Parametric Hamilton’s equations for stochastic systems

Abstract

Abstract In the context of instanton method for stochastic system this paper purposes a modification of the arclength parametrization of the Hamilton’s equations allowing for an arbitrary instanton speed. The main results of the paper are: (i) it generalizes the parametrized Hamilton’s equations to any speed required. (ii) Corrects the parametric action on the occasion that the Hamiltonian is small but finite and how it adjusts to the probability density function (pdf). (iii) Improves instanton approximation to pdf by noise and propagator renormalization. As an application of the above set up we evaluate the instanton and predict the statistics of two models: Ornstein–Uhlenbeck and passive scalar gradients in a Lagrangian model for turbulence, namely the scalar gradient recent fluid deformation closure.