Given a S.D.E and the dual of its infinitesimal generator $\cal L^*$ (as given below), are there general conditions known ("iff"?) when this $\cal L^*$ would be,
- self-adjoint i.e $\int f ({\cal L}^* h) = \int h ({\cal L}^* f)$
- and negative-definite i.e $\int f ({\cal L}^* f) \leq 0$ ?
Given ``nice" functions $b : \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$ and $\sigma : \mathbb{R}^n \rightarrow \mathbb{R}^{n \times n}$, we consider a homogeneous S.D.E $d{X_t} = b(X_t)d{t} + \sigma(X_t)d{B_t}$ and its corresponding infinitesimal generator ${\cal L}$ is s.t it acts on test functions $f$ as, ${\cal L}f(x) = \sum_{i=1}^n b_i(x) \cdot \partial_i f + \frac{1}{2} \sum_{i,j=1}^n (\sigma \sigma^\top)_{ij} \cdot \partial_i \partial_j f$
Given ${\cal L}$ as above, we define the operator ${\cal L}^*$ s.t for any two bounded measurable $f$ and $h$ we have, $\int ({\cal L}f)h = \int f ({\cal L}^* h)$.
Then it follows that, $\mathcal{L}^{*} f (x) = - \sum_{i=1}^{n}\partial_{i}\left(f \cdot b_{i}(x)\right) + \frac{1}{2}\sum_{i,j=1}^{n}\partial_{i}\partial_{j}\left((\sigma(x)(\sigma(x))^{\top})_{i,j} \cdot f\right)$
