Ovidiu Calin

Heat Kernels for Elliptic and Sub-elliptic Operators

Part I. Traditional Methods for Computing Heat Kernels.- Introduction.- Stochastic Analysis Method.- A Brief Introduction to Calculus of Variations.- The Path Integral Approach.- The Geometric Method.- Commuting Operators.- Fourier Transform Method.- The Eigenfunctions Expansion Method.- Part II. Heat Kernel on Nilpotent Lie Groups and Nilmanifolds.- Laplacians and Sub-Laplacians.- Heat Kernels for Laplacians and Step 2 Sub-Laplacians.- Heat Kernel for Sub-Laplacian on the Sphere S^3.- Part III. Laguerre Calculus and Fourier Method.- Finding Heat Kernels by Using Laguerre Calculus.- Constructing Heat Kernel for Degenerate Elliptic Operators.- Heat Kernel for the Kohn Laplacian on the Heisenberg Group.- Part IV. Pseudo-Differential Operators.- The Psuedo-Differential Operators Technique.- Bibliography.- Index.

FUNDAMENTAL SOLUTIONS FOR HERMITE AND SUBELLIPTIC OPERATORS

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □J in a family of quadratic submanifolds.

Geometric modeling in probability and statistics

Part I: The Geometry of Statistical Models.- Statistical Models.- Explicit Examples.- Entropy on Statistical Models.- Kullback-Leibler Relative Entropy.- Informational Energy.- Maximum Entropy Distributions.- Part II: Statistical Manifolds.- An Introduction to Manifolds.- Dualistic Structure.- Dual Volume Elements.- Dual Laplacians.- Contrast Functions Geometry.- Contrast Functions on Statistical Models.- Statistical Submanifolds.- Appendix A: Information Geometry Calculator.

On SubRiemannian Geodesics

We consider a subRiemannian geometry induced by a step 3 subelliptic partial differential operator in ℝ3. Our main result is the characterization of a canonical submanifold through the origin, all of whose points are connected to the origin by infinitely many (subRiemannian) geodesics.

Non-Connectivity example in subRiemannian geometry

The article deals with a step 2 example of subRiemannian geometry which locally resembles the Grusin example. We show that while locally the connectivity by geodesics holds and it behaves very similar to the Grusin case, globally the connectivity does not hold.

Generalized Hamilton—Jacobi Equation and Heat Kernel on Step Two Nilpotent Lie Groups

We study geometrically invariant formulas for heat kernels of subelliptic differential operators on two step nilpotent Lie groups and for the Grusin operator in ℝ2. We deduce a general form of the solution to the Hamilton—Jacobi equation and its generalized form in ℝn × ℝm. Using our results, we obtain explicit formulas of the heat kernels for these differential operators.

Geodesics with constraints on Heisenberg manifolds

We provide a qualitative description for the solutions of Euler-Lagrange equations associated to Lagrangians with linear and quadratic constraints. An important role is played by the natural metric induced by the Heisenberg manifold. In the second part we arrive at a formula which is the analog for the Gauss’ formula for the Heisenberg group.

Geodesics on a Certain Step 2 Sub-Riemannian Manifold

The goal of this paper is to consider a step 2 sub-Riemannian manifold where the connectivity bynormal geodesics between distant points fails.

Heat kernels for a class of degenerate elliptic operators using stochastic method

Formulas for heat kernels are found for degenerate elliptic operators by finding the probability density of the associated Ito diffusion. The formulas involve an integral of a product between a volume function and an exponential term.

The Fourier Transform Method

The Fourier transform has been known as one of the most powerful and useful methods of finding fundamental solutions for operators with constant coefficients. Sometimes the application of a partial Fourier transform might be more useful than the full Fourier transform. In this chapter, by the application of the partial Fourier transform, we shall reduce the problem of finding the heat kernel of a complicated operator to a simpler problem involving an operator with fewer variables. After solving the problem for this simple operator, the inverse Fourier transform provides the heat kernel for the initial operator represented under an integral form. In general, this integral cannot be computed explicitly, but in certain particular cases it actually can be worked out. We shall also apply this method to some degenerate operators.