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Etude probabiliste des équations de Smoluchowski ; Schéma d'Euler pour des fonctionnelles ; Amplitude du mouvement brownien avec dérive

Abstract : This thesis çonsists of three independent parts. The first part is a probabilist study of the Smoluchowski's coagulation equations. A representation of the solutions is established thanks to branching processes of type Galton-Watson. One shows besides a connection between additive and multiplicatif kernels. The asymptotique behavior of the solutions after renormalisation is also studied. Finally, one builds a process, a solution of a non-linear S.D.E. governed by a Poisson process, temporal of which marginal are solutions of the eqnations of Smoluchowski. This process allows to obtain estimates with a particles system. In the second part, we estimate the error committed by replacing a regular diffusion by its estimate obtained with Euler's scheme to compute the expectation of sorne irregular functional of the trajectory of this diffusion. We notably obtain the optimal speed of convergence in the case of the integral of a function only measurable and bounded of the trajectory. . In the third part, we study the process of the range of a brownian motion with drift. We give a decomposition of trajectories by using the successive extremums by "raising" the time. The results are notably obtained by means of techniques of filtrations enlargement.
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Theses
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http://docnum.univ-lorraine.fr/prive/SCD_T_2001_0178_TANRE.pdf
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https://hal.univ-lorraine.fr/tel-01746627
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Submitted on : Thursday, March 29, 2018 - 10:41:31 AM
Last modification on : Tuesday, December 15, 2020 - 12:26:45 PM

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Etienne Tanré. Etude probabiliste des équations de Smoluchowski ; Schéma d'Euler pour des fonctionnelles ; Amplitude du mouvement brownien avec dérive. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2001. Français. ⟨NNT : 2001NAN10178⟩. ⟨tel-01746627⟩

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