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

## TL;DR

This paper investigates the conditions under which solutions to a class of second order stochastic differential equations exhibit uniqueness or blowup, revealing critical thresholds for the parameter  and initial conditions.

## Contribution

It establishes new criteria for solution uniqueness and finite-time blowup in second order SDEs with multiplicative noise, highlighting the role of the parameter  and initial states.

## Key findings

- Solutions are nonunique if 0<<1 and initial state is zero.
- Solutions are unique if 1/2<<1 and initial state is nonzero.
- Finite-time blowup occurs if >1 and initial state is nonzero.

## Abstract

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 $(x_0,y_0)\neq(0,0)$.