scholarly journals A pure jump Markov process with a random singularity spectrum

2010 ◽  
Vol 38 (5) ◽  
pp. 1924-1946 ◽  
Author(s):  
Julien Barral ◽  
Nicolas Fournier ◽  
Stéphane Jaffard ◽  
Stéphane Seuret
Keyword(s):  
Markov Process ◽  
Singularity Spectrum ◽  
Jump Markov Process

scholarly journals Jump Markov models and transition state theory: the quasi-stationary distribution approach

2016 ◽  
Vol 195 ◽  
pp. 469-495 ◽  
Author(s):  
Giacomo Di Gesù ◽  
Tony Lelièvre ◽  
Dorian Le Peutrec ◽  
Boris Nectoux
Keyword(s):  
Markov Process ◽  
Transition State ◽  
State Space ◽  
Stationary Distribution ◽  
Transition State Theory ◽  
State Theory ◽  
Jump Markov Process ◽  
Continuous State Space ◽  
Continuous State ◽  
Quasi Stationary Distribution

We are interested in the connection between a metastable continuous state space Markov process (satisfyinge.g.the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring–Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring–Kramers formula to build kinetic Monte Carlo or Markov state models.

scholarly journals Adiabatic reduction of a model of stochastic gene expression with jump Markov process

2013 ◽  
Vol 68 (5) ◽  
pp. 1051-1070 ◽  
Author(s):  
Romain Yvinec ◽  
Changjing Zhuge ◽  
Jinzhi Lei ◽  
Michael C. Mackey
Keyword(s):  
Gene Expression ◽  
Markov Process ◽  
Stochastic Gene Expression ◽  
Jump Markov Process

Pure jump shock models in reliability

1986 ◽  
Vol 18 (02) ◽  
pp. 423-440 ◽  
Author(s):  
James W. Drosen
Keyword(s):  
Markov Process ◽  
Failure Rate ◽  
Failure Time ◽  
Jump Markov Process ◽  
Damage Level ◽  
Random Threshold ◽  
Shock Models ◽  
Environmental Shocks

There are many examples of a device suffering damage from random environmental shocks. We model the damage level of such a device as a pure jump Markov process, where the incremental damage caused by a shock depends both on the magnitude of the shock and on the damage level just before the shock. We also look at the time until failure of the device, which occurs when the damage level exceeds a random threshold. The distribution of the failure time and the failure rate are examined, and conditions for the failure rate to be increasing or to have an increasing average are found.

Representing stochastic damage evolution in disordered media as a jump Markov process using the fiber bundle model

2016 ◽  
Vol 26 (1) ◽  
pp. 147-161 ◽  
Author(s):  
Sohan Kale ◽  
Martin Ostoja–Starzewski
Keyword(s):  
Markov Process ◽  
Fiber Bundle ◽  
Damage Evolution ◽  
Probability Distributions ◽  
Failure Process ◽  
Lattice Models ◽  
Disordered Media ◽  
Jump Markov Process ◽  
Fiber Bundle Model ◽  
Stochastic Damage

The damage evolution in quasi-brittle materials is inherently stochastic due to the presence of strong disorder in the form of heterogeneities, voids, and microcracks. The final macroscopic failure is foreshadowed by accumulation of a significant amount of distributed damage that results in precursory events observed as avalanches in experiments and simulations. Simulations on spring lattice models of disordered media have been widely used to understand the collective nature of the quasi-brittle material failure process. In this study, we use the jump Markov process to model stochastic damage evolution, which is informed by the avalanche size distributions for a given material. The jump Markov process is defined based on the probability distributions of the jump sizes, the wait-time between consecutive jumps, and the failure strength. The fiber bundle model is used as an example to obtain the required inputs and test the viability of the proposed approach. The stochasticity and size-dependence of the damage evolution process is inherently captured through the inputs provided for the jump Markov process. The avalanche and strength distributions are used to describe the effect of microscopic information present in the form of disorder, on the macroscopic damage evolution behavior.

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