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Forward, backward, and weighted stochastic bridges.

Peter D Drummond1

  • 1Physics Department, Swinburne University of Technology, Melbourne 3122, Victoria, Australia.

Physical Review. E
|January 20, 2018
PubMed
Summary

We introduce generalized stochastic bridges, unifying forward and backward time processes with path-dependent weights. This new method offers a robust numerical algorithm for sampling complex stochastic paths, including rare events.

Area of Science:

  • Stochastic processes
  • Mathematical physics
  • Computational methods

Background:

  • Stochastic bridges are conditional distributions of stochastic paths between two points in phase-space.
  • Existing theories often rely on forward or backward stochastic differential equations, which are not always equivalent.
  • Path-dependent weights add complexity to defining and analyzing these processes.

Purpose of the Study:

  • To generalize the theory of stochastic bridges to include time-reversed and weighted stochastic processes.
  • To demonstrate the equivalence of stochastic bridges derived from forward and backward time processes.
  • To develop a robust and easily implementable numerical algorithm for sampling these generalized distributions.

Main Methods:

  • Generalization of stochastic bridge theory to incorporate time-reversed processes.

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  • Inclusion of arbitrary path-dependent weights in stochastic differential equations.
  • Development of a numerical sampling algorithm using partial stochastic equations.

    • Main Results:

    • Stochastic bridges derived from forward or backward time processes are shown to be identical.
    • The numerical algorithm is robust, easily implemented, and capable of sampling complex distributions.
    • Results confirm a conjecture for stochastic equations without gradient drift and generalize it to weighted cases.

    Conclusions:

    • The generalized theory provides a unified framework for stochastic bridges.
    • The numerical method effectively handles complex stochastic paths, including large excursions in tunneling and escape events.
    • This approach offers a powerful tool for analyzing systems with path-dependent dynamics.