C∞− regularization of ODEs perturbed by noise

2021 ◽  
pp. 2140010
Author(s):  
Fabian Andsem Harang ◽  
Nicolas Perkowski
Keyword(s):  
Differential Equations ◽  
Gaussian Process ◽  
Ordinary Differential Equations ◽  
Unique Solution ◽  
Vector Fields ◽  
Local Times ◽  
Schwartz Distributions ◽  
Smooth Flow ◽  
Sample Paths ◽  
Regularity Properties

We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space–time regularity properties of local times, and solutions are constructed using the approach of Catellier–Gubinelli based on nonlinear Young integrals.

scholarly journals Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

2017 ◽  
Vol 28 (11) ◽  
pp. 1750080
Author(s):  
Hassan Azad ◽  
Indranil Biswas ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Second Order ◽  
Semisimple Lie Algebra ◽  
Semisimple Algebras ◽  
Generic Orbit ◽  
Cartan Subalgebras ◽  
Local Vector

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.

scholarly journals Painleve Test and Discrete Boltzmann Equations

1989 ◽  
Vol 42 (1) ◽  
pp. 1 ◽  
Author(s):  
N Euler ◽  
W-H Steeb
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Bäcklund Transformations ◽  
Painlevé Analysis ◽  
Painlevé Test ◽  
Boltzmann Equations ◽  
Backlund Transformations ◽  
Painleve Analysis

The Painleve test for various discrete Boltzmann equations is performed. The connection with integrability is discussed. Furthermore the Lie symmetry vector fields are derived and group-theoretical reduction of the discrete Boltzmann equations to ordinary differentiable equations is performed. Lie Backlund transformations are gained by performing the Painleve analysis for the ordinary differential equations

Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points

1995 ◽  
Vol 125 (4) ◽  
pp. 759-776 ◽  
Author(s):  
Teresa Faria ◽  
Luis T. Magalhães
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Equilibrium Points ◽  
Functional Differential Equations ◽  
Finite Dimensional ◽  
Retarded Functional Differential Equations ◽  
Functional Differential ◽  
Necessary And Sufficient

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

scholarly journals Ordinary differential equations and their exponentials

2006 ◽  
Vol 4 (1) ◽  
pp. 64-81 ◽  
Author(s):  
Anders Kock ◽  
Gonzalo Reyes
Keyword(s):  
Differential Geometry ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Function Space ◽  
Vector Fields ◽  
Synthetic Differential Geometry

AbstractIn the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.

scholarly journals LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL

2013 ◽  
Vol 10 (02) ◽  
pp. 235-282 ◽  
Author(s):  
FRANÇOIS BOUCHUT ◽  
GIANLUCA CRIPPA
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Singular Integral ◽  
Vector Fields ◽  
Transport Equations ◽  
Well Posedness ◽  
Quantitative Stability ◽  
Quantitative Estimates

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.

scholarly journals Limit theorems for stochastic difference-differential equations

1992 ◽  
Vol 127 ◽  
pp. 83-116 ◽  
Author(s):  
Tsukasa Fujiwara ◽  
Hiroshi Kunita
Keyword(s):  
Stochastic Process ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Theorems ◽  
Vector Fields ◽  
Mixing Conditions ◽  
Deterministic Function ◽  
Stochastic Ordinary Differential Equations

There are extensive works on the limit theorems for sequences of stochastic ordinary differential equations written in the form:where is a stochastic process and is a deterministic function, both of which take values in the space of vector fields. The case where {ftn} n satisfies certain mixing conditions has been studied by Khas’minskii [7], Kesten-Papanicolaou [6] and others.

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