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On the pitchfork bifurcation for the Chafee–Infante equation with additive noise

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2023

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Blumenthal, Alex
Engel, Maximilian

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U.S. National Science Foundation (NSF): DMS - 2009431

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Probability Theory and Related Fields. Springer. 2023, 187(3-4), pp. 603-627. ISSN 0178-8051. eISSN 1432-2064. Available under: doi: 10.1007/s00440-023-01235-3

Zusammenfassung

We investigate pitchfork bifurcations for a stochastic reaction diffusion equation perturbed by an infinite-dimensional Wiener process. It is well-known that the random attractor is a singleton, independently of the value of the bifurcation parameter; this phenomenon is often referred to as the “destruction” of the bifurcation by the noise. Analogous to the results of Callaway et al. (AIHP Prob Stat 53:1548–1574, 2017) for a 1D stochastic ODE, we show that some remnant of the bifurcation persists for this SPDE model in the form of a positive finite-time Lyapunov exponent. Additionally, we prove finite-time expansion of volume with increasing dimension as the bifurcation parameter crosses further eigenvalues of the Laplacian.

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510 Mathematik

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ISO 690BLUMENTHAL, Alex, Maximilian ENGEL, Alexandra BLESSING-NEAMTU, 2023. On the pitchfork bifurcation for the Chafee–Infante equation with additive noise. In: Probability Theory and Related Fields. Springer. 2023, 187(3-4), pp. 603-627. ISSN 0178-8051. eISSN 1432-2064. Available under: doi: 10.1007/s00440-023-01235-3
BibTex
@article{Blumenthal2023pitch-69740,
  year={2023},
  doi={10.1007/s00440-023-01235-3},
  title={On the pitchfork bifurcation for the Chafee–Infante equation with additive noise},
  number={3-4},
  volume={187},
  issn={0178-8051},
  journal={Probability Theory and Related Fields},
  pages={603--627},
  author={Blumenthal, Alex and Engel, Maximilian and Blessing-Neamtu, Alexandra}
}
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