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Windings of the planar Brownian motion and Green’s formula

Abstract : We study the windings of the planar Brownian motion around points, following the previous works of Wendelin Werner in particular. In the first chapter, we motivate this study by the one of smoother curves. We prove in particular a Green formula for Young integration, without simplicity assumption for the curve. In the second chapter, we study the area of the set of points around which the Brownian motion winds at least N times. We give an asymptotic estimation for this area, up to the second order, both in the almost sure sense and in the Lp spaces, when N goes to infinity.The third chapter is devoted to the proof of a result which shows that the points with large winding are distributed in a very balanced way along the trajectory. In the fourth chapter, we use the results from the two previous chapters to give a new Green formula for the Brownian motion. We also study the averaged winding around randomly distributed points in the plan. We show that, almost surely for the trajectory, this averaged winding converges in distribution, not toward a constant (which would be the Lévy area), but toward a Cauchy distribution centered at the Lévy area. In the last two chapters, we apply the ideas from the previous chapters to define and study the Lévy area of the Brownian motion, when the underlying area measure is not the Lebesgue measure anymore, but instead a random and highly irregular measure. We deal with the case of the Gaussian multiplicative chaos in particular, but the tools can be used in a much more general framework.
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Submitted on : Thursday, September 22, 2022 - 11:25:53 AM
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  • HAL Id : tel-03783516, version 1

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Isao Sauzedde. Windings of the planar Brownian motion and Green’s formula. Probability [math.PR]. Sorbonne Université, 2021. English. ⟨NNT : 2021SORUS437⟩. ⟨tel-03783516⟩

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