In this thesis, we derive a multivariate analogue of Ruijsenaars’s 2F1-generalisation R. We use Hopf algebra representation theory of the modular double of sl.2/, a Hopf algebra structure strongly related to quantum groups, to relate the function R to overlap coefficients of eigenfunctions. Using properties of the algebra and the representation, we derive an Askey-Wilson type difference equation. We moreover recover Ruijsenaars’s unitary transformation kernel E.

Expanding on the Hopf algebra structure, we extend our derivations to the multivariate version of R. Employing representation theory, we obtain multivariate difference equations. Furthermore, we demonstrate that the multivariate function enables the definition of a unitary transformation on multivariate functions in L2..0; ∞/N/.

In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the three classical polynomials can be described as matrix coefficients of the representations. Using this connection of the polynomials to Lie groups, we derive various properties of the polynomials from the unitarity of the representation and the associated Lie algebra representation. Next, the representations are generalised to higher dimensional spaces. Doing so, a new connection is shown between the Lie group SU(d+1) and the multivariate Krawtchouk polynomials, extending the known theory for the univariate polynomials. Another new result that will be established is the connection between the multivariate Meixner polynomials and Lie theory. This will be done by defining a representation of SU(1,d) in the Bergman space of the d-dimensional unit ball. Similar as for the univariate polynomials, we will derive the orthogonality, recurrence relations and difference equations from the associated Lie theory.

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets.
Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time stochastic processes are constructed as solutions to a certain class of fractional-order parabolic stochastic partial differential equations. Until now, the solution concepts, that have been formulated for these stochastic equations, either assume zero initial conditions for the process itself and (if existent) for all of its (mean-square temporal) derivatives, or set all (existing) higher-order derivatives at the starting time to zero. The aim of this thesis is twofold: Firstly, we generalize this class of stochastic processes in such a way that non-zero initial conditions (for the process and its temporal derivatives) can be meaningfully incorporated. Secondly, we show that for certain (integer) values of the parameter corresponding to the fractional power of the parabolic operator, the solutions satisfy certain Markov properties.
To this end, we first generalize the integration domain of stochastic Hilbert-space-valued Itô integrals to the whole real line. Afterwards, we present the modified spatiotemporal stochastic processes that facilitate incorporating nonzero initial conditions for the process and its derivatives at a given initial time. Lastly, we show that this proposed process satisfies a so-called multiple Markov property under certain conditions. Markov properties are desirable in numerical applications, because a Markov process allows for a much faster numerical approximation of its covariance operator.

In 2017 Martijn Caspers, Fedor Sukochev and Dmitriy Zanin published a paper which generalises the proof of Davies' 1988 paper, and thus resolves the Nazarov-Peller conjecture. The proofs of these papers have been presented in this thesis. They have been expanded with a proof that generalises the conjecture to arbitrary Schatten classes. The optimality of the estimates in the conjecture is also studied, following the example of the 2016 paper by Coine et al. In addition, the quantum mechanical context is provided to interpret the presented results.

In this thesis, we introduce the quantum groups Uq(SL(2,C)) and Aq(SL(2,C)) as Hopf algebras. We study their representations, including their similarities and differences with the classical theory. We show that the eigenvectors of Koorwinder's twisted primitive elements of Uq(SU(2)) are dual q-Krawtchouk polynomials. We use this explicit expression to define generalised matrix elements and spherical functions in Aq(SL(2,C)). Then we use the Haar functional to show that these generalised matrix elements are Askey-Wilson polynomials with two continuous and two discrete parameters. Next, we show a new result. Namely, two twisted primitive elements of Uq(SL(2,C)) generate Zhedanov's Askey-Wilson algebra AW(3). Consequently, AW(3) is embedded as a subalgebra into Uq(SL(2,C)). We use this to show that overlap functions of twisted primitive elements in Uq(SU(2)) are q-Racah polynomials. With that, we derive a summation formula connecting q-Racah and dual q-Krawtchouk polynomials.

In 2011 Avsec showed strong solidity of the q-Gaussian algebras, building upon previous results of Houdayer and Shlyakhtenko, and Ozawa and Popa. In this work we study this result as well as the necessary literature and q-mathematics needed to replicate the proof. The literature is combined within this thesis, whilst additionally filling in gaps in Avsec's proofs. Overall, the thesis aims to present an improved and more accessible proof of the strong solidity of the q-Gaussian algebras.

Orthogonality relations of q-Meixner polynomials, polynomials in terms of basic hypergeometric series, will be proved by using spectral measures and a difference operator.

The spherical transform maps the orthogonal basis of symmetric Jacobi-type polynomials to an orthogonal basis of (symmetric) Wilson polynomials. The spherical transform is closely related to the Cherednik-Opdam transform, as it is essentially its symmetric version. The symmetric Jacobi-type polynomials can be composed from the non-symmetric Jacobi-type polynomials. These relations, between the symmetric and non-symmetric theory, give an incentive to consider the Cherednik-Opdam transform of non-symmetric Jacobi-type polynomials. This work gives an overview of the symmetric theory about the spherical transform of Jacobi-type polynomials and lays down the groundwork for the Cherednik-Opdam transform of the non-symmetric Jacobi-type polynomials.

Symmetric and nonsymmetric Macdonald polynomials associated to root systems are very general families of orthogonal polynomials in multiple variables. Their definition is quite complex, but in certain cases one can define so-called interpolation polynomials that have a surprisingly simple definition and are related to the Macdonald polynomials by a binomial formula. In this thesis we will discuss such formulas for two kinds of root systems: type A and type (C^∨,C). For the latter case, there are still some open questions that remain unanswered.

In this report we study Markov processes on compact and connected Riemannian manifolds. We define a random walk on such manifolds and give a direct proof of the invariance principle. This principle says that under some conditions on the jumping distributions (i.e. the distributions of single steps), the random walk converges to Brownian motion when space is scaled by 1/N, time by N^2 and N tends to infinity (which has been shown with more general methods by Jorgensen and by Blum). To prove this, we show convergence of the generators on the set of smooth functions and we apply the Trotter-Kurtz theorem (as has been done by Blum, in a rather sketchy way and in a slightly different setting). We also show convergence of the corresponding Dirichlet forms. Then we show that the conditions on the jumping distributions are satisfied if they are compactly supported and have mean 0 and a covariance matrix which is invariant under orthogonal transformations.
Next, we define random grids on a Riemannian manifold and we define random walks on them. We show that their Dirichlet forms converge to the Dirichlet form of Brownian motion, using the results above. We also prove a result that is a bit weaker than convergence of the generators in this case.
Finally, these grids allow us to define the Symmetric Exclusion Process (SEP) on a manifold. Using the convergence results above, we follow the line of a proof of Seppäläinen to show that the hydrodynamic limit of the SEP satisfies the heat equation. Some details still need to be filled in, but we believe that this method will allow us to study interacting particle systems and their hydrodynamic limits on Riemannian manifolds.
Before all of this we start with an introduction to Markov processes, their semigroups and generators. In particular we focus on time-reversible (or symmetric) processes and the Dirichlet form with its properties. We also give an introduction to Riemannian manifolds and related notions.