User:Carl Boettiger/Notebook/Stochastic Population Dynamics/2010/06/02

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Actually starting at Arkin Lab

Meeting with M. Samoilov

  • Kurtz Theorems: two limits, timescale and system size, often taken in the wrong order in practice.
  • Einstein-Smoluchowski debate regarding Langevin correspondence to Ito vs Statonvich representations resolved: [theorem]. (Statonvich is the more natural interpretation of the way in which we measure data to compare to Langevin. Of course Ito and Stratonvich can be mapped back and forth anyway).
  • Of course this difference doesn't exist, and nothing goes wrong / no stochastic surprises if dynamics are linear.
  • Necessary limits of applicability not often checked in SDE famework.
  • Deterministic limit often works better than these limits would imply, as the same deterministic equation arises from multiple cases
  • Hamiltonian-Jacobi derivation sheds some light on why deterministic result is more general (requires fewer assumptions/weaker limits than SDE approach)
  • Intrinsic + extrinsic noise is hard. Environmental noise poorly defined, hard to make physical. Can be incorporated into the master equation directly in a more meaningful way, still very hard problem. Still, environmental noise often not needed to capture the behavior once we go back to original master equation.
  • References and examples to come.

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