Global solutions to elliptic and parabolic ${\Phi }^4$ models in Euclidean space

Abstract

We prove existence of global solutions to singular SPDEs on ${\mathbb{R}}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the ${\Phi }_d^4$ Euclidean quantum field theory via Parisi--Wu stochastic quantization, while the elliptic equations are linked to the ${\Phi }_{d-2}^4$ Euclidean quantum field theory via the Parisi--Sourlas dimensional reduction mechanism.

arXiv comment63 pages

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Cited by in corpus (17)

• A variational method for ${\Phi }_3^4$
• Global dynamics for the two-dimensional stochastic nonlinear wave equations
• Stochastic quantization associated with the $\exp (\Phi {)}_2$-quantum field model driven by space-time white noise on the torus
• Large $N$ limit of the $O(N)$ linear sigma model in 3D
• Elliptic stochastic quantization
• Some recent progress in singular stochastic PDEs
• Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain
• A priori bounds for the ${\Phi }^4$ equation in the full sub-critical regime
• A stochastic PDE approach to large N problems in quantum field theory: a survey
• Variational methods for a singular SPDE yielding the universality of the magnetization ripple
• Construction of a non-Gaussian and rotation-invariant ${\Phi }^4$-measure and associated flow on ${\mathbb{R}}^3$ through stochastic quantization
• Commutator estimates from a viewpoint of regularity structures
• The microlocal irregularity of Gaussian noise
• Ergodicity of 2D singular stochastic Navier-Stokes equations
• Anderson Hamiltonians with singular potentials
• Global dynamics for the stochastic nonlinear beam equations on the four-dimensional torus
• Multiscale coupling and the maximum of $\mathcal{P}(\varphi {)}_2$ models on the torus