I am reading a paper ('Hydrodynamics of the N-BBM Process', by De Masi, Ferrari, Presutti, Soprano-Loto) which quotes the 'Brownian representation formula' to represent the solution of a free boundary problem as: $$ \int_a^\infty u(r,t)dr = e^t \int \rho(r)\mathbb{P}(B_t^r > a, \tau^{r,L}>t)dr $$ (this is equation (6) in the paper). However I can find no reference to the 'Brownian representation formula' which shows this anywhere. Whilst it seems fully intuitive, it also seems like a non-trivial statement to prove. Does anyone know what result 'Brownian representation formula' refers to or where it can be found in the literature.
1 Answer
$\begingroup$ $\endgroup$
To me this looks like it follows from the standard Feynman-Kac formula, e.g. Theorem 3.7 in 'Continuous Time Markov Processes' by Liggett. I guess you could call this a Brownian motion representation of solutions.
They apply it, in Liggett's notation, with $h(x)=1$ and $f(x)=1_{x>a}$. Translating from Brownian starting in a fixed point (Liggett) to a random one with density $\rho$ gives you the integrals. The boundary condition $u(L_t, t) =0$ means that you actually have a killed Brownian motion which yields the $\{\tau^{r,L}>t\}$ on the right-hand side, formally this is hidden in the generator in Liggett.
