Take home questions


  1. 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}).
  2. Compute \Gamma_2(f)=\frac{1}{2}( L\Gamma(f,f)-2\Gamma(f,Lf)) for the previous operator.
  3. 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.
  4. 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.
  5. Show that if the vector fields V_i‘s are globally Lipschitz, then L^g is essentially self-adjoint.
  6. 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.
  7. Deduce a criterion for the stochastic completeness of the semigroup P^g_t=e^{tL^g}.
  8. 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.
  9. 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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