"The properties of a general class of Markov processes have been thoroughly investigated. This has been achieved as a combination of two approaches. First, we have introduced and used a novel, general framework which allows to link the properties of every such Markov process to the ones of a well-known classical Markov process. Second, for this particular study we have established a link between the investigated Markov processes and a group of Markov processes, that we have a priori understood. Despite these useful links the work performed is based on detailed yet challenging studies. These results are included in three papers:
1.""Spectral expansions of non-selfadjoint generalized Laguerre semigroups"", Patie, P. and Savov, M.,162 pages, second round of refereeing;
2.""Cauchy problem of the non-selfadjoint Gauss-Laguerre semigroups and uniform bounds for generalized Laguerre polynomials"", Patie, P. and Savov, M., Journal of Spectral Theory, 33 pages, accepted;
3.""Intertwining, Excursion Theory and Krein Theory of Strings for Non-selfadjoint Markov Semigroups"", Patie, P., Savov, M. and Zhao Y., 33 pages, submitted.
As an application of these results and methods we have developed the spectral theory for an important class of self-similar Markov processes. This gives a wealth of new information for these objects and includes in its domain a number of results that appear in the literature.
The derivation of the results for the Markov processes also requires a thorough investigation of the class of random variables called the exponential functionals of Levy processes. To achieve this we introduce a class of special functions that we call Bernstein-Gamma functions, whose analytic properties reveal the most important features of the exponential functionals. For this purpose a separate and self-contained paper has been written on the topic:
4. ""Bernstein-Gamma functions and exponential functionals of Levy processes"", Patie, P. and Savov, M., 84 pages, submitted.
We note that the Bernstein-Gamma functions appear in various probabilistic contexts. For example, the pricing of Asian options and perpetuities are investigated with the help of these functions and the development of other novel tools.
As a result of networking two separate publications beyond the scope of the project have been refined and published. First, with the better understanding and employment of spectral theory an open problem has been resolved in :
5. ""Conditional survival distributions of Brownian trajectories in a one dimensional Poissonian environment in the critical case"", M. Kolb and M. Savov, Electronic Journal of Probability, (2017), 1--27.
Also, we have managed to improve on results on the behaviour of the classical Brownian motion with limited return to the origin, now published under the title:
6. ""Transience and recurrence of a Brownian path with limited local time"", Kolb, M. and Savov, M., Annals of Probability, (2016), Vol. 44, No. 6, 4083--4132.
"
