Let $\{X_t^{(\mu)},t\ge 0\}$ be a L\'evy process on $R^d$ whose distribution at time 1 is $\mu$, and let $f$ be a nonrandom measurable function on $(0, a), 0 < a\leq \infty$. Then we can define a mapping $\Phi_f(\mu)$ by the…

Let $\{X_t^\mu,t\geq0\}$ be a Lévy process on $\mathbb{R}^d$ whose distribution at time $1$ is a $d$-dimensional infinitely distribution $\mu$. It is known that the set of all infinitely divisible distributions on $\mathbb{R}^d$, each of which is represented by the law…