论文标题
过渡半群的梯度公式，对应于由独立莱维过程系统驱动的随机方程
Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes
论文作者
论文摘要
令$（p_t）$是Markov家族的过渡半群$（x^x（t））$由Sde $$ d x = b（x）dt + d z，\ qquad x（0）= x，$ z = \ lest（z_1，z__1，\ ldots，z__d \ right）^*$是独立的lul lul lul lul lul lul lul lul lul lul lul。使用malliavin conculus，我们建立以下渐变公式$$ \ nabla p_tf（x）= \ mathbb {e} \，f \ left（x^x（x x（t）\ right）y（t，x），\ qquad f \ in b_b in b_b（\ mathbb {r} r}^d in $ y $ y $还给出了$ \ nabla p_tf（x）$的尖锐估计，当$ z_1，\ ldots，z_d $为$α$ - 稳定过程，$α\ in（0,2）$时。
Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$ d X= b(X) dt + d Z, \qquad X(0)=x, $$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula $$ \nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\right) Y(t,x), \qquad f\in B_b(\mathbb{R}^d), $$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\nabla P_tf(x)$ when $Z_1, \ldots , Z_d$ are $α$-stable processes, $α\in (0,2)$, are also given.
