In this work we consider a finite dimensional stochastic differential equation(SDE) driven by a Lévy noise L(t)=Lt, t>0. The transition probability density pt, t>0 of the semigroup associated to the solution ut, t⩾0 of the SDE is given by a power series expansion. The series expansion of pt can bere-expressed in terms of Feynman graphs and rules. We will also prove that pt, t>0 has an asymptotic expansion in power of a parameter β>0, and it can be given by a convergent integral. A remark on some applications will be given in this work.

中文翻译：

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与Lévy噪声驱动的SDE相关的半群过渡密度的渐近展开

在这项工作中，我们考虑由Lévy噪声L（t）= Lt，t> 0驱动的有限维随机微分方程（SDE）。与SDE的解ut，t⩾0相关的半群的转移概率密度pt，t> 0由幂级数展开给出。可以根据Feynman图和规则重新表达pt的级数展开。我们还将证明，pt，t> 0的幂幂为β> 0的渐近展开，并且可以由收敛积分给出。在这项工作中将对某些应用程序进行说明。