Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields.

Area of Science:

• Stochastic Analysis
• Probability Theory
• Partial Differential Equations

Background:

• Gaussian random fields are fundamental in probability and physics.
• Understanding their correlation properties is crucial for analyzing complex systems.
• Previous work by Hairer and Xu explored related models.

Purpose of the Study:

• To provide a self-contained proof for uniform bounds on multi-point correlations of trigonometric functions of Gaussian random fields.
• To establish improved estimates compared to existing literature.
• To demonstrate the convergence of composite Gaussian fields with more general functions.

Main Methods:

• Development of a self-contained proof technique.
• Analysis of multi-point correlations.
• Investigating convergence properties of composite functions.

Main Results:

• A uniform bound on multi-point correlations of trigonometric functions of Gaussian random fields was established.
• Improved estimates were achieved for a special case of interface fluctuation models.
• Convergence of composite Gaussian fields with general functions was demonstrated.

Conclusions:

• The derived bounds and convergence results are key components for proving weak universalities in singular stochastic partial differential equations (SPDEs).
• This work contributes to the theoretical framework for analyzing singular SPDEs.