论文标题
Hölder规律性和梯度估计由圆柱$α$稳定过程驱动的SDE的Hölder规律性和梯度估计值
Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $α$-stable processes
论文作者
论文摘要
我们建立了Hölder的规律性和梯度估算的解决方案的过渡分组到以下sde：$$ {\ rm d} x_t =σ（t，x_ {t-}）{\ rm d} z_t} z_t+b（t，t+b（t，t，x_t，x_t）{\ rm d} {\ rm d} $$ $（z_t）_ {t \ geq 0} $是$ d $ - 二维圆柱$α$ - 稳定过程，$α\ in（0，2）$，$σ（t，x）：{\ \ mathbb r} r}^d \ otimes {\ mathbb r}^d $是有限的，可以测量，统一的非排效和Lipschitz在$ x $中连续$ x $在$ t $中，$ b（t，x）：{\ mathbb r} $β$-Hölder连续$ x $在$ t $中连续$ t $，$β\在[0,1] $满足$α+β> 1 $的[0,1] $中。此外，我们还显示了$ x（t，x）$的分销密度的存在和规律性。我们的证明是基于Littlewood-Paley的理论。
We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $α$-stable process with $α\in (0, 2)$, $σ(t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $β$-Hölder continuous in $x$ uniformly in $t$ with $β\in[0,1]$ satisfying $α+β>1$. Moreover, we also show the existence and regularity of the distributional density of $X (t, x)$. Our proof is based on Littlewood-Paley's theory.
