scholarly journals Exponential functionals of Brownian motion and disordered systems

1998 ◽  
Vol 35 (2) ◽  
pp. 255-271 ◽  
Author(s):  
Alain Comtet ◽  
Cécile Monthus ◽  
Marc Yor
Keyword(s):  
Free Energy ◽  
Brownian Motion ◽  
Continuous Time ◽  
Disordered Systems ◽  
Heat Bath ◽  
One Dimensional ◽  
Disordered Models ◽  
Continuous Time Finance ◽  
Explicit Expressions

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.

scholarly journals Exponential functionals of Brownian motion and disordered systems

1998 ◽  
Vol 35 (02) ◽  
pp. 255-271 ◽  
Author(s):  
Alain Comtet ◽  
Cécile Monthus ◽  
Marc Yor
Keyword(s):  
Free Energy ◽  
Brownian Motion ◽  
Continuous Time ◽  
Disordered Systems ◽  
Heat Bath ◽  
One Dimensional ◽  
Disordered Models ◽  
Continuous Time Finance ◽  
Explicit Expressions

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.

scholarly journals Limiting stochastic processes of shift-periodic dynamical systems

2019 ◽  
Vol 6 (11) ◽  
pp. 191423
Author(s):  
Julia Stadlmann ◽  
Radek Erban
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Real Line ◽  
Continuous Time ◽  
Initial Distribution ◽  
Dynamical Behaviour ◽  
One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

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