I am going through a paper called "The KPP Equation as a Scaling Limit of Locally Interacting Brownian Particles", https://arxiv.org/abs/2101.01031v2. We construct a process by defining a generator and then apply Ito's Lemma to a function of the process. My question is how. We don't have an SDE which the process solves (or at least I don't see it), so I don't understand how we can apply Ito's Lemma. The exact situation is as follows:
We consider a process of particles moving in $\mathbb R^d$ according to Brownian Motion with given birth and death rates. A configuration of the process is a sequence $\eta = (x_n, a_n)_{n\in \mathbb N}\in (\mathbb R^d \times \{L,N\})^\mathbb N$, where $a_n$ is $L$ if the corresponding particle is alive, and $N$ if it is not. We add the constraint, that the number of live particles is always finite. All sums in the following are over the number of live particles, so they are all finite sums.
We consider test functions $F$ which depend only on a finite number of coordinates and are smooth in their position and define the infinitesimal generator, denoting by $\Delta_x$ the second derivative in $x$ $$ (\mathcal L F)(\eta)=\frac 1 2\sum_j \Delta_{x_j}F(\eta)+\sum_j\left( F(\eta^j)-F(\eta)\right) + \sum_{j,k} \theta(x_j-x_k)\left(F(\eta^{-j})-F(\eta)\right), $$ where $\eta^j$ signifies the birth of a particle at position $x_j$, while $\eta^{-j}$ signifies deleting particle $j$. $\theta$ is a given, non-negative, Hölder continiuous and compactly supported function with $\int \theta =1$. The support of $\theta$ models a neighborhood of $x_j$. The deathe rate of a particle at $x_j$ is proportional to the number of particles in a neighborhood of $x_j$.
Is it already clear from this how we can apply Ito's Lemma to a function of $\eta(t)$? Is it clear what SDE the process $\eta(t)$ solves? The first sum of course looks very much like the generator of Brownian Motion but what about the two thereafter?
Specifically, the paper goes on to define $Q(t,\eta):=\sum_j\phi(t,x_j)$ for a given $\phi \in C_c^{1,2}([0,T)\times \mathbb R^d)$ and claims that by the Ito Lemma applied to $Q(t,\eta(t))$ we have $$ Q(T,\eta(T))-Q(0,\eta(0))=\int_0^T\sum_j (\partial_t +\frac 1 2 \Delta_{x_j}) \phi(t,x_j(t))dt \\ + \int_0^T\sum_j (1-\sum_k \theta(x_j(t)-x_k(t))\phi(t,x_j(t))dt +M_T $$ for some martingale $M$.
How do we get this? Thanks in advance for any help.