Instantaneous blowup for interacting SDEs with superlinear drift

TL;DR

This paper demonstrates that certain interacting stochastic differential equations with superlinear drift exhibit instantaneous blowup for specific initial conditions, using comparison techniques with one-dimensional blowup SDEs.

Contribution

It establishes the occurrence of instantaneous blowup in interacting SDEs with superlinear drift under particular initial decay conditions, employing the splitting-up method for analysis.

Findings

Instantaneous blowup occurs for initial profiles decaying slower than a specific exponential rate.

Comparison with one-dimensional SDEs shows blowup behavior in the interacting system.

The splitting-up method effectively analyzes blowup phenomena in complex stochastic systems.

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