Dans cet article, nous pénalisons la loi d’une araignée brownienne ${\left(A_t\right)}_{t\ge 0}$ prenant ses valeurs dans un ensemble fini $E$ de demi-droites concourantes, avec un poids égal à $\frac{1}{Z_t}exp({\alpha }_{N_t}{X}_t+\gamma L_t)$, où $t$ est un réel positif, ${\left({\alpha }_k\right)}_{k\in E}$ une famille de réels indexés par $E$, $\gamma$ un paramètre réel, $X_t$ la distance de $A_t$ à l’origine, $N_t$ ($\in E$) la demi-droite sur laquelle se trouve $A_t$, $L_t$ le temps local de ${\left(X_s\right)}_{0\le s\le t}$ à l’origine, et $Z_t$ la constante de normalisation. Nous montrons que la famille des mesures de probabilité obtenue par ces pénalisations converge vers...

In this article we tackle the problem of inverse non linear ill-posed problems from a statistical point of view. We discuss the problem of estimating an indirectly observed function, without prior knowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show...

In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of $(N,d)$-Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the Brownian...