The Most Probable Transition Paths of Stochastic Dynamical Systems: Equivalent Description and Characterization

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Abstract
This work is devoted to show a sufficient and necessary description for the most probable transition paths of stochastic dynamical systems with Brownian noise. The equivalence is established by studying the relations between Markovian bridge measures and Onsgaer-Machlup action functionals. This cannot be achieved by existing methods because Markovian bridge measures are not quasi translation invariant. We overcome this problem by showing that the Onsager-Machlup action functional can be derived from a Markovian bridge process. We then demonstrate that, for some special cases of linear stochastic systems and nonlinear stochastic systems with small noise, the most probable transition path can be determined by a first order deterministic differential equation with an initial value.

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