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Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. (English) Zbl 1032.35114
Birman, Michael Sh. (ed.) et al., Nonlinear problems in mathematical physics and related topics II. In honour of Professor O. A. Ladyzhenskaya. New York, NY: Kluwer Academic Publishers. Int. Math. Ser., N.Y. 2, 243-265 (2002).
The note is an useful collection of many results concerning second-order linear and nonlinear Kolmogorov operators applied, for istance, in probability and diffusion theory. Mathematical models that use Kolmogorov-type equations are also considered in finance problems.
The starting point is to define Lie groups and metric structures then are studied linear equations at first with constant coefficients, then with Hölder continuous coefficients and finally if the coefficients belong to the Sarason class of vanishing mean oscillation functions. Moreover the authors recall some recent properties for ultraparabolic equations with nonlinear total derivatives terms and present new results related to existence, uniqueness and regularity of the classical solution to a Cauchy problem.
For the entire collection see [Zbl 1005.00022].

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K15 Initial value problems for second-order parabolic equations