Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

2016 ◽  
Vol 11 (2) ◽  
pp. 461-496 ◽  
Author(s):  
Shaoqin Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Harnack Inequality ◽  
Stochastic Partial Differential Equations ◽  
Integration By Parts ◽  
Partial Differential ◽  
Integration By Parts Formula

scholarly journals The Caccioppoli ultrafunctions

2017 ◽  
Vol 8 (1) ◽  
pp. 946-978
Author(s):  
Vieri Benci ◽  
Luigi Carlo Berselli ◽  
Carlo Romano Grisanti
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Calculus Of Variations ◽  
Integration By Parts ◽  
Partial Differential ◽  
Gauss Theorem ◽  
Integration By Parts Formula ◽  
New Space ◽  
Class Of Functions

Abstract Ultrafunctions are a particular class of functions defined on a hyperreal field {\mathbb{R}^{\ast}\supset\mathbb{R}} . They have been introduced and studied in some previous works [2, 6, 7]. In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to Partial Differential Equations and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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