Etude de processus de diffusion

Abstract : The first part contains a study of a large deviation phenomenon. Our approach generalizes the result of Freidlin and Wentzell. Here we emphases the asymptotic behaviour of a solution of a stochastic differential equation which diffusion coefficient tends to zero, and which associated ordinary differential equation satisfies a Peano phenomenon. Proofs are based on probabilistic (large deviations theory) and analytic (viscosity solutions for Hamilton-Jacobi equations) tools. In the second part, we study a system of self-stabilizing processes. This system is obtained by taking the limit in a system of two kinds of particules. These particules satisfy stochastic differential equations. Moreover, two particules of the same family attract each other and a particule pushes a particule of the other family away. We let the number of particule tend to infinity and prove that the system admits an unique solution which converges to the stationnary distribution as the time is getting large. In the last part, we study a stochastic differential equation with long memory. The drift of this diffusion, which depends on ail the past, attracts the process to the place where it has spent the most time. We prove then that the behaviour of this process, when the time tends to infinity, depends on the behaviour of the interaction function in the neighbourhood of the origin. In some particular cases the process converges almost surely, in other cases the process does not converge but remains bounded. Proofs are based on comparison results.
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Samuel Herrmann. Etude de processus de diffusion. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2001. Français. ⟨NNT : 2001NAN10026⟩. ⟨tel-01748134⟩

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