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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.
These notes are based on a course which I gave during the academic
year 1983-84 at the University of Colorado. My intention was to
provide both my audience as well as myself with an introduction to
the theory of 1arie deviations * The organization of sections 1)
through 3) owes something to chance and a great deal to the
excellent set of notes written by R. Azencott for the course which
he gave in 1978 at Saint-Flour (cf. Springer Lecture Notes in
Mathematics 774). To be more precise: it is chance that I was
around N. Y. U. at the time'when M. Schilder wrote his thesis. and
so it may be considered chance that I chose to use his result as a
jumping off point; with only minor variations. everything else in
these sections is taken from Azencott. In particular. section 3) is
little more than a rewrite of his exoposition of the Cramer theory
via the ideas of Bahadur and Zabel. Furthermore. the brief
treatment which I have given to the Ventsel-Freidlin theory in
section 4) is again based on Azencott's ideas. All in all. the
biggest difference between his and my exposition of these topics is
the language in which we have written. However. another major
difference must be mentioned: his bibliography is extensive and
constitutes a fine introduction to the available literature. mine
shares neither of these attributes. Starting with section 5).
The central and distinguishing feature shared by all the
contributions made by K. Ito is the extraordinary insight which
they convey. Reading his papers, one should try to picture the
intellectual setting in which he was working. At the time when he
was a student in Tokyo during the late 1930s, probability theory
had only recently entered the age of continuous-time stochastic
processes: N. Wiener had accomplished his amazing construction
little more than a decade earlier (Wiener, N. , "Differential
space," J. Math. Phys. 2, (1923)), Levy had hardly begun the
mysterious web he was to eventually weave out of Wiener's P~!hs,
the generalizations started by Kolmogorov (Kol mogorov, A. N. ,
"Uber die analytische Methoden in der Wahrscheinlichkeitsrechnung,"
Math Ann. 104 (1931)) and continued by Feller (Feller, W. , "Zur
Theorie der stochastischen Prozesse," Math Ann. 113, (1936))
appeared to have little if anything to do with probability theory,
and the technical measure-theoretic tours de force of J. L. Doob
(Doob, J. L. , "Stochastic processes depending on a continuous
parameter, " TAMS 42 (1937)) still appeared impregnable to all but
the most erudite. Thus, even at the established mathematical
centers in Russia, Western Europe, and America, the theory of
stochastic processes was still in its infancy and the student who
was asked to learn the subject had better be one who was ready to
test his mettle.
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