(P)$ and with $b=0$.

\[H3\] We assume that $P_{t}(x,dy)=p_{t}(x,y)dy$ with $p_{t}\in
C^{\infty }({\mathbb{R}}^{d}\times {\mathbb{R}}^{d})$ and the blow up of $%
p_{t}\rightarrow \infty $ as $t\rightarrow 0$ is controlled in the following way. For every fixed $q\in {\mathbb{N}},\kappa \geq 0$ there exist some constants $C\geq 1,0<\lambda \leq 1$ and $\eta >0$ such that for every two multi-indexes $\alpha ,\beta $ with $\left\vert \alpha \right\vert
+\left\vert \beta \right\vert \leq q$$$(H_{3})\quad \left\vert \partial _{x}^{\alpha }\partial _{y}^{\beta
}p_{t}(x,y)\right\vert \leq \frac{C}{(\lambda t)^{\theta _{0}(q+\theta _{1})}%
}\times \frac{\psi _{\eta }(x)}{\psi _{\kappa }(x-y)}  \label{J6c}$$Here $\theta _{i}\geq 0,i=0,1$ are some fixed parameters.

We construct now the perturbed semigroup. We consider a Poisson process $%
N(t) $ of parameter $\rho >0$ and we denote by $T_{k}$, $k\in {\mathbb{N}}$, its jump times. We also consider a sequence $Z_{k}$, $k\in {\mathbb{N}}$, of independent random variables of law $\nu ,$ on a measurable space $(E,%
\mathcal{E}),$ which are independent of $N(t)$ as well. Moreover we take a function $\phi :E\times {\mathbb{R}}^{d}\rightarrow {\mathbb{R}}^{d}$ and we denote $\phi _{z}(x)=\phi (z,x).$ We will precise in a moment the hypothesis on $\phi .$ We associate the operator$$U_{z}f(x)=f(\phi _{z}(x))  \label{J1}$$and we define $\overline{P}_{t}$ to be the perturbation of $P_{t}$ in the following way. Conditionally to $T_{k}$ and $Z_{k},k\in {\mathbb{N}}$ we define$$\begin{array}{ll}
P_{t}^{N,Z}=P_{t} & \quad \mbox{for}\quad t<T_{1}, \\ 
P_{T_{k}}^{N,Z}=U_{Z_{k}}P_{T_{k}-}^{N,Z}, &  \\ 
P_{t}^{N,Z}=P_{t-T_{k}}P_{T_{k}}^{N,Z} & \quad \mbox{for}\quad T_{k}\leq
t<T_{k+1}%
\end{array}
\label{J2}$$The second equality reads $P_{T_{k}}^{N,Z}f(x)=P_{T_{k}-}^{N,Z}f(\phi
_{Z_{k}}).$ Essentially (\[J2\]) means that on $[T_{k-1},T_{k})$ we have a process which evolves according to the semigroup $P_{t}$ and at time $T_{k}$ it jumps according to $\phi _{Z_{k}}$. Then we define$$\overline{P