Definizione generatore della dinamica dato un sistema di EDO
Buongiorno,
sto leggendo un articolo di fisica matematica in cui viene presentato un sistema di EDO:
$d \varphi_j (t) = - \frac{K}{N} \sum_{i=1}^{N} \sin(\varphi_j(t)-\varphi_i(t))dt+\sigma d\omega_j(t)$,
viene poi scritto(pag 3) che il generatore della dinamica associata è dato da:
$L_{K,N} F(\varphi) = \frac{\sigma^2}{2} \sum_{i=1}^{N} \frac{\partial^2 F (\varphi)}{\partial \varphi_i^2} - K \sum_{i=1}^{N} \frac{\partial H_N (\varphi)}{\partial \varphi_i} \frac{\partial F (\varphi)}{\partial \varphi_i}, \quad \forall F \in C^2$.
Vorrei capire come si passi dal sistema al suddetto generatore e la ragione per cui si chiami così l'operatore $L_{K,N}$.
Come sempre vi ringrazio!
sto leggendo un articolo di fisica matematica in cui viene presentato un sistema di EDO:
$d \varphi_j (t) = - \frac{K}{N} \sum_{i=1}^{N} \sin(\varphi_j(t)-\varphi_i(t))dt+\sigma d\omega_j(t)$,
viene poi scritto(pag 3) che il generatore della dinamica associata è dato da:
$L_{K,N} F(\varphi) = \frac{\sigma^2}{2} \sum_{i=1}^{N} \frac{\partial^2 F (\varphi)}{\partial \varphi_i^2} - K \sum_{i=1}^{N} \frac{\partial H_N (\varphi)}{\partial \varphi_i} \frac{\partial F (\varphi)}{\partial \varphi_i}, \quad \forall F \in C^2$.
Vorrei capire come si passi dal sistema al suddetto generatore e la ragione per cui si chiami così l'operatore $L_{K,N}$.
Come sempre vi ringrazio!
Risposte
Trovato! E' il generatore infinitesimale, da wikipedia [1]:
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}) \, 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
$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} ( \sigma (x) \sigma (x)^{\top} )_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x).$
[1]: https://en.wikipedia.org/wiki/Infinites ... _processes)
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}) \, 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
$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} ( \sigma (x) \sigma (x)^{\top} )_{i, j} \frac{\partial^{2} f}{\partial x_{i} \, \partial x_{j}} (x).$
[1]: https://en.wikipedia.org/wiki/Infinites ... _processes)
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