Stochastic Regularization in Nonlinear Transport

Nonlinear transport equations often exhibit severe gradient amplification when subjected to super-linear feedback, leading to instability along characteristic trajectories. Classical stochastic perturbations introduced through additive noise typically disperse trajectories without modifying the underlying transport operator responsible for this amplification.&nbsp; <div> In this work we investigate the stabilizing role of stochastic transport perturbations introduced through the Stochastic Advection by Lie Transport (SALT) framework. By applying the Stratonovich-Itô conversion to the stochastic Lagrangian flow, we show that SALT dynamics generate an emergent deterministic second-order spatial operator in the Eulerian formulation. When the covariance tensor of the transport perturbations is uniformly nondegenerate, this operator introduces a dissipative contribution that suppresses gradient amplification in expectation.&nbsp; </div> <div> In the limit of isotropic spatial noise, the stochastic transport operator converges formally to a Laplacian with effective viscosity determined by the trace of the transport covariance tensor. Numerical experiments illustrate the convergence of the ensemble-averaged stochastic dynamics toward the corresponding viscous limit.&nbsp; </div> <div> These results provide a structural link between stochastic transport models and classical dissipative regularization mechanisms in nonlinear transport systems. </div>