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With each methodology treated in its own chapter, this monograph is a thorough exploration of several theories that can be used to find explicit formulas for heat kernels for both elliptic and sub-elliptic operators. The authors show how to find heat kernels for classical operators by employing a number of different methods. Some of these methods come from stochastic processes, others from quantum physics, and yet others are purely mathematical. Depending on the symmetry, geometry and ellipticity, some methods are more suited for certain operators rather than others.

What is new about this work is the sheer diversity of methods that are used to compute the heat kernels. It is interesting that such apparently distinct branches of mathematics, including stochastic processes, differential geometry, special functions, quantum mechanics and PDEs, all have a common concept - the heat kernel. This unifying concept, that brings together so many domains of mathematics, deserves dedicated study.

One of the large classes of operators studied in this book is the sum of squares operators. These operators might be either elliptic or sub-elliptic. The methods for finding the heat kernel depend on the commutativity condition of the operators. Another class of operators investigated in this book is the sum between a second partially differential operator and a smooth potential. The authors demonstrate that the case of linear and quadratic potentials can be solved explicitly either by path integral methods, or by Van Vleck's formula, or by geometric methods that encounter classical action and volume function. They can also be solved by means of psuedo-differential operators. The book contains most of the heat kernels computable by means of elementary functions. Future research in this field can consider the possibility of closed-form expressions of heat kernels involving elliptic functions and hyperelliptic functions.

Heat Kernels for Elliptic and Sub-elliptic Operators is an ideal resource for graduate students, researchers, and practitioners in pure and applied mathematics as well as theoretical physicists interested in understanding different ways of approaching evolution operators.