Dirichlet Forms: Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Varenna, Italy, June 8-19, 1992

The theory of Dirichlet forms has witnessed recently some

very important developments both in theoretical foundations

and in applications (stochasticprocesses, quantum field

theory, composite materials,...). It was therefore felt

timely to have on this subject a CIME school, in which

leading experts in the field would present both the basic

foundations of the theory and some of the recent

applications. The six courses covered the basic theory and

applications to:

- Stochastic processes and potential theory (M. Fukushima

and M. Roeckner)

- Regularity problems for solutions to elliptic equations in

general domains (E. Fabes and C. Kenig)

- Hypercontractivity of semigroups, logarithmic Sobolev

inequalities and relation to statistical mechanics (L. Gross

and D. Stroock).

The School had a constant and active participation of young

researchers, both from Italy and abroad.