scholarly journals Hamiltonian Feynman measures, Kolmogorov integral and infinite dimensional pseudo-differential operators

2019 ◽  
Vol 488 (3) ◽  
pp. 243-247
Author(s):  
O. G. Smolyanov ◽  
N. N. Shamarov
Keyword(s):  
Differential Operators ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Feynman Measure ◽  
Kolmogorov Integral

Properties of infinite dimensional pseudo-differential operators (PDO) are discussed; in particular, the connection between two definitions of the PDO is considered: one given in terms of the Hamiltonian Feynman measure, and another introduced in this work in terms of the Kolmogorov integral.

scholarly journals The martingale problem for pseudo-differential operators on infinite-dimensional spaces

1999 ◽  
Vol 153 ◽  
pp. 101-118 ◽  
Author(s):  
V. Bogachev ◽  
P. Lescot ◽  
M. Röckner
Keyword(s):  
Differential Operators ◽  
Martingale Problem ◽  
Existence Of A Solution ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators

AbstractA martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional “stable-like” processes are presented.

Pseudo-Differential Operators $(\psi{\cal D}{\cal O})$ Algebra and Certain Infinite-Dimensional Lie Algebras

1997 ◽  
Vol 12 (22) ◽  
pp. 1589-1595 ◽  
Author(s):  
E. H. El Kinani
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Differential Operators ◽  
Quantum Plane ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Physics Literature

The class of pseudo-differential operators Lie algebra [Formula: see text] on the quantum plane [Formula: see text] is introduced. The embedding of certain infinite-dimensional Lie algebras which occur in the physics literature in [Formula: see text] is discussed as well as the correspondence between [Formula: see text] and [Formula: see text] as k→+∞ is examined.

scholarly journals WEIGHTED TRACES ON ALGEBRAS OF PSEUDO-DIFFERENTIAL OPERATORS AND GEOMETRY ON LOOP GROUPS

2002 ◽  
Vol 05 (04) ◽  
pp. 503-540 ◽  
Author(s):  
A. CARDONA ◽  
C. DUCOURTIOUX ◽  
J. P. MAGNOT ◽  
S. PAYCHA
Keyword(s):  
Elliptic Operator ◽  
Differential Operators ◽  
Loop Groups ◽  
Linear Functionals ◽  
Geometric Framework ◽  
Infinite Dimensional ◽  
Type Formula ◽  
Pseudo Differential Operators ◽  
Set Up ◽  
The Way

Using weighted traces which are linear functionals of the type [Formula: see text] defined on the whole algebra of (classical) pseudo-differential operators (P.D.Os) and where Q is some admissible invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces.

RENORMALIZED TRACES AS A LOOKING GLASS INTO INFINITE-DIMENSIONAL GEOMETRY

2001 ◽  
Vol 04 (02) ◽  
pp. 221-266 ◽  
Author(s):  
SYLVIE PAYCHA
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Differential Operators ◽  
Index Theorem ◽  
Wodzicki Residue ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators ◽  
Geometric Concepts ◽  
The One ◽  
Definition Of

This paper, based on results obtained in recent years with various coauthors,1–3,13,53 presents a proposal to extend some classical geometric concepts to a class of infinite-dimensional manifolds such as current groups and to a class of infinite-dimensional bundles including the ones arising in the family index theorem. The basic idea is to extend the notion of trace underlying many geometric concepts using renormalized traces which are linear functionals on pseudo-differential operators. The definition of "renormalized traces" involves extra data on the manifolds or vector bundles, namely a weight given by a field of elliptic operators which becomes part of the geometric data, leading to the notion of weighted manifold and weighted vector bundle. This weight is a source of anomaly arising typically as a Wodzicki residue of some pseudo-differential operator. We investigate the anomalies that arise when trying to extend to the infinite-dimensional setting classical results of finite-dimensional geometry such as a Weitzenböck formula, Chern–Weil invariants or the relation between the first Chern form on a complex vector bundle and the curvature on the associated determinant bundle. When comparable, we relate our approach to the one adopted for similar problems in noncommutative geometry.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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