Homogenization of cadlag processes
Let \(L\) be a \(d\)-dimensional Lévy type operator: \[ \begin{multlined} Lf(x)={1\over 2}\sum_{i,j=1}^ d a_{ij}(x)\partial_{x_ i}\partial{x_ j}f(x)+\sum_{i=1}^ d b_ i(x)\partial_{x_ i}f(x)+\\ +\int_{\mathbb{R}^ d}\left\{f(x+y)-f(x)-\sum_{i=1}^ d y_ i\partial_{x_ i}f(x)\right\} \nu(x,dy), \end{multlined} \] where \(\partial_{x_ i}=\partial/\partial x_ i,\) \(a(x)=(a_{ij}(x))\) is a nonnegative definite symmetric \(d\times d\) matrix, \(b(x)=(b_ i(x))\) is a \(d\)-vector, and \(\nu(x,dy)\) is a Lévy measure on \(\mathbb{R}^ d\) for each \(x\in\mathbb{R}^ d\). Denote by \(\{X^ L(t)\}\) a cadlag process on \(\mathbb{R}^ d\) governed by \(L\). We consider a homogenization problem associated with \(\{X^ L(t)\}\), under the condition of periodicity of \(a(x)\), \(b(x)\) and \(\nu(x,dy)\) in \(x\) and some additional condition. The essential assumption is that there exists the limit Lévy measure \(\nu^*(\cdot)=\lim_{\varepsilon\downarrow 0}\int_{\mathbb{T}^ d}\nu^ \varepsilon(x,\cdot)\mu(dx)\), where \(\mu\) is the invariant probability measure of the cadlag process on \(\mathbb{T}^ d\) governed by \(L\), and \(\nu^ \varepsilon(x,\cdot)=\nu(x,\cdot/\varepsilon)/\varepsilon^ \alpha K(1/\varepsilon)\) for some \(\alpha\in(1,2)\) and slowly varying function \(K\). Let \(\{X^{L^ \varepsilon}(t)\}\) and \(\{X^{L^*}(t)\}\) be the cadlag processes governed by \(L^ \varepsilon\) and \(L^*\), respectively: \[ \begin{aligned} L^ \varepsilon f(x) &= {1\over 2}{\varepsilon^{2-\alpha} \over K(1/\varepsilon)} \sum_{i,j=1}^ d a_{ij}(x/\varepsilon)\partial_{x_ i}\partial_{x_ j}f(x)+ {\varepsilon^{1-\alpha} \over K(1/\varepsilon)}\sum_{i=1}^ d b_ i (x/\varepsilon) \partial_{x_ i}f(x)+\\ &+ \int_{\mathbb{R}^ d}\left\{f(x+y)-f(x)- \sum_{i=1}^ d y_ i\partial_{x_ i}f(x)\right\} \nu^ \varepsilon (x/\varepsilon,dy),\\ L^* f(x) &= \int_{\mathbb{R}^ d}\left\{f(x+y)-f(x)-\sum_{i=1}^ d y_ i\partial_{x_ i}f(x) \right\}\nu^*(dy). \end{aligned} \] Let \(P_ x^ \varepsilon\) and \(P^*_ x\) be the probability measure on \(W\equiv D([0,\infty)\to\mathbb{R}^ d)\) induced by \(\{X^{L^ \varepsilon}(t)\}\) and \(\{X^{L^*}(t)\}\) starting at \(x\), respectively. Then \(P^ \varepsilon_ x\) converges to \(P^*_ x\) as \(\varepsilon\downarrow 0\).
scientific article; zbMATH DE number 4216361