is another viable method.

The Level-Crossing Problem: First-Passage, Escape and Extremes

2014 ◽  
Vol 13 (04) ◽  
pp. 1430001 ◽  
Author(s):  
Jaume Masoliver
Keyword(s):  
Brownian Motion ◽  
Diffusion Processes ◽  
Extreme Values ◽  
Level Crossing ◽  
First Passage ◽  
One Dimensional ◽  
Absolute Value ◽  
Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (04) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

Diffusion-reaction in one dimension

1988 ◽  
Vol 25 (4) ◽  
pp. 733-743 ◽  
Author(s):  
David Balding
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Particle Systems ◽  
Diffusion Reaction ◽  
One Dimension ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Special Cases ◽  
Initial Arrangement ◽  
Number Of Particles

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.

scholarly journals LQG Homing in a Finite Time Interval

2011 ◽  
Vol 2011 ◽  
pp. 1-3 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Fixed Constant

LetX(t)be a controlled one-dimensional diffusion process having constant infinitesimal variance. We consider the problem of optimally controllingX(t)until timeT(x)=min{T1(x),t1}, whereT1(x)is the first-passage time of the process to a given boundary andt1is a fixed constant. The optimal control is obtained explicitly in the particular case whenX(t)is a controlled Wiener process.

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