  We investigate a linear kinetic equation derived from a velocity jump process
modelling bacterial chemotaxis in the presence of an external chemical signal
centered at the origin. We prove the existence of a positive equilibrium
distribution with an exponential decay at infinity. We deduce a hypocoercivity
result, namely: the solution of the Cauchy problem converges exponentially fast
towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot,
and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass,
Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to
the null spaces of the collision operator and of the transport operator. From a
modelling viewpoint it is related to the observation that exponential
confinement is generated by a spatially inhomogeneous bias in the velocity jump
process.
