Generalized diffusion processes: construction, limit theorems, approximation methods

The talk is devoted to the notion of generalized difusion process, which is an extension of the Kolmogorov's classical definition of a diffusion process, with the drift vector and diffusion matrix possibly being generalized functions. Analytic background of the theory of generalized diffusion processess will be outlined, and a probabilistic approach to the construction of generalized diffusion processes will be presented in details. Within the probabilistic approach, the required process is constructed as the strong solution to an SDE which involves the local times of the unknown process. Various limit theorems for generalized difusion processes will be presented, together with perspectives in simulation procedures for SDEs involving local times of the unknown process.