On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type only involving gradient operator. Then, we obtain local solution, blow-up criterion, and global existence. interplay between stability exiting times continuous dependence of solution initial data is also studied multiplicative case.