This book deals with equations that have played a central role in
the interplay between partial differential equations and
probability theory. Most of this material has been treated
elsewhere, but it is rarely presented in a manner that makes it
readily accessible to people whose background is probability
theory. Many results are given new proofs designed for readers with
limited expertise in analysis. The author covers the theory of
linear, second order, partial differential equations of parabolic
and elliptic types. Many of the techniques have antecedents in
probability theory, although the book also covers a few purely
analytic techniques. In particular, a chapter is devoted to the De
Giorgi-Moser-Nash estimates, and the concluding chapter gives an
introduction to the theory of pseudodifferential operators and
their application to hypoellipticity, including the famous theorem
of Lars Hormander.
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