Local time for run and tumble particle

We investigate the local time $(T_{loc})$ statistics for a run and tumble particle in an one dimensional inhomogeneous medium. The inhomogeneity is introduced by considering position dependent rate of form $R(x) = \gamma \frac{|x|^{\alpha}}{l^{\alpha}}$ with $\alpha \geq 0$. For =0$, we derive probability distribution $T_{loc}$ exactly which expressed as series $\delta$-functions coefficients can be interpreted multiple revisits to origin starting from origin. general $\alpha$, show that typical fluctuations scale $T_{loc} \sim t^{\frac{1+\alpha}{2+\alpha}}$ large $t$ their possesses scaling behaviour described function have computed analytically. In second part, study till RTP makes first passage $x=M~(>0)$. this case also, sum all values $\alpha~(\geq 0)$ appearing appropriate exit problems. All our analytical findings are supported numerical simulations.