Morse inequalities, a probabilistic approach

Abstract

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.

(cont.) The case of the degenerate Bott-Morse function requires a bit more work due to the fact that the geometry near the critical submanifolds is in general not trivial. After some standard constructions, we have two choices of the connection around critical submanifolds. One is the Levi-Civita and the other is Bismut's connection. The main step in this analysis is to prove that the heat kernels of certain operators with respect to Levi-Civita connection and the Bismut connection stay bounded when the parameters involved become large. This is achieved by a fiberwise version of the argument given in the nondegenerate case. Using the boundedness, one can prove the basic comparison. Finally, the rest is just a fiberwise harmonic oscillator problem.