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Instantaneous blowup for interacting SDEs with superlinear drift

Mathew Joseph, Shubham Ovhal

TL;DR

This paper proves instantaneous, lattice-wide blowup for a system of interacting SDEs with a superlinear drift under an Osgood-type condition. The authors develop a splitting-up (alternating) scheme to relate the ISDE to a one-dimensional SDE with drift $b$ and multiplicative noise, and they prove convergence of the Alternating Process to truncated ISDE solutions. A key technical contribution is establishing instantaneous spatial blowup in the driftless case and then propagating blowup to the full system via Mueller's comparison principle and the splitting method. The results illuminate sharp blowup phenomena for stochastic systems with superlinear drift and multiplicative noise, with implications for blowup behavior in SPDEs and related stochastic lattice models.

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.

Instantaneous blowup for interacting SDEs with superlinear drift

TL;DR

This paper proves instantaneous, lattice-wide blowup for a system of interacting SDEs with a superlinear drift under an Osgood-type condition. The authors develop a splitting-up (alternating) scheme to relate the ISDE to a one-dimensional SDE with drift and multiplicative noise, and they prove convergence of the Alternating Process to truncated ISDE solutions. A key technical contribution is establishing instantaneous spatial blowup in the driftless case and then propagating blowup to the full system via Mueller's comparison principle and the splitting method. The results illuminate sharp blowup phenomena for stochastic systems with superlinear drift and multiplicative noise, with implications for blowup behavior in SPDEs and related stochastic lattice models.

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 . We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up.

Paper Structure

This paper contains 6 sections, 24 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The underlying SDE
  3. The Alternating Process
  4. Comparison of the Alternating Process with the underlying SDE.
  5. Spatial growth of the driftless interacting SDEs
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Consider eq:ISDE with the above assumptions on $b$ and $\mathscr L$. Let $U_t(x)$ be a minimal solution to eq:ISDE with initial profile $U_0$ satisfying u0:decay. Then we have instantaneous everywhere blowup of the system, that is for each $t>0$

Theorems & Definitions (46)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:explosion']}
  • Remark 2
  • Proposition 2
  • ...and 36 more