[Paper Review] On moments of the parabolic Anderson model

Alexei Borodin, Ivan Corwin|arXiv (Cornell University)|Nov 30, 2012

advanced mathematical theories3 citations

TL;DR

This paper derives contour integral formulas for all moments of the parabolic Anderson model with nearest-neighbor jumps and space-time white noise in discrete space and continuous time. It computes the second moment Lyapunov exponent for the general case and establishes moment Lyapunov exponents of all orders for the right-jump-only model, providing exact analytical expressions for moment growth rates in stochastic PDEs with multiplicative noise.

ABSTRACT

We study the parabolic Anderson model with nearest neighbor jumps and space-time white noise (discrete space / continuous time). We prove a contour integral formula for the second moment and compute the second moment Lyapunov exponent. For the model with only jumps to the right, we prove a contour integral formula for all moments and compute moment Lyapunov exponents of all orders.

Motivation & Objective

• To analyze the statistical behavior of the parabolic Anderson model under nearest-neighbor jumps and space-time white noise.
• To derive exact expressions for the second moment using contour integral representations.
• To extend the moment analysis to all orders in the special case where jumps occur only to the right.
• To compute the moment Lyapunov exponents for all orders in the right-jump model, quantifying long-time moment growth rates.

Proposed method

• Derives a contour integral formula for the second moment using generating functions and complex analysis techniques.
• Applies residue calculus to evaluate the second moment and extract the Lyapunov exponent from the asymptotic growth rate.
• Extends the contour integral method to all moments in the right-jump-only model via combinatorial and generating function methods.
• Uses the structure of the right-jump model to enable exact computation of moment Lyapunov exponents of all orders.
• Relies on the Markov property and transition kernel structure of the underlying jump process to derive moment equations.
• Employs Laplace transform and inverse transform techniques to connect moment generating functions to contour integrals.

Experimental results

Research questions

• RQ1 What is the exact asymptotic growth rate of the second moment in the parabolic Anderson model with nearest-neighbor jumps and space-time white noise?
• RQ2 Can a closed-form contour integral representation be derived for the second moment in this model?
• RQ3 Does the moment Lyapunov exponent of all orders exist and can it be computed exactly in the right-jump-only case?
• RQ4 How does the structure of the jump mechanism (e.g., only to the right) affect the moment behavior and Lyapunov exponents?
• RQ5 What is the role of complex analysis in deriving exact moment formulas for stochastic PDEs with multiplicative noise?

Key findings

• A contour integral formula is derived for the second moment of the parabolic Anderson model with nearest-neighbor jumps and space-time white noise.
• The second moment Lyapunov exponent is computed explicitly using the asymptotic behavior of the contour integral.
• For the right-jump-only model, a contour integral formula is established for all moments, enabling full moment analysis.
• The moment Lyapunov exponents of all orders are computed exactly in the right-jump model, revealing precise long-time moment growth rates.
• The method demonstrates that the right-jump structure allows for exact solvability beyond the second moment, unlike the general case.
• The results provide exact analytical expressions for the exponential growth rates of moments, advancing the understanding of intermittency in stochastic PDEs.