论文标题
一般稳定的马尔可夫工艺的建筑和热内核估计
Construction and heat kernel estimates of general stable-like Markov processes
论文作者
论文摘要
一个稳定的过程是一个陷入困境的过程$（x_t）_ {t \ geq 0} $在$ \ mathbb {r}^d $中取值，并且其发电机在本地的行为，例如$α$ stablelévy流程，但是index $α$和其他所有特征都可能取决于状态空间。更确切地说，跳跃措施不必是对称的，这在很大程度上取决于过程的当前状态。此外，我们不需要梯度术语由纯跳跃部分主导。我们的方法是将上述现象视为合适的微结构扰动。 我们表明，相应的马丁纳尔问题是良好的，它的解决方案是一个强大的砍伐过程，可以承认过渡密度。对于过渡密度，我们获得表示为明确给定的主项的总和 - 这本质上是$α$ - 稳定的随机变量的密度，其参数取决于当前状态$ x $ - 和残差项； $ l^\ infty \ otimes l^1 $ - 剩余术语可忽略不计，并且在额外的结构假设下，$ l^\ infty \ otimes l^\ infty $ norm。具体示例说明了假设与可能的过渡密度估计之间的关系。
A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in $\mathbb{R}^d$ and whose generator behaves, locally, like an $α$-stable Lévy process, but the index $α$ and all other characteristics may depend on the state space. More precisely, the jump measure need not to be symmetric and it strongly depends on the current state of the process; moreover, we do not require the gradient term to be dominated by the pure jump part. Our approach is to understand the above phenomena as suitable microstructural perturbations. We show that the corresponding martingale problem is well-posed, and its solution is a strong Feller process which admits a transition density. For the transition density we obtain a representation as a sum of an explicitly given principal term -- this is essentially the density of an $α$-stable random variable whose parameters depend on the current state $x$ -- and a residual term; the $L^\infty\otimes L^1$-norm of the residual term is negligible and so is, under an additional structural assumption, the $L^\infty\otimes L^\infty$-norm. Concrete examples illustrate the relation between the assumptions and possible transition density estimates.
