Statistics Seminar

Yinghui Shi
Jiangsu Normal University
Intrinsic Ultracontractivity of Laplacian and Fractional Laplacian Perturbed by Non-local Operator
Abstract: The intrinsic ultracontractivity (Abbr. I.U.) of the subprocesses $X_D^b$ of two kinds of special Markov processes $X^b$ upon leaving any bounded open set $D \subset \mathbb{R}^d$ will be given in this talk. Here the processes $X^b$ are associated with the operator $\mathcal{L}^b= \Delta^{\alpha/2} +\mathcal{S}^b$ with $d \geq 1$ and $0 < \beta < \alpha \leq 2$, where $$\mathcal{S}^bf(x):=\int_{\mathbb{R}^d}(f(x+z)-f(x)-\nabla f(x)\cdot z \mathbb{1}_{\{|z|\leq 1\}})\frac{b(x,z)}{|z|^{d+\beta}}dz$$ and $b(x,z)$ is a bounded Borel function on $\mathbb{R}^d\times \mathbb{R}^d$ with $b(x,z)=b(x,-z)$ for $x,z\in \mathbb{R}^d$. The operator $\mathcal{L}^b$ can be seen as the Laplacian ($\alpha = 2$) or the fractional Laplace operator ($0<\alpha <2$) with a lower order perturbation $\mathcal{S}^b$.
Our main results are proved under the frame of the I.U. for non-symmetric Levy processes. We discuss the transition density function for $X_D^b$ firstly and then we get its dual process under some reference measure. At last, we prove that I.U. stands under the conditions as follows: for any compact subset $K, L \subset\mathbb{R}^d$, $\inf_{x\in K}\inf_{z\in L}b(x,z)>0$ in the case of $\alpha=2$ and $\inf_{x\in K}\inf_{z\in L}(1+\frac{b(x,z)}{\mathcal{A}(d,\alpha)})|z|^{\alpha-\beta}>0$ in the case of $0<\alpha<2$. (Joint work with Yi, Bingji and Song, Renming)
Wednesday August 30, 2017 at 4:00 PM in SEO 636
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