Take home questions | Research and Lecture notes

Take home questions

Posted on December 13, 2016 by Fabrice Baudoin

Let
$L=\Delta +\langle \nabla U, \nabla \cdot\rangle,$
where $U$ is a smooth function on $\mathbb{R}^n$ and $\Delta$ the usual Laplace operator on $\mathbb{R}^n$ . Show that with respect to the measure $\mu(dx)=e^{U(x)} dx$ , the operator $L$ is essentially self-adjoint on $\mathcal{C}_c (\mathbb{R}^n,\mathbb{R})$ .
Compute $\Gamma_2(f)=\frac{1}{2}( L\Gamma(f,f)-2\Gamma(f,Lf))$ for the previous operator.
Show that if, as a bilinear form, $\mathbf{Hess} U \ge \rho$ for some $\rho \in \mathbb{R}$ , then the semigroup $P_t=e^{tL}$ is stochastically complete.
Let now $V_1,\cdots,V_n$ be smooth vector fields on $\mathbb{R}^n$ such that $V_1(x),\cdots, V_n(x)$ is a basis of $\mathbb{R}^n$ for every $x$ . We denote by $\Delta^g$ the Laplace-Beltrami operator associated with the corresponding Riemannian metric and the diffusion operator $L^g f =\Delta^g f +\sum_{i=1}^n V_i U V_i f,$ where $U$ is a smooth function. Show that $L^g$ is symmetric with respect to a measure that shall be computed.
Show that if the vector fields $V_i$ ‘s are globally Lipschitz, then $L^g$ is essentially self-adjoint.
Compute $\Gamma^g_2(f)=\frac{1}{2}( L^g\Gamma^g(f,f)-2\Gamma^g(f,L^gf))$ where $\Gamma^g$ is the carre du champ of $L^g$ .
Deduce a criterion for the stochastic completeness of the semigroup $P^g_t=e^{tL^g}$ .
Let $K$ be an elliptic diffusion operator with smooth coefficients on $\mathbb{R}^n$ . Show that $K$ can be written as $K=L^g+Z$ , where $Z$ is a smooth vector field and $\Delta^g$ is the Laplace Beltrami operator of some Riemannian metric.
Show that if $K$ is symmetric with respect to a measure equivalent to the Lebesgue measure, then it can be written as $L^g$ (see question 4) for some $g$ and some $U$ .

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