A quasilinear approach to fully nonlinear parabolic (S)PDEs on ℝ^d

2018/02/18 by Antonio Agresti, Agresti, Antonio
Economics, Econometrics and Finance · Mathematics · #35K55 (Primary) #60H30 (Secondary) #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1802.06395

openalex publication_date 2018/02/18 · openalex created_date 2018/03/06 · openalex updated_date 2026/07/28

Abstract

We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable blow-up criterion. The key idea is the use of the paradifferential operator calculus in order to reduce the fully nonlinear problem into an abstract quasilinear (stochastic) parabolic equation. This allows us to use some recent results on abstract quasilinear (stochastic) evolution equations in Banach spaces. To do so, we analyse the properties of the paradifferential operator, in light of known results on the boundedness of the H^∞-calculus for pseudodifferential operator. Finally, we extend the theory just developed to cover high order fully nonlinear parabolic (S)PDEs.

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