On smoothing properties of transition semigroups associated to a class of SDEs with jumps
On smoothing properties of transition semigroups associated to a class of SDEs with jumps
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DOI：
10.1214/13-aihp559
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发表时间：
2012-08
影响因子：
1.5
通讯作者：
S. Kusuoka;Carlo Marinelli
中科院分区：
文献类型：
--
作者：
S. Kusuoka;Carlo Marinelli
证明了$\mathbb{R} ^{d}$中一类由加性纯跳Levy噪声驱动的随机微分方程的非局部迁移半群的光滑性质.特别地，我们假设驱动该过程的Levy过程是从属Wiener过程$Y$的和。（即，$Y=W\circ T$，其中$T$是从零开始的递增纯跳Levy过程，并且独立于Wiener过程$W$）和独立于$Y$的任意Levy过程，漂移系数是连续的（但不一定是Lipschitz连续的），增长速度不超过多项式，并且该多项式允许Feller弱解。利用概率和分析相结合的方法，给出了与该半群相关联的马氏半群是强Feller的充分条件以及L_{p}（\mathbb{R} ^{d}）映射到连续有界函数的充分条件.一个关键的中间步骤是正则化性质的研究的过渡半群与$Y$的负时刻的从属$T$。
We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $\mathbb{R} ^{d}$ driven by additive pure-jump Levy noise. In particular, we assume that the Levy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y=W\circ T$, where $T$ is an increasing pure-jump Levy process starting at zero and independent of the Wiener process $W$) and of an arbitrary Levy process independent of $Y$, that the drift coefficient is continuous (but not necessarily Lipschitz continuous) and grows not faster than a polynomial, and that the SDE admits a Feller weak solution. By a combination of probabilistic and analytic methods, we provide sufficient conditions for the Markovian semigroup associated to the SDE to be strong Feller and to map $L_{p}(\mathbb{R} ^{d})$ to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to $Y$ in terms of negative moments of the subordinator $T$.