Parametric Hamilton's equations for Lagrangian model for passive scalar gradients

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 predict the statistics of passive scalar gradients in a Lagrangian model for turbulence, namely the scalar gradient Recent Fluid Deformation Closure (sgRFD).