Fundamental solutions for linear parabolic systems and matrix processes

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In this thesis we use Lie symmetry methods and integral transforms to obtain fundamental matrices for systems of PDEs of the form [Production note: due to software limitations equation can only been found in thesis] for functions 𝑔𝑖(𝑥), and ƒ𝑖(𝑥) satisfying some necessary conditions. We also provide the methodology to obtain these matrices for a wider range of systems. We then turn to the Lie symmetry study of the Kolmogorov Backwards equation associated to the process of the eigenvalues of a Wishart process. We focus on 2- dimensional Wishart processes with eigenvalues X𝑡 > Y𝑡 ≥ 0 for most of our work. We obtain the cosine transform of the transition density function of the difference X𝑡 ― Y𝑡, as well as some integral expressions for E[X𝑡], E[Y𝑡]. We also obtain some bounds for the variances of X𝑡 and Y𝑡 and the expected values for a wide range of functions of these eigenvalues including, among many others, the expected value for any symmetric polynomia the variables X𝑡, Y𝑡. These results are all new, to the best of our knowledge.
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