The continuous Anderson hamiltonian in $d\le 3$

Cyril Labbé (1, 2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Université Paris Dauphine-PSL

We construct the continuous Anderson hamiltonian on $(-L,L)^d$ driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension $d\le 3$ and relies on the theory of regularity structures: it yields a self-adjoint operator in $L^2\big((-L,L)^d\big)$ with pure point spectrum. In $d\ge 2$, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for $d=3$ these estimates show that the eigenvalues do not have exponential moments.

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Cyril Labbé. The continuous Anderson hamiltonian in $d\le 3$. Journal of Functional Analysis, 2019, 277 (9), ⟨10.1016/j.jfa.2019.05.027⟩. ⟨hal-01898384⟩