Hypoelliptic Laplacian and probability
The purpose of this article is to describe the theory of the hypoelliptic Laplacian in a relatively simple framework, together with its underlying probabilistic aspects. In its simplest form, the hypoelliptic Laplacian is a second-order differential operator \(M^X_b\) on the tangent bundle \(T X\) of a compact Riemannian manifold \(X\), namely, \(M^X_b :=\frac{A}{ b^2} + \frac Ub\), where \(A = -\frac12 \Delta + \triangledown\) is the Ornstein-Uhlenbeck operator (hence a harmonic oscillator) acting on the fibers, \(U\) denotes the vector field on \(T X\) that generates the geodesic flow, and \(b\) is an interpolation positive parameter. Then, the corresponding diffusion on \(T X\) is a geometric Langevin process solving the stochastic Stratonovitch differential equation \(b^2{\ddot{ x}} + \dot{x} = \dot{w}\) for some Brownian motion \(w\) (along fibers, transported by means of the Levi-Civita connection). A first and key result, to which the author provides different proofs and also intuitive explanations, asserts that as \(b \rightarrow 0\), \(M^X_b\) collapses to the half Laplacian \(-\frac12 \Delta^X\) on the base space \(X\), whereas as \(b \rightarrow \infty\), \(M^X_b\) goes to a perturbed form of the vector field \(U\) (of the geodesic flow). Moreover, these convergences occur in a strong sense, in particular at the level of the kernels and their derivatives, and then of their traces. The author emphasizes the connections of this basic result with the central limit theorem for the occupation measure by the geometric Brownian motion. He also emphasizes the connections to index theory, related to the rigidity (with respect to \(b\)) of a part of the spectrum of \(M^X_b\). References are provided for results which are not proved here. The above convergences are robust under projection onto tori and hold, in particular, over locally symmetric spaces, which is also explained and justified in this article. This theory is even more robust and extends to more general geometric frameworks, including general bundles. As a striking example of application, the author indicates roughly how, by considering the supertrace of the heat kernel, the above convergences (in the suitable framework) can entail the celebrated Selberg trace formula.
diffusion processes and stochastic analysis on manifolds
scientific article; zbMATH DE number 6529322
scientific article