Semi-discrete semi-linear parabolic spdes

Nicos Georgiou, Mathew Joseph, Davar Khoshnevisan, Shang Yuan Shiu

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Consider an infinite system (eqution presented) of interacting Itǒ diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solution under standard regularity assumptions on the nonlinearity σ. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the kth moment Lyapunov exponent is frequently of sharp order κ2, in contrast to the continuous-space stochastic heat equation whose kth moment Lyapunov exponent can be of sharp order κ3. When the underlying walk is transient and the noise level is sufficiently low, we prove also that the solution is a.s. uniformly dissipative provided that the initial profile is in 1(Zd ).

Original languageEnglish
Pages (from-to)2959-3006
Number of pages48
JournalAnnals of Applied Probability
Volume25
Issue number5
DOIs
StatePublished - 1 Oct 2015

Keywords

  • BDG inequality
  • Comparison principle
  • Discrete space
  • Dissipative behavior
  • Interacting diffusions
  • Lyapunov exponents
  • Parabolic Anderson model
  • Semi-discrete stochastic heat equation
  • The stochastic heat equation

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