We establish a large deviation theory for a velocity jump process, where new
random velocities are picked at a constant rate from a Gaussian distribution.
The Kolmogorov forward equation associated with this process is a linear
kinetic transport equation in which the BGK operator accounts for the changes
in velocity. We analyse its asymptotic limit after a suitable rescaling
compatible with the WKB expansion. This yields a new type of Hamilton Jacobi
equation which is non local with respect to velocity variable. We introduce a
dedicated notion of viscosity solution for the limit problem, and we prove
well-posedness in the viscosity sense. The fundamental solution is explicitly
computed, yielding quantitative estimates for the large deviations of the
underlying velocity-jump process {\em \`a la Freidlin-Wentzell}. As an
application of this theory, we conjecture exact rates of acceleration in some
nonlinear kinetic reaction-transport equations