Multi-shockpeakons for the stochastic Degasperis-Procesi equation

The deterministic Degasperis-Procesi equation admits weak multi-shockpeakon solutions of the form

$$u(x,t)=\sum _{i=1}^n{m}_i(t)e^{-|x-x_i(t)|}-\sum _{i=1}^n{s}_i(t)\mathrm{s}\mathrm{g}\mathrm{n}(x-x_i(t))e^{-|x-x_i(t)|},$$

where $\mathrm{s}\mathrm{g}\mathrm{n}(x)$ denotes the signum function with $\mathrm{s}\mathrm{g}\mathrm{n}(0)=0$, if and only if the time-dependent parameters $x_i(t)$ (positions), $m_i(t)$ (momenta) and $s_i(t)$ (shock strengths) satisfy a system of $3n$ ordinary differential equations. We prove that a stochastic perturbation of the Degasperis-Procesi equation also has weak multi-shockpeakon solutions if and only if the positions, momenta and shock strengths obey a system of $3n$ stochastic differential equations.