## Homework 1. MA5016. Due September 13 in class

Exercise 1. Let $L: \mathcal{C}^{\infty} (\mathbb{R}^n) \rightarrow \mathcal{C} (\mathbb{R}^n)$ be a linear operator such that:

• $L$ is a local operator, that is if $f=g$ on a neighborhood of $x$ then $Lf(x)=Lg(x)$;
• If $f \in \mathcal{C}^{\infty} (\mathbb{R}^n)$ has a global maximum at $x$ with $f(x)\ge 0$ then $Lf (x) \le 0$.

Show that for $f \in \mathcal{C}^{\infty} (\mathbb{R}^n)$ and $x \in \mathbb{R}^n$,

$Lf(x)=\sum_{i,j=1}^n \sigma_{ij} (x) \frac{\partial^2 f}{ \partial x_i \partial x_j} +\sum_{i=1}^n b_i (x)\frac{\partial f}{\partial x_i} -c(x)f(x),$

where $b_i$, $c$ and $\sigma_{ij}$ are continuous functions on $\mathbb{R}^n$ such that for every $x \in \mathbb{R}^n$, $c(x) \ge 0$ and the matrix $(\sigma_{ij}(x))_{1\le i,j\le n}$ is symmetric and nonnegative.

Exercise 2.

• Show that if $f,g :\mathbb{R}^n \rightarrow \mathbb{R}$ are $C^1$ functions and $\phi_1,\phi_2: \mathbb{R} \rightarrow \mathbb{R}$ are also $C^1$ then,
$\Gamma (\phi_1 (f), \phi_2 (g))=\phi'_1 (f) \phi_2'(g) \Gamma(f,g).$
• Show that if $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a $C^2$ function and $\phi: \mathbb{R}\rightarrow \mathbb{R}$ is also $C^2$,
$L \phi (f)=\phi'(f) Lf+\phi''(f) \Gamma(f,f).$

Exercise 3 On $\mathcal{C}_c (\mathbb{R}^n,\mathbb{R})$, let us consider the diffusion operator $L=\Delta +\langle \nabla U, \nabla \cdot \rangle,$ where $U: \mathbb{R}^n \rightarrow \mathbb{R}$ is a $C^1$ function. Show that $L$ is symmetric with respect to the measure $\mu (dx)=e^{U(x)} dx$.

Exercise 4 (Divergence form operator). On $\mathcal{C}_c (\mathbb{R}^n,\mathbb{R})$, let us consider the operator $Lf=\mathbf{div} (\sigma \nabla f),$ where $\mathbf{div}$ is the divergence operator defined on a $C^1$ function $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by
$\mathbf{div} \text{ } \phi=\sum_{i=1}^n \frac{\partial \phi_i}{\partial x_i}$
and where $\sigma$ is a $C^1$ field of non negative and symmetric matrices. Show that $L$ is a diffusion operator which is symmetric with respect to the Lebesgue measure.

Exercise 5: On $\mathbb{R}^n$, we consider the divergence form operator

$Lf=\mathbf{div} (\sigma \nabla f),$
where $\sigma$ is a smooth field of positive and symmetric matrices that satisfies
$a \|x \|^2 \le \langle x , \sigma x \rangle \le b \|x \|^2, \quad x \in \mathbb{R}^n,$
for some constant $0 < a \le b$. Show that with respect to the Lebesgue measure, the operator $L$ is essentially self-adjoint on $\mathcal{C}_c (\mathbb{R}^n,\mathbb{R})$

This entry was posted in Diffusions on manifolds. Bookmark the permalink.