  We investigate the run and tumble particle (RTP), also known as persistent
Brownian motion, in one dimension. A telegraphic noise $\sigma(t)$ drives the
particle which changes between $\pm 1$ values with some rates. Denoting the
rate of flip from $1$ to $-1$ as $R_1$ and the converse rate as $R_2$, we
consider the position and direction dependent rates of the form
$R_1(x)=\left(\frac{\mid x \mid}{l}\right)
^{\alpha}\left[\gamma_1~\theta(x)+\gamma_2 ~\theta (-x)\right]$ and
$R_2(x)=\left(\frac{\mid x \mid}{l}\right)
^{\alpha}\left[\gamma_2~\theta(x)+\gamma_1 ~\theta (-x)\right]$ with $\alpha
\geq 0$. For $\gamma_1 >\gamma_2$, we find that the particle exhibits a
steady-state probability distriution even in an infinite line whose exact form
depends on $\alpha$. For $\alpha =0$ and $1$, we solve the master equations
exactly for arbitrary $\gamma_1$ and $\gamma_2$ at large $t$. From our explicit
expression for time-dependent probability distribution $P(x,t)$ we find that it
exponentially relaxes to the steady-state distribution for $\gamma_1 >
\gamma_2$. On the other hand, for $\gamma_1<\gamma_2$, the large $t$ behaviour
of $P(x,t)$ is drastically different than $\gamma_1=\gamma_2$ case where the
distribution decays as $t^{-\frac{1}{2}}$. Contrary to the latter, detailed
balance is not obeyed by the particle even at large $t$ in the former case. For
general $\alpha$, we argue that the approach to the steady state in
$\gamma_1>\gamma_2$ case is exponential which we numerically demonstrate....
