1. PRELUDE 7
Dawson [41, 42, 43], Dawson and Salehi , Itˆ o [67, 66, 65, 64, 63], Iwata
, Krylov and Rozovski˘ ı [83, 80, 81, 79], Kuo [84, 85, 87, 86], Kuo and Piech
, Marcus [99, 100], M´ etivier and Viot , Pardoux [109, 108, 110], Viot
[134, 135, 136], and Yor [146, 145]; for early relations to classical P (φ)1 field
theories see Parisi and Wu .
For a representative list of more recent related results, see the volume edited by
Carmona and Rozovskii , as well as the books and papers by Chow , Corwin
, Daletski˘ ı and Fomin , Da Prato and Zabczyk [32, 31, 30], Dalang et
al , Gawarecki and Mandrekar , Hairer , Kotelenez , Krylov ,
Nualart [105, 104], Peszat and Zabczyk , Pr´ evˆ ot and R¨ ockner , Quastel
, and Sanz–Sol´ e . There is a great deal more; I have, for example, added
very little bibliography for SPDEs that are driven by “colored noise.” The analysis
of such models leads to a wonderful topic of both theoretical as well as applied
interest. But I will not have anything to say about that topic in these lecture
notes, mainly due to space-time limitations.
Finally, let me add a small, choice, selection of books and papers that con-
tain illuminating discussions on the physical notion of intermittency: There is the
standard classic book by Mandelbrot , as well as related works by some of the
early pioneers of the subject, Shandarin and Zel dovich , and Zel dovich et al
[147, 148, 149].
I would like to conclude this prologue with a few words of gratitude.
Many thanks are due to the National Science Foundation and the Conference
Board of Mathematical Sciences for providing a terrific venue for these lectures, and
to the University of Utah for providing with an excellent research environment.
Particularly warm thanks are due to the organizers, Professors Yimin Xiao and
V. S. Mandrekar, for their kind invitation and for making this excellent lecture
A number of colleagues have made comments, corrections, and many sugges-
tions for improvements. I thank them all. Among those, special thanks are owed
to Tom Alberts, Pavel Bezdek, Le Chen, Wei-Dan Chen, Robert Dalang, Mathew
Joseph, Paul Jung, Kunwoo Kim, Sheu-Tang Li, Sergey Lototsky, Atma Mandrekar,
Carl Mueller, David Nualart, Andreas Roy, Shannon Starr, Jason Swanson, Frederi
Viens, Yimin Xiao, and Xinghua Zheng. All remaining errors are mine.
Salt Lake City, Utah