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arXiv 1303.3552math.PR
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Dirichlet生成元的Levy-Khintchine型表示与半Dirichlet型

Levy-Khintchine type representation of Dirichlet generators and Semi-Dirichlet forms

Wei Sun, Jing Zhang

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摘要：

设$U$是$\mathbb{R}^n$的开集，$m$是$U$上的正Radon测度，使得${\rm supp}[m]=U$，$(P_t)_{t>0}$是$L^2(U;m)$上的强连续收缩子马氏半群.我们研究了$(P_t)_{t>0}$的结构。(i)分别用$(A,D(A))$和$(\hat A,D(\hat A))$表示$(P_t)_{t>0}$的生成元和共生成元。在假设$C^{\infty}_0(U)\subset D(A)\cap D(\hat A)$的条件下，给出了$A$在$C^{\infty}_0(U)$上的显式Lévy-Khintchine型表示. (ii)若$(P_t)_{t>0}$是一个解析半群，并由此与一个半Dirichlet型$({\cal E}, D({\cal E}))$相关联，在假定$C^{\infty}_0(U)$.我们还提出了一般状态空间上正则半Dirichlet型的扩散部分的LeJan型变换规则。

英文摘要：

Let $U$ be an open set of $\mathbb{R}^n$, $m$ a positive Radon measure on $U$ such that ${\rm supp}[m]=U$, and $(P_t)_{t>0}$ a strongly continuous contraction sub-Markovian semigroup on $L^2(U;m)$. We investigate the structure of $(P_t)_{t>0}$. (i) Denote respectively by $(A,D(A))$ and $(\hat A,D(\hat A))$ the generator and the co-generator of $(P_t)_{t>0}$. Under the assumption that $C^{\infty}_0(U)\subset D(A)\cap D(\hat A)$, we give an explicit Lévy-Khintchine type representation of $A$ on $C^{\infty}_0(U)$. (ii) If $(P_t)_{t>0}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form $({\cal E}, D({\cal E}))$, we give an explicit characterization of ${\cal E}$ on $C^{\infty}_0(U)$ under the assumption that $C^{\infty}_0(U)\subset D({\cal E})$. We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

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