Random time changes of Feller processes

We show that the SDE $d{X}_t=\sigma (X_{t-})d{L}_t$, $X_0\sim \mu$ driven by a one-dimensional symnmetric $\alpha$-stable L'evy process $(L_t){t \geq 0}$,$\alpha \in (0,2]$,hasauniqueweaksolutionforanycontinuousfunction$\sigma: $\mathbb{R}$ \to (0,\infty)$ which grows at most linearly. Our approach relies on random time changes of Feller processes. We study under which assumptions the random-time change of a Feller process is a conservative $C_b$-Feller process and prove the existence of a class of Feller processes with decomposable symbols. In particular, we establish new existence results for Feller processes with unbounded coefficients. As a by-product, we obtain a sufficient condition in terms of the symbol of a Feller process $(X_t){t \geq 0}$fortheperpetualintegral$\int_{(0,\infty)} f(X_{s}) , ds$ to be infinite almost surely.