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Author: Schmid, Patrick ×

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    Random walks conditioned to stay in Weyl chambers of type C and D
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) König, Wolfgang; Schmid, Patrick
    We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob $h$-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and König who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an $h$-transform in the Weyl chamber of type C.
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    Brownian motion in a truncated Weyl chamber
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) König, Wolfgang; Schmid, Patrick
    We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.