Lattice approximation to the dynamical Φ₃⁴ model

2015/08/23 by Rongchan Zhu, Zhu, Rongchan, Xiangchan Zhu +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1508.05613

openalex publication_date 2015/08/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the lattice approximations to the dynamical Φ⁴₃ model by paracontrolled distributions proposed in [GIP13]. We prove that the solutions to the lattice systems converge to the solution to the dynamical Φ₃⁴ model in probability, locally in time. The dynamical Φ₃⁴ model is not well defined in the classical sense. Renormalisation has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation which leads to adding a drift term in the lattice systems.

Citations

• Approximating three-dimensional Navier-Stokes equations driven by space-time white noise
• Convergence of the two-dimensional dynamic Ising-Kac model to Φ⁴₂
• Renormalization Group and Stochastic PDE's

Related

• Piecewise linear approximation for the dynamical Φ⁴₃ model
• Convergent Perturbation Theory for the lattice ϕ⁴-model
• Weak universality of the dynamical Φ₃⁴ model on the whole space
• Paracontrolled Distributions and the 3-dimensional Stochastic Quantization Equation
• The dynamical Φ₃⁴ model by an approximation of Laplacian