Position distribution in a generalized run-and-tumble process

We study a class of stochastic processes the type $\frac{d^n x}{dt^n}= v_0\, \sigma(t)$ where $n>0$ is positive integer and $\sigma(t)=\pm 1$ represents an `active' telegraphic noise that flips from one state to other with constant rate $\gamma$. For $n=1$, it reduces standard run tumble process for active particles in dimension. This can be analytically continued any including non-integer values. compute exactly mean squared displacement at time $t$ all show late times while grows as $\sim t^{2n-1}$ $n>1/2$, approaches $n<1/2$. In marginal case $n=1/2$, very slowly \ln t$. Thus undergoes {\em localisation} transition $n=1/2$. also position distribution $p_n(x,t)$ remains time-dependent even $n\ge 1/2$, but stationary time-independent form The tails exhibit large deviation form, $p_n(x,t)\sim \exp\left[-\gamma\, t\, \Phi_n\left(\frac{x}{x^*(t)}\right)\right]$, $x^*(t)= t^n/\Gamma(n+1)$. function $\Phi_n(z)$ numerically using importance sampling methods, finding excellent agreement between them. three special values $n=2$ $n=1/2$ we exact cumulant generating $t$.