Instantaneous blowup for interacting SDEs with superlinear drift

Abstract

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 $\exp \left(-\sqrt{|\log |x||}\right)$. We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up.

Expanded Section 4 to include non-constant initial profiles