Survival probability and position distribution of a run and tumble particle in $U(x)=\alpha |x|$ potential with an absorbing boundary

We study the late time exponential decay of the survival probability $S_\pm(t,a|x_0)\sim e^{-\theta(a)t}$, of a one-dimensional run and tumble particle starting from $x_0<a$ with an initial orientation $\sigma(0)=\pm 1$, under a confining potential $U(x)=\alpha|x|$ with an absorbing boundary at $x=a>0$. We find that the decay rate $\theta(a)$ of the survival probability has strong dependence on the location $a$ of the absorbing boundary, which undergoes a freezing transition at a critical value $a=a_c=(v_0-\alpha)\sqrt{v_0^2-\alpha^2}/(2\alpha\gamma)$, where $v_0>\alpha$ is the self-propulsion speed and $\gamma$ is the tumbling rate of the particle. For $a>a_c$, the value of $\theta(a)$ increases monotonically from zero, as $a$ decreases from infinity, till it attains the maximum value $\theta(a_c)$ at $a=a_c$. For $0<a<a_c$, the value of $\theta(a)$ freezes to the value $\theta(a)=\theta(a_c)$. We also obtain the propagator with the absorbing boundary condition at $x=a$. Our analytical results are supported by numerical simulations.