Probability of Transition to Turbulence in a Reduced Stochastic Model of Pipe Flow

We study the phenomenon of turbulence initiation in pipe flow under different noise structures by estimating the probability of initiating metastable transitions. We establish lower bounds on turbulence transition probabilities using linearized models with multiplicative noise near the laminar state. First, we consider the case of stochastic perturbations by It\^o white noise; then, through the Stratonovich interpretation, we extend the analysis to noise types such as white and red noise in time. Our findings demonstrate the viability of detecting the onset of turbulence as rare events under diverse noise assumptions. The results also contribute to applied SPDE theory and offer valuable methodologies for understanding turbulence across application areas.