Kolmogorov equations are a set of second-order elliptic and parabolic equations in a Hilbert H. space.
Compared to classical theory the Kolmogorov equations are characterized by the fact that the coefficients of the differential operators under consideration can be limited and quite irregular. Furthermore, during their study a great deal of attention is dedicated to the connections with differential stochastic equations.
There are a number of reasons to study Kolmogorov equations, for example:
- a natural mathematical interest in extending classical results without supposing the usual hypotheses of regularity and limit in the coefficients;
- the possibility to use the results obtained in the Kolmogorov equations for the study of partial derivate stochastic equations and associated control problems;
The possibility to apply the results obtained to mathematical models which intervene in various fields, such as economics and finance, fluid dynamics (especially turbulence), population dynamics, statistic mechanics and field theory (stochastic quantization).