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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