  We prove that linear and weakly non-linear run and tumble equations converge
to a unique steady state solution with an exponential rate in a weighted total
variation distance. In the linear setting, our result extends the previous
results to an arbirtary dimension $d \geq 1$ while relaxing the assumptions.
The main challenge is that even though the equation is a mass-preserving,
Boltzmann-type kinetic-transport equation, the classical $L^2$ hypocoercivity
methods, e.g., by Dolbeault, Mouhot, Schmeiser (Tans. Amer. Math. Soc.,
367(6):3807-3828, 2015) are not applicable for dimension $d >1$. We overcome
this difficulty by using a probabilistic technique, known as Harris's theorem.
We also introduce a weakly non-linear model via a non-local coupling on the
chemoattractant concentration. This toy model serves as an intermediate step
between the linear model and the physically more relevant non-linear models. We
build a stationary solution for this equation and provide a hypocoercivity
result.
