scholarly journals Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

Annals of PDE ◽  
2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Nathan E. Glatt-Holtz ◽  
David P. Herzog ◽  
Jonathan C. Mattingly
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Control Problems ◽  
Dimensional Control ◽  
Partial Differential ◽  
Infinite Dimensional

scholarly journals On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

2021 ◽  
Author(s):  
David Criens ◽  
Moritz Ritter
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Strong Uniqueness ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Dual Version ◽  
Martingale Problems ◽  
Weak Uniqueness ◽  
Strong Existence

AbstractWe consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

scholarly journals The dual Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations

2021 ◽  
Vol 105 (0) ◽  
pp. 51-68
Author(s):  
S. Tappe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Mild Solutions ◽  
Moving Frame ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Dual Result ◽  
Path Dependent

We provide the dual result of the Yamada–Watanabe theorem for mild solutions to semilinear stochastic partial differential equations with path-dependent coefficients. An essential tool is the so-called “method of the moving frame”, which allows us to reduce the proof to infinite dimensional stochastic differential equations.

scholarly journals On the existence of optimal control for controlled stochastic partial differential equations

1989 ◽  
Vol 115 ◽  
pp. 73-85 ◽  
Author(s):  
Noriaki Nagase
Keyword(s):  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Admissible Control ◽  
Order Differential Operator ◽  
Control Problems ◽  
Partial Differential ◽  
The Cauchy Problem ◽  
Existence Of Optimal Control

In this paper we are concerned with stochastic control problems of the following kind. Let Y(t) be a d’-dimensional Brownian motion defined on a probability space (Ω, F, Ft, P) and u(t) an admissible control. We consider the Cauchy problem of stochastic partial differential equations (SPDE in short)where L(y, u) is the 2nd order elliptic differential operator and M(y) the 1st order differential operator.

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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