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A linear kinetic transport equation with velocity-integral gain $K_t$ and nonnegative loss coefficient $a$. Its characteristic integral formulation combines the damped free-transport evolution with the Boltzmann Volterra operator. A uniform square-integrable velocity kernel yields bounded gain on $L_{x,v}^2$ and a globally convergent Volterra series for the linear Boltzmann equation on every finite time interval.

Denote the damped free-transport evolution from time $s$ to time $t$ byBy the argument in (b), $\Vert U_a(t,s)g{\Vert }_2\le \Vert g{\Vert }_2$. The integral operator is the Boltzmann Volterra operatorThe Minkowski integral inequality and (a) give the useful stronger pointwise boundTherefore the requested estimate isFor strongly measurable bounded $L^2$-valued $f$, these are Bochner integrals; their finite norm bounds establish existence of the integrals.

Work in the Banach space ${\mathcal{B}}_T$ of bounded strongly measurable maps $[0,T]\to L_{x,v}^2$, with norm $\Vert g{\Vert }_{{\mathcal{B}}_T}={\sup }_{0\le t\le T}\Vert g(t){\Vert }_2$. Keeping actual representatives at every time matches the pointwise-in-time mild formulation. The Boltzmann Volterra operator is bounded on this space, and the factorial bound for a Volterra iterate givesThus the Volterra series for the linear Boltzmann equation converges in operator norm for every finite $T$, even when $CT\ge 1$. SetFor its partial sums, $(I-\tau ){\sum }_{n=0}^N{\tau }^{n}F=F-{\tau }^{N+1}F$. The remainder tends to zero by the factorial estimate. Hence $(I-\tau )f=F$, precisely the required characteristic integral equation.

The norm bound in (b) gives an explicit choice of the existence constant:Also $f(0)=f_0$, since every term with $n\ge 1$ vanishes at zero and the damping interval has length zero. This proves existence in the paper's weak, characteristic-integral sense. With merely measurable nonnegative $a$, that sense does not itself require a continuous initial trace; that trace follows under the additional local characteristic-integrability condition described in (b).