Wave-breaking and weak instability for the stochastic modified two-component Camassa-Holm equations

In this paper, we consider the stochastic modified two-component Camassa-Holm equations. For the periodic boundary value problem for this SPDE, we first study the local existence, uniqueness and blow-up criterion of a solution in Sobolev spaces H-s with s > 5/2. Particularly, for the linear non-autonomous noise case, we study the wave-breaking phenomenon. When wave-breaking occurs, we estimate the corresponding probability and the breaking rate of the solutions. Finally, we study the noise effect on the dependence on initial data. It is shown that the noise cannot improve the stability of exiting times and the continuity of solution map at the same time.