Extensions of Brownian motion to a family of Grushin-type singularities

Abstract

We consider a one-parameter family of Grushin-type singularities on surfaces, and discuss the possible diffusions that extend Brownian motion to the singularity. This gives a quick proof and clear intuition for the fact that heat can only cross the singularity for an intermediate range of the parameter. When crossing is possible and the singularity consists of one point, we give a complete description of these diffusions, and we describe a "best" extension, which respects the isometry group of the surface and also realizes the unique symmetric one-point extension of the Brownian motion, in the sense of Chen-Fukushima. This extension, however, does not correspond to the bridging extension, which was introduced by Boscain-Prandi, when they previously considered self-adjoint extensions of the Laplace-Beltrami operator on the Riemannian part for these surfaces. We clarify that several of the extensions they considered induce diffusions that are carried by the Marin compactification at the singularity, which is much larger than the (one-point) metric completion. In the case when the singularity is more than one-point, a complete classification of diffusions extending Brownian motion would be unwieldy. Nonetheless, we again describe a "best" extension which respects the isometry group, and in this case, this diffusion corresponds to the bridging extension. A prominent role is played by Bessel processes (of every real dimension) and the classical theory of one-dimensional diffusions and their boundary conditions.

Product formulas and convolutions for two-dimensional Laplace-Beltrami operators: beyond the trivial case

“…which belongs to a family of Laplace-Beltrami operators on conic surfaces that has been the object of recent studies [6,7,36]. We show that the eigenfunctions ω k,λ of L can be written in terms of confluent hypergeometric (or Whittaker) functions, and that the eigenfunction expansion…”

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

“…Concerning, instead, the heat flow, a satisfactory interpretation of the heatconfinement in the Grushin cylinder is known in terms of Brownian motions [45] and random walks [3]: roughly speaking, random particles are lost in the infinite area accumulated along Z: the latter, in practice, acts as a barrier. Clearly, whereas curvature Laplacians are meaningful in the above context of inducing a non-confining (transmitting) Schrödinger flow on two-step two-dimensional almost-Riemannian manifolds (including the Grushin cylinders), thus making quantum and classical picture more alike and well connected by semi-classics, this has no direct meaning instead in application to the heat flow on Riemannian or almost Riemannian manifolds.…”

“…The study of a quantum particle on degenerate Riemannian manifolds, and the problem of the purely geometric confinement away from the singularity locus of the metric, as opposite to the dynamical transmission across the singularity, has recently attracted a considerable amount of attention in relation to Grushin structures (defined, e.g., in [54], as well as in Section 5.1) and to the induced confining effective potentials on cylinder, cone, and plane (as in the works by Nenciu and Nenciu [193], Boscain and Laurent [44], Boscain, Prandi and Seri [47], Prandi, Rizzi and Seri [201], Boscain and Prandi [46], Franceschi, Prandi and Rizzi [104], Gallone, Michelangeli and Pozzoli [114,115,116], Boscain and Neel [45], Pozzoli [199], Beschastnnyi, Boscain and Pozzoli [37], Gallone and Michelangeli [112]), as well as, more generally, on two-step two-dimensional almost-Riemannian structures (Boscain and Laurent [44], Beschastnnyi, Boscain and Pozzoli [37], Beschastnnyi [36]), or also generalisations to almost-Riemannian structures in any dimension and of any step, and even to sub-Riemannian geometries, provided that certain geometrical assumptions on the singular set are taken (Franceschi, Prandi, and Rizzi [104], Prandi, Rizzi, and Seri [201]). On a related note, a satisfactory interpretation of the heat-confinement in the Grushin cylinder is known in terms of Brownian motions (Boscain and Neel [45]) and random walks (Agrachev, Boscain, Neel, and Rizzi [3]).…”

Uniform, localized asymptotics for sub-Riemannian heat kernels and diffusions

“…Thus this is not a smooth sub-Laplacian, and indeed, the Léandre asymptotics fail dramatically. For recent work in this direction on Grushin and related structures, see [21,20,26,25].…”

“…We express the leading term of the nth logarithmic derivative is given as an nth-order joint cumulant. In particular, we can define a family of probability measures m t on Γ ε in terms of a ratio of Laplace integrals, see Equation (20), which are sub-sequentially compact, see Theorem 36. In terms of the m t , we have the following expression for the log-derivatives of p t .…”