In this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using... Show moreIn this paper we consider continued fraction (CF) expansions on intervals different from [0,1]. For every x in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss–Kuzmin–Lévy based approximation method is used. Convergence of this method follows from [32] but the speed of convergence remains unknown. For a lot of known densities the method gives a very good approximation in a low number of iterations. Finally, a subfamily of the N-expansions is studied. In particular, the entropy as a function of a parameter α is estimated for N=2 and N=36. Interesting behavior can be observed from numerical results. Show less
This PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the... Show moreThis PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the classical Hilbert space setting to the setting of general CAT(0) spaces. In the third chapter, the non-symmetric Fokker-Planck equation is treated as a flow on the Wasserstein-2 space of probability measures, and it is proven that its semigroup of solutions possesses similar properties to those of the gradient flow semigroups. In the forth chapter, a general theory of maximal monotone operators and the induced flows on Wasserstein-2 spaces is developed. This theory generalizes the theory of gradient flows by Ambrosio-Gigli-Savare. In the final fifth chapter the problem of the existence of an invariant measure for stochastic delay equations is proven. The diffusion coefficient has delay, and is assumed to be locally Lipschitz and bounded. Show less
