Instantaneous blowup for interacting SDEs with superlinear drift

We consider a system of interacting SDEs on the integer lattice with multiplicative noise and a drift satisfying the finite Osgood’s condition. We show instantaneous everywhere blowup for initial profiles decaying slower than $\exp (-\sqrt{\mid \log \mathrm{\mid }x\mathrm{\mid }\mid })$. We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up.