Stochastic Processes

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• Stochastic Variables
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• Random Events
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• Stochastic Processes
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• Markov Processes
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• The Master Equation
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• One-Step Processes
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• Chemical Reactions
  Concepts | Exercises
• The Fokker-Planck Equation
  Concepts | Exercises
• The Langevin Approach
  Concepts | Exercises
• Expansion of the Master Equation
  Concepts | Exercises
• The Diffusion Type
  Concepts | Exercises
• First-Passage Problems
  Concepts | Exercises
• Unstable Systems
  Concepts | Exercises
• Fluctuations in Continuous Systems
  Concepts | Exercises
• The Statistics of Jump Events
  Concepts | Exercises
• Stochastic Differential Equations
  Concepts | Exercises
• Stochastic Behavior of Quantum Systems
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Established April 20, 2006
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A random set of points is a recording of the number and timing of random events of the same kind along the time axis. No causal link is presupposed between the cause of occurrence of the random events and the effects these events may drive downstream of their occurrence. The sample space for a random set of points is a reunion of states. The state for a random set of points (which is an instance of the sample space) is determined by:

1. a non-negative integer $s$ describing the number of points the random set of the state considered has
2. a state-subset of real numbers ${\tau }_s$ obeying $-\infty <{\tau }_1<{\tau }_2<\dots <{\tau }_s<\infty$ describing the timing distribution of occurrence of the points in the random set

The probability distribution over the states is $Q_s\left({\tau }_1,{\tau }_2\dots {\tau }_s\right)$, the probability of occurrence of $s$ events with timing sequence ${\tau }_1,{\tau }_2\dots {\tau }_s$. If $Q_s\left({\tau }_1,{\tau }_2\dots {\tau }_s\right)$ is a symmetric function of its variables, i.e. $Q_s\left({\tau }_1,{\tau }_2\dots {\tau }_s\right)=Q_s\left({\tau }_2,{\tau }_1\dots {\tau }_s\right)$ the ordering $-\infty <{\tau }_1<{\tau }_2<\dots <{\tau }_s<\infty$ can be removed and each ${\tau }_i$ can vary from $-\infty$ to $\infty$. Then the normalization condition writes: