-3. chen:16:third

The third moment for the parabolic Anderson model

Le Chen

University of Kansas

Abstract: In this paper, we study the parabolic Anderson model starting from the Dirac delta initial data:

\[\left(\frac{\partial}{\partial t} -\frac{\nu}{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = λ u(t,x) \dot{W}(t,x), \qquad u(0,x)=\delta_0(x), \quad x\in\mathbb{R},\]

where \(\dot{W}\) denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [BC14c], we obtain some explicit formulas for \(\mathbb{E}[u(t,x)^3]\). One application of these formulas is given to show the exact phase transition for the intermittency front of order three. These moment formulas enable us to give another characterization for the intermittency fronts based on some integrability conditions, from which we are able to obtain some almost sure results.

MSC 2010 subject classifications: Primary 60H15; Secondary 35R60.

Keywords: stochastic heat equation, parabolic Anderson model, Dirac delta initial condition, space-time white noise, moment formula, intermittency fronts, growth indices

Preprint

[Che16a] Le Chen (2016) ‘The third moment for the parabolic Anderson model’, Preprint arXiv:1609.01005

@article{chen:16:third,
  title         = {The third moment for the parabolic Anderson model},
  author        = {Le Chen},
  year          = {2016},
  month         = {September},
  journal       = {Preprint arXiv:1609.01005},
  url           = {https://www.arxiv.org/abs/1609.01005}
}

Figures:

\(λ=3\)

\(λ=4\)

\(λ=5\)

\(λ=6\)

\(λ=7\)

\(λ=8\)

References: Bertini and Cancrini [BC95a]; Borodin and Corwin [BC14c]; Carmona and Molchanov [CM94]; Chen and Dalang [CD15b]; Chen et al. [CHN17]; Chen and Kim [CK17]; Chen and Kim [CK19]; Chen [Che15]; Foondun and Khoshnevisan [FK09]; Kardar [Kar87]; König [Kon16]; Olver et al. [OLBC10]; Walsh [Wal86];

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