Reflected BSDE and Locally Periodic Homogenization of Semilinear PDEs with Nonlinear Neumann Boundary Condition

We study the homogenization of semilinear partial differential equations (PDEs) with nonlinear Neumann boundary condition, locally periodic coefficients, and highly oscillating drift and nonlinear term. Our method is entirely probabilistic, as in a periodic case by Ouknine and Pardoux [14 Ouknine , Y. , and Pardoux , É. 2002 . Homogenization of PDEs with non linear boundary condition, Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progresses of Probability, 52, Birkhäuser, Basel , pp. 229 – 242 . [Google Scholar]] and builds on our earlier work [5 Diakhaby , A. , and Ouknine , Y. 2006 . Locally periodic homogenization of reflected diffusion . Journal of Applied Mathematics and Stochastic Analysis . [Google Scholar]], which gives us the locally periodic counterpart of Theorem 2.2 in Tanaka [21 Tanaka , H. 1984 . Homogenization of diffusion processes with boundary conditions . Stochastic Analysis and Applications 7 : 411 – 437 . Advanced Probability and Related Topics 7, Dekker, New York . [Google Scholar]].     

Sur quelques résultats de convergence dans l'inférence d'une classe de processus ponctuelsConvergence results for point processes estimators

Our aim is to generalize some results obtained for a Poisson point process in [7], to a general point process. Those results are in field of complete convergence of two like Parzen–Rosenblatt estimates of density of mean measure function and regression curves. Those estimates are defined from the superposition of n i.i.d. point processes as:

$\text{f}{}_{\mathrm{n}}\text{(x)=}\text{1}\text{nh}\text{∑}\text{i=1}\text{m}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{and}\text{Ψ}{}_{\mathrm{n}}\text{(x)=}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{Y}{}_{\mathrm{i}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{∑}{}_{\mathrm{i=1}}{}^{\mathrm{m}}\text{K}\text{x?X}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{h(n)}\text{,}$

where m is the number of seem generics points of the superposition. We give some sufficient conditions for the convergence of those kernel-like estimators. To cite this article: A. Diakhaby, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 597–602.