Séminaire de Probabilités XLII

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...103 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...