engleski

Homogenization of periodic diffusion with small jumps

In this talk, we will discuss the problem of homogenization of a class of diffusions with jumps, that is, Feller processes generated by an integro-differential operator of the following type \begin{; ; align*}; ; \mathcal{; ; A}; ; f(x)=& \frac{; ; 1}; ; {; ; 2}; ; {; ; \rm Tr}; ; \, C(x)D^{; ; 2}; ; f(x) +\int_{; ; \mathbb{; ; R}; ; ^{; ; d}; ; }; ; \left(f(y+x)- f(x)-\langle y, \nabla f(x)\rangle1_{; ; \{; ; z:|z|\leq1\}; ; }; ; (y)\right)\nu(x, dy).\end{; ; align*}; ; Under the assumptions that the underlying diffusion with jumps (i) has periodic coefficients, (ii) it admits only ``small jumps" (that is, $\sup_{; ; x\in\mathbb{; ; R}; ; ^{; ; d}; ; }; ; \int_{; ; \mathbb{; ; R}; ; ^{; ; d}; ; }; ; |y|^{; ; 2}; ; \nu(x , dy) <\infty$) (iii) and under certain additional regularity conditions, we show that the homogenized process is a Brownian motion. The presented results generalize the classical and well-known results related to the homogenization of diffusion processes.

diffusion with jumps ; homogenization ; semimartingale characteristics

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