On Uniqueness and Blowup Properties for a Class of Second Order SDEs

As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha<1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha<1$ and $(x_0,y_0)\neq(0,0)$. We also show that blowup in finite time holds if $\alpha>1$ and $(x_0,y_0)\neq(0,0)$.