Morse inequalities, a probabilistic approach

In this thesis we give a probabilistic proof of the Morse inequalities in the nondegenerate and degenerate case. For the nondegenerate case the kernel associated with the Witten Laplacian has an expression via the Malliavin calculus. The first step is the analysis of this heat kernel at a point away the critical set. Using Markov property, an iteration procedure and estimates on exit times from balls, everything is reduced to the estimation of a solution to a parabolic initial-boundary problem on a ball in the Euclidean space. We achieve that by constructing a supersolution. For the case the point is close to the critical set, we use an integration by parts in the Malliavin calculus and split the analysis for paths staying inside a given distance from the critical point or exiting the corresponding ball. For the paths exiting, again an iterative Markov property argument reduces the problem to a parabolic initial-boundary value problem that can be handled by the construction of the supersolution mentioned above. For the quantity involving the paths staying inside a given ball around the critical point, we can reverse the argument, this time with the Euclidean space playing the role of the original manifold and reduce the problem to one in the Euclidean settings. This turns out to be an elementary harmonic oscillator problem that finishes the argument.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2004

Author dc:creator

• Popescu, Ionel, 1974-

Advisor dc:contributor.advisor

• Daniel W. Stroock.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/32247

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/32247