I'm reading this article , where an (S)ODEs system is presented:
$d \varphi_j (t) = - \frac{K}{N} \sum_{i=1}^{N} \sin(\varphi_j(t)-\varphi_i(t))dt+\sigma d\omega_j(t)$,
Page 3, there's written that the generator of the dynamics is:
$L_{K,N} F(\varphi) = \frac{\sigma^2}{2} \sum_{i=1}^{N} \frac{\partial^2 F (\varphi)}{\partial \varphi_i^2} - K \sum_{i=1}^{N} \frac{\partial H_N (\varphi)}{\partial \varphi_i} \frac{\partial F (\varphi)}{\partial \varphi_i}, \quad \forall F \in C^2$.
I don't understand the definition of "dynamic generator" and how I can pass from that operator to the system and viceversa.
Many thanks.
I found the answer. Basically the "dynamic generator" is the infinitesimal generator, in stochastic terms.
==Definition by wikipedia==
Let $X:[0,+∞) \times Ω→R$ defined on a probability space $(Ω,Σ,P)$ be an Itô diffusion satisfying a stochastic differential equation of the form:
$ dX_{t} = b(X_{t}) \, dt + \sigma (X_{t}) \, \mathrm{d} B_{t},$
where $B$ is an $m$-dimensional Brownian motion and $b$: $R^n → R^n$ and $σ: R^n → R^{n \times m}$ are the drift and diffusion fields respectively. For a point $x \in R^n$, let $P^x$ denote the law of $X$ given initial datum $X^0 = x$, and let $E^x$ denote expectation with respect to $P^x$.
The infinitesimal generator (what I called the dynamic generator) of $X$ is the operator $A$, which is defined to act on suitable functions $f:R^n→R$ by
$A f (x) = \lim_{t → 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}.$
The set of all functions $f$ for which this limit exists at a point ''x'' is denoted $D_A(x)$, while $D_A$ denotes the set of all $f$ for which the limit exists for all $x \in R^n$. One can show that any compact support $C^2$ function $f$ lies in $D_A$ and that
$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} \big( \sigma (x) \sigma (x)^{\top} \big)_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x).$