Solution

Separate the transported initial datum from the gain term. Define

F (f_{0}) (t, x, v) = e^{- t} f_{0} (x - t v, v),

(τ f) (t, x, v) = \int_{0}^{t} e^{- (t - s)} \int_{R^{d}} k (s, x - (t - s) v, v, v_{*}) f (s, x - (t - s) v, v_{*}) d v_{*} d s .

The Duhamel principle makes the linear Boltzmann equation equivalent, whenever the integrals and solution are legitimate, to

f = F (f_{0}) + τ f .

The gain samples velocities

v_{*}

at the backward spatial position of the outgoing velocity

v

; replacing that spatial shift by one depending on

v_{*}

would give a different equation.

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