Vincent Beffara; Wendelin Werner; Université de Paris-Sud.
|注意：||Thèse rédigée majoritairement en anglais.|
|描述：||135 p. : ill. ; 30 cm.|
|責任：||Vincent Beffara ; sous la dir. de Wendelin Werner.|
This thesis is dedicated to the study of various geometric properties of planar Brownian motion and the SLE process (also known as stochastic Loewner evolution). We prove that, on a typical planar Brownian path, there almost surely exist "pivoting" points, i.e. cut-points around which one half of the curve can rotate by a positive angle without ever intersecting the other half of the path; the set of all pivoting points of a given positive (small enough) angle is then of positive Hausdorff dimension. About SLE, the main result we obtain in this thesis is the computation of the Hausdorff dimension of the curve generating it (the dimension is equal to one plus one eighth of the parameter), for any positive parameter smaller than eight and different from four. We also study the problem of the generalization of the SLE process to non-simply connected surfaces; we show that the construction is doable for two particular values of the parameter (six and eight thirds), but the universality property of usual SLE 1S then lost.