scholarly journals Safety Verification of Continuous-Space Pure Jump Markov Processes

2016 ◽  
pp. 147-163 ◽  
Author(s):  
Sadegh Esmaeil Zadeh Soudjani ◽  
Rupak Majumdar ◽  
Alessandro Abate
Keyword(s):  
Markov Processes ◽  
Safety Verification ◽  
Continuous Space ◽  
Jump Markov Processes

scholarly journals Poincaré-Type Inequalities for Compact Degenerate Pure Jump Markov Processes

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 518 ◽  
Author(s):  
Pierre Hodara ◽  
Ioannis Papageorgiou
Keyword(s):  
Membrane Potential ◽  
Point Process ◽  
Markov Processes ◽  
Relevant Information ◽  
Neural System ◽  
Brain Neurons ◽  
Actual Position ◽  
The Past ◽  
Poincaré Type Inequalities ◽  
Jump Markov Processes

We aim to prove Poincaré inequalities for a class of pure jump Markov processes inspired by the model introduced by Galves and Löcherbach to describe the behavior of interacting brain neurons. In particular, we consider neurons with degenerate jumps, i.e., which lose their memory when they spike, while the probability of a spike depends on the actual position and thus the past of the whole neural system. The process studied by Galves and Löcherbach is a point process counting the spike events of the system and is therefore non-Markovian. In this work, we consider a process describing the membrane potential of each neuron that contains the relevant information of the past. This allows us to work in a Markovian framework.

Solutions of ordinary differential equations as limits of pure jump markov processes

1970 ◽  
Vol 7 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Thomas G. Kurtz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Radioactive Decay ◽  
Dynamic Processes ◽  
Deterministic Models ◽  
Image Position ◽  
Jump Markov Processes

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

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