Infinite–dimensional elliptic and stochastic equations with hölder-continuous coefficients

1999 ◽  
Vol 17 (3) ◽  
pp. 487-508
Author(s):  
Lorenzo Zambotti
Keyword(s):  
Stochastic Equations ◽  
Hölder Continuous ◽  
Infinite Dimensional

DIRICHLET PROBLEMS IN A HALF-SPACE OF A HILBERT SPACE

2002 ◽  
Vol 05 (02) ◽  
pp. 257-291 ◽  
Author(s):  
ENRICO PRIOLA
Keyword(s):  
Hilbert Space ◽  
Half Space ◽  
Regularity Result ◽  
Interpolation Theory ◽  
Dirichlet Problems ◽  
Optimal Regularity ◽  
Hölder Continuous ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Coefficients

We study a homogeneous infinite dimensional Dirichlet problem in a half-space of a Hilbert space involving a second-order elliptic operator with Hölder continuous coefficients. Thanks to a new explicit formula for the solution in the constant coefficients case, we prove an optimal regularity result of Schauder type. The proof uses nonstandard techniques from semigroups and interpolation theory and involves extensive computations on Gaussian integrals.

scholarly journals Stochastic model reduction: convergence and applications to climate equations

2021 ◽  
Author(s):  
Sigurd Assing ◽  
Franco Flandoli ◽  
Umberto Pappalettera
Keyword(s):  
Stochastic Model ◽  
Model Reduction ◽  
Hilbert Spaces ◽  
Evolution Equations ◽  
Climate Modeling ◽  
Stochastic Equations ◽  
Weak And Strong Convergence ◽  
Uhlenbeck Process ◽  
Infinite Dimensional ◽  
Reduced Equations

AbstractWe study stochastic model reduction for evolution equations in infinite-dimensional Hilbert spaces and show the convergence to the reduced equations via abstract results of Wong–Zakai type for stochastic equations driven by a scaled Ornstein–Uhlenbeck process. Both weak and strong convergence are investigated, depending on the presence of quadratic interactions between reduced variables and driving noise. Finally, we are able to apply our results to a class of equations used in climate modeling.

scholarly journals Banach manifold structure and infinite-dimensional analysis for causal fermion systems

2021 ◽  
Author(s):  
Felix Finster ◽  
Magdalena Lottner
Keyword(s):  
Continuous Functions ◽  
Mathematical Framework ◽  
Banach Manifold ◽  
Hölder Continuous ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis ◽  
Fermion Systems ◽  
Hölder Continuous Functions ◽  
A Chain ◽  
Derivatives Of

AbstractA mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

STOCHASTIC EQUATIONS AND QUASI-INVARIANCE ON INFINITE PRODUCT GROUPS

1999 ◽  
Vol 02 (02) ◽  
pp. 283-288 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ALEXEI DALETSKII
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Lie Group ◽  
Existence And Uniqueness ◽  
Infinite Product ◽  
Stochastic Equations ◽  
Compact Lie Group ◽  
Infinite Dimensional ◽  
Countable Power ◽  
Infinite Dimensional Lie Group

A stochastic differential equation on an infinite-dimensional Lie group G constructed as the countable power of a compact Lie group G is considered. The existence and uniqueness of the solutions and quasi-invariance of their distribution are proved.

A COMPARISON THEOREM FOR STOCHASTIC EQUATIONS IN INFINITE DIMENSIONS AND APPLICATIONS

2010 ◽  
Vol 10 (02) ◽  
pp. 197-210
Author(s):  
NIKOLAOS HALIDIAS ◽  
MARIUSZ MICHTA
Keyword(s):  
Existence Theorem ◽  
Banach Spaces ◽  
Comparison Theorem ◽  
Stochastic Equations ◽  
Dimensional Case ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Drift Coefficient

In this paper we consider stochastic equations in Banach spaces. Our first result is a comparison theorem. As an application we prove an existence theorem in the case when the drift coefficient is nonsmooth. The present studies extend some results both for deterministic and stochastic equations in infinite dimensional case.

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