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Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation

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  • Aguirre, Natham
  • Kowalczyk, Michał

Abstract

In this work we consider the problem of pattern formation modeled by a one dimensional stochastic reaction–diffusion equation with time periodic coefficients. In particular, we apply Large Deviations methods to obtain lower bounds on the probability that certain evenly spaced patterns will develop. Our estimates are optimized when the number of interfaces scales as (ϵT)−1, where ϵ is the length-scale and T is the time-scale. For large times T=ρ|lnϵ| our lower bound is of order exp(−ϵ2ρ), suggesting high likelihood for evenly spaced patterns whose number of interfaces is of order (ϵρ|lnϵ|)−1. Numerical simulations provide support to the idea that the more likely number of interfaces, even among unevenly spaced patterns, follows this law.

Suggested Citation

  • Aguirre, Natham & Kowalczyk, Michał, 2022. "Large deviations approach to a one-dimensional, time-periodic stochastic model of pattern formation," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    • Handle: RePEc:eee:chsofr:v:165:y:2022:i:p1:s0960077922010244
    DOI: 10.1016/j.chaos.2022.112845
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    References listed on IDEAS

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    1. Martin Heida & Matthias Röger, 2018. "Large deviation principle for a stochastic Allen–Cahn equation," Journal of Theoretical Probability, Springer, vol. 31(1), pages 364-401, March.
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