Pénalisations de marches aléatoires

Pierre Debs 1, 2
2 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : The subject of my thesis is the theory of penalisation originaly developed by B .Roynette, P. Vallois and M. Yor in the case of the brownian motion. In a few words, it consists in putting a weight on the probability measure to favorise trajectories with probability measure equals to zero.
The first part of my thesis is the discrete counterpart of their work:
let $\left(\Omega,\,\left(X_n,\,\mathcal F_n,\,n\geq0\right),\mathcal F_\infty=\bigvee_{n\geq0}\mathcal F_n,\,\p\right)$ the symmetric random walk where $\mathcal F_n$ is the canonical filtration.
For some adapted and positive functionals $G:\mathbb N\times\Omega\time\Omega\rightarrow\mathbb R^+$, I study $\forall n\in\mathbb N,\,\forall\Lambda_n\in\mathcal F_n$, the limit when $p\rightarrow\infty$ of the quantity:
\begin{equation*}
\frac{\e_x[\mathds{1}_{\Lambda_n}G_p]}{\e_x[G_p]}
\end{equation*}
When this limit exists, it is equal to $Q\left(\Lambda_n\right):=\e_x[\mathds{1}_{\Lambda_n}M_n]$ where $\left(M_n,n\geq0\right)$ is a positive non uniformly integrable martingale. The definition of $Q$ induces a new probability on $\left(\Omega,\,\mathcal F_\infty\right)$ and then I study $\left(X_n,n\geq0\right)$ under $Q$.
In a second part, I try to expend this theory to birth and death Markov processes. Recall that these processes have the property that, after an exponential random length of time, only transitions to neighbouring states are possible. Precisely, I penalize the distribution of the transient birth and death process by the number of visits at the state 0 (which is like local time type penalization). When I force the process to visit an infinitely often the state zero %it is reasonable to think that what I get is a recurrent chain. Indeed, I prove that, under the new probability measure induced by penalization, the process behaves as a recurrent birth and death process.
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Pierre Debs. Pénalisations de marches aléatoires. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2007. Français. ⟨NNT : 2007NAN10091⟩. ⟨tel-01748273v2⟩

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