Séminaire de Probabilités XLII

This book offers an introduction to rough paths. Coverage also includes the interface between analysis and probability to special processes, Levy processes and Levy systems, representation of Gaussian processes, filtrations and quantum probability. The tradition of specialized courses in the Séminaires de Probabilités is continued with A. Lejay's Another introduction to rough paths. Other topics from this 42nd volume range from the interface between analysis and probability to special processes, Lévy processes and Lévy systems, branching, penalization, representation of Gaussian processes, filtrations and quantum probability. Nine volumes ago, in S' eminaire de Probabilit' es XXXIII, a series of advanced courses was started; nine such courses have appeared since. Two of them are due to Antoine Lejay, including his Introduction to rough paths in v- umeXXXVII. Thisunrepentantrecidivistnowstrikesagain,withYetanother introduction to rough paths, which sheds a more algebraic light on the same matter. The various contributions which constitute the rest of the volume ex- plify the r ole the S' eminaire intends to play on the probabilistic stage: junior authors go side by side with older contributors, with a predominance from French or francophile ones; short notes mix with real research articles; and the themes are well in the traditional spirit of the S' eminaire, ranging over the broad spectrum of interest of the readership of the S' eminaire. ' Catherine Donati-Martin, Michel Emery, Alain Rouault, Christophe Stricker vii Contents Yet Another Introduction to Rough Paths Antoine Lejay...1 Monotonicity of the Extremal Functions for One-dimensional Inequalities of Logarithmic Sobolev Type Laurent Miclo...1 03 Non-monotone Convergence in the Quadratic Wasserstein Distance Walter Schachermayer, Uwe Schmock, and Josef Teichmann...131 On the Equation ? =S ??? t t Fangjun Xu...137 Shabat Polynomials and Harmonic Measure Philippe Biane...147 Radial Dunkl Processes Associated with Dihedral Systems Nizar Demni...153 Matrix Valued Brownian Motion and a Paper by P' olya Philippe Biane...171 On the Laws of First Hitting Times of Points for One-dimensional Symmetric Stable L' evy Processes Kouji Yano, Yuko Yano, and Marc Yor...