Leseprobe
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The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)
Jean-Michel Bismut, Gilles Lebeau
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Naturwissenschaften, Medizin, Informatik, Technik / Geometrie
Beschreibung
This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion.
The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained.
The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions.
Kundenbewertungen
Martingale (probability theory), Principal bundle, Trace class, Matrix calculus, Fokker–Planck equation, Derivative, Determinant, Taylor series, Integration by parts, Explicit formulae (L-function), Ricci curvature, Wave equation, Mellin transform, Parameter, Supertrace, Explicit formula, Hilbert space, Stochastic calculus, Equation, Self-adjoint, Fredholm determinant, Dirac operator, Asymptotic expansion, Formal power series, Translational symmetry, Estimation, Self-adjoint operator, Sobolev space, Cohomology, Probabilistic method, Berezin integral, Computation, Eigenvalues and eigenvectors, Classical Wiener space, Projection (linear algebra), Commutator, Spectral theory, Transversality (mathematics), Fiber bundle, Summation, De Rham cohomology, Invertible matrix, Alexander Grothendieck, Hodge theory, Brownian motion, Heat kernel, Resolvent set, Holomorphic function, Bijection, Vector bundle, Feynman–Kac formula, Notation, Riemann–Roch theorem, Hypoelliptic operator, Euclidean space, Malliavin calculus, Eigenform, Covariance matrix, Variable (mathematics), Parametrix, Morse theory, Logarithm, Fourier series, Brownian dynamics, Tangent space, Theorem, Asymptote, Analytic function, Girsanov theorem, Ellipse