论文标题
具有无限漂移的超临界SDE的加热内核
Heat kernel of supercritical SDEs with unbounded drifts
论文作者
论文摘要
令$α\ in（0,2）$和{\ Mathbb n} $中的$ d \。请考虑以下在$ {\ Mathbb r}^d $：$$ {\ rm d} x_t = b（t，x_t）{\ rm d} t+a（t，x_ {t - }）{\ rm d} lm d} l^{（α）} _ t，t， a $ d $ - 二维旋转不变$α$稳定过程，$ b：{\ Mathbb r} _+\ times {\ Mathbb r}^d \ to {\ Mathbb r}^d $ and $ a：{\ Mathb r} r}^d \ otimes {\ mathbb r}^d $是h {Ö} lder在太空中的连续函数，分别订单$β，γ\ in（0,1）$，使得$（β\wedgeγ）+α> 1 $，在$ t $中均匀地均匀。 在这里$ b $可能是无限的。我们还为对数衍生物建立了锋利的上限。特别是，我们涵盖了（0,1）$的整个超临界范围$α\。您的证明是基于临时参数扩展和概率技术。
Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X_t(x)$ of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range $α\in (0,1) $.Our proof is based on ad hoc parametrix expansions and probabilistic techniques.
