Elliptic stochastic quantization. (English) Zbl 1446.60041

Albeverio, Sergio; De Vecchi, Francesco C.; Gubinelli, Massimiliano

Ann. Probab. 48, No. 4, 1693-1741 (2020).

The authors first introduce the notions of strong and weak solutions to two-dimensional elliptic multidimensional stochastic partial differential equations and prove the existence of strong solutions (and thus also of weak solutions) under some hypothesis. The authors also provide a representation of weak solutions via the theory of transformation of measures on abstract Wiener spaces developed by A. S. Üstünel and M. Zakai [Transformation of measure on Wiener space. Berlin: Springer (2000; Zbl 0938.46045)]. Further, the authors prove a dimensional reduction result for the solution to elliptic stochastic partial differential equations and show the convergence of weak solutions using the convergence of potentials. The authors also propose a brief introduction to supersymmetry and supersymmetric Gaussian fields.

Reviewer: Udhayakumar Ramalingam (Sathyamangalam)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
81Q60	Supersymmetry and quantum mechanics
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35D30	Weak solutions to PDEs
35D35	Strong solutions to PDEs

Keywords:

stochastic quantization; elliptic stochastic partial differential equations; dimensional reduction; Wiener space; supersymmetry; Euclidean quantum field theory

Citations:

Zbl 0938.46045

References:

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