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Homogénéisation pour des processus associés à des frontières perméables. (French) Zbl 0549.60070

The author studies the stochastic processes with penetrable boundaries in the plane [cf. B. Grigelionis and R. Mikulyavichyus, Mathematical statistics and probability theory, Lect. Notes Stat. 2, 152- 169, (1980; Zbl 0446.62099), see also Litov. Mat. Sb. 20, No.2, 27-40 (1980; Zbl 0448.60037)]. The main result for the boundaries with lines is that if \(\delta_ n(y)\) is a continuous function with period 1 and if \(n(1-\delta_ n(y))\) converges uniformly to \(\alpha(y)\) (\(\leq\) some constant) on [-\({1\over2}\),\({1\over2}]\) as \(n\to\infty \), then the sequence \(P^ n\), which is the unique solution of the following martingale problem
(*) \(P^ n(X_ 0=0,\quad Y_ 0=0)=1,\quad\int^{t}_{0}1_{X_ s\in Z^ n}ds=0\quad P^ n\) a.s., where \(Z^ n=\{x^ k_ n=k/n;\quad k\in {\mathbb{Z}}\},\)
(**) For any \(f(x,y)\) of class \(C^ 2\) \[ \begin{split} f(X_ t,Y_ t)-f(0,0)-2^{- 1}\int^{t}_{s}\Delta f(X_ s,Y_ s)a(nX_ s,nY_ s)ds- \\ -2^{- 1}\sum_{k\in Z}\int^{t}_{0}(f'_ x(k^+n^{-1},\gamma_ s)- \delta_ n(nY_ s)\cdot f'_ x(k^{-}n^{-1},\gamma_ s))dL_ s^{k/n} \end{split} \] is \(P^ n\)-martingale, converges weakly to \(P_{(0,0)}\) which is associated with the Markov process generated by \[ L: \quad Lf(x,y)=2^{-1}K[\Delta f(x,y)+(\int_{\pi 2({\mathcal T})}\alpha (y)dy)\cdot f'_ x(x,y)]. \] Another result concerns the case of circle boundaries.
Reviewer: C.Wu

MSC:

60J60 Diffusion processes
60F05 Central limit and other weak theorems
60G35 Signal detection and filtering (aspects of stochastic processes)
60G46 Martingales and classical analysis
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