Links

Watch: Physics - Physical Applications of Stochastic Processes by Prof. V. Balakrishnan

Stochastic processes

Probability theory

Martingales, Martingales Through Measure Theory

Examples

• Brownian motion:
  • Paths appear non-differentiable, in continuous space-time limit!
  • In path description, the noise term is chosen to be Gaussian white noise.

Classification of models

• Discreteness
  • Discrete space-time
    • One-step process (can be continuous time too)
      • Unbiased random walk
      • Biased random walk
      • Persistent random walk (it can be seen as having memory, i.e. being non-Markovian).
    • Multi-step processes
  • Continuous space-time
    • Diffusion
    • Kramers-Moyal expansion. To get continuous models from discrete ones.
    • Orientation space. See Stochastic dynamics of orientation.
  • Continuous space-discrete time
  • Discrete space-continuous time
• Memory
  • Memoryless (only dependent on current state) $\rightarrow$ Markov process. Most of the stochastic processed used and studied in practice are Markov.

Descriptions

All these generally are Markov processes

• Continuous space-time
  • Path description
    • Stochastic differential equations. Like: Langevin equation
    • Path integrals for stochastic processes
  • Probability description. Obtained from path description by averaging, essentially. Gives differential equations for probability density, like:
    • Fokker-Planck equation
• Discrete space
  • Probability description $\rightarrow$ Master equation
    • Discrete time $\rightarrow$ Difference equation. Discrete time master equation.
    • Continuous time $\rightarrow$ Differential Continuous time master equation.
• Continuous space-discrete time ??. An example is the beginning of the derivation for Brownian motion by Einstein

Important results

• Dissipation-fluctuation relation. Friction and dissipation are due to the random movements of particles. Fluctuations are too. The coefficients describing them (diffusion coefficient and viscosity) should be related.

Computational methods

Monte Carlo method

Other mathematical aspects

https://en.wikipedia.org/wiki/It%C3%B4_calculus

Applications

Chemistry

Chemical kinetics

Oscillating chemical reactions

Biology

Enzyme kinetics

General phenomena

Number fluctuations

Others

Telegraph noise

Complex systems

Recent paper by Ramin Golestanian (26th Feb 2016): http://pubs.acs.org/doi/pdf/10.1021/acs.nanolett.5b04372 on power spectrum for electric-field-driven ion transport through nanopores. Apparently Pink noise (noise that has power law power spectrum, instead of flat, as for white), is common place in situations with electric fields, and underlying mechanism not totally understood.

https://en.wikipedia.org/wiki/Point_process

Stochastic processes with JS: https://www.npmjs.com/package/stochastic