Math 728 — Spring 2001 Seminar in Partial Differential Equations: Dirichlet Spaces and Symmetric Semigroups of Operators

The notion of a Dirichlet space was introduced by A. Beurling and J. Deny in 1959 as a function space which is continuously embedded into a space of locally integrable functions and on which every normal contraction operates. The prototype of a Dirichlet space is the Sobolev space of functions that are square integrable and have square integrable first order derivatives.

Any Dirichlet space yields a symmetric, positivity preserving semigroup of operators on a Hilbert space of square integrable functions. In the classical case, this semigroup is the heat diffusion semigroup.

Because they encompass so many important examples, Dirichlet spaces and their analysis have been the subject of many investigations. Questions such as heat kernel estimates, Harnack inequalities, Sobolev and Nash inequalities find a natural development in the context of Dirichlet spaces.

This course will give an introductory and comprehensive account of the theory of Dirichlet spaces, Dirichlet forms and their applications. A number of examples will be presented in order to illustrate the general results.

Key words: Dirichlet spaces, Dirichlet form, Dirichlet integral, Markov semigroups, ultracontractivity, Dirichlet problem, elliptic operator, harmonic functions, potentials, Harnack inequality, intrinsic distance, capacity.

References:

• M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter, Studies in Mathematics, 19, 1994
• Z. M. Ma and M. Röckner, Introduction to the theory of (non symmetric) Dirichlet forms, Springer-Verlag, 1992.
• A. Bendikov, Potential Theory on infinitely dimensional Abelian groups de Gruyter, Studies in Mathematics, 21, 1995.
• E. B. Davies, Heat kernels and spectral theory, Cambridge University Press, 1989.