Resumen:
It is well known that some (one dimensional) Markov processes can be studied by using spectral methods, especially those related with orthogonal polynomials. Such are the cases of random walks and birth and death processes, if the state space S is taken to be discrete, and di ffusion processes, where the state space S is a real interval. In this talk I will explain how the matrix orthogonality plays an important role in order to study spectral properties of bivariate Markov processes assuming values in Sx{1,2,...,N}, where N is a nonnegative integer (number of phases). I will focus in the case where both components are dependent each other and give some examples where S is discrete or continuous. The key point to apply spectral methods is the theory of matrix-valued orthogonal functions, especially those that are eigenfunctions at the same time of a second-order di fference and di fferential operator. At the end I will focus on future projects and other questions related to matrix orthogonality and applications.
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