SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the ‘cut locus’ case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ rough path theory.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Refined estimates on the scalar heat kernel for bounded b

This chapter obtains uniform bounds for the kernels rb,tX and another rb,tX for bounded b > 0, with the proper decay at infinity on X or ̂X. These bounds will be used to obtain corresponding bounds for the kernel qb,tX in the next chapter. Furthermore, the arguments developed in Chapter 12, which connect the hypoelliptic heat kernel with the wave equation, play an important role in proving the required estimates. Hence, this chapter first establishes estimates on the Hessian of the distance function on X. Then, the chapter obtains bounds on the first heat kernel rb,tX and establishes the bounds on another heat kernel rb,tX.

The action functional and the harmonic oscillator

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b → 0 and when b → +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action. The aforementioned variational problem has already been considered previously as a warm up to the more general problem on Riemannian manifolds, in order to study in detail the small time asymptotics of the hypoelliptic heat kernel.

The heat kernel qb,tX′ for b large

This chapter establishes the estimates of Chapter 9 on the hypoelliptic heat kernel q−b,tX. More precisely, this chapter shows that as b → +∞, q−b,tX exhibits the proper decay away from ̂Fᵧ = îₐN(k⁻¹) ⊂ ̂X. It first proves similar estimates on scalar hypoelliptic heat kernels over X. Such estimates are then extended to scalar hypoelliptic heat kernels over ̂X. Ultimately, this chapter extends the estimates to the kernel q−b,tX using the Feynman-Kac formula. The term ½ ∣ [Yᶰ, Y TX] ∣² in the right-hand side of an equation from Chapter 2 for ℒbX plays a crucial role in proving the estimates.

The heat kernel qb,tX for bounded b

This chapter establishes the estimates of a theorem discussed in Chapter 4 for the hypoelliptic heat kernel qb,tX((x,Y),(x′,Y′)) on ̂X. More precisely, this chapter shows that for bounded b > 0, the heat kernel verifies uniform Gaussian type estimates. Also, the limit is studied as b → 0 of this heat kernel. The method consists in using the techniques of chapters 12 and 13 for the scalar heat kernels over X and ̂X, and to control the kernel qb,tX by using a Feynman-Kac formula. In particular, the contribution of the representation ρ‎ᴱ is obtained by using results of chapter 12 applied to the symmetric space attached to the complexification K C of K.