A singular parabolic {A}nderson model. (English) Zbl 1071.60055

Mueller, Carl; Tribe, Roger

Electron. J. Probab. 9, Paper No. 5, 98-144 (2004).

The authors consider solutions \(u=u(t,x)\) of the parabolic Anderson model, which is given by the following linear stochastic partial differential equation with multiplicative noise: \[ \partial_t u=\frac12 \Delta u +\kappa u \dot{F} \] for \(x\in\mathbb{R}^d\) in dimension \(d\geq3\), where \(\dot{F}\) is a mean zero Gaussian noise with singular covariance \[ \mathbf{E} (\dot{F}(t,x)\dot{F}(s,y)) =\delta(t-s)| x-y| ^{-2}\;. \] For sufficiently small \(\kappa>0\) and under suitable assumptions on the initial conditions, the authors show that solutions of the corresponding martingale problem only exist as singular measures, which is different from the case of \(\delta(t-s)| x-y| ^{-p}\) with \(0<p<2\). Furthermore, the solutions are unique in law and various properties of the solutions are shown, like extinction for time \(t\to\infty\), lower bounds on the Hausdorff dimension of the support, and density of the support.
Some of the key technical tools are scaling, Wiener-chaos expansion, self-duality, and moment formulae. The resctriction to \(\kappa\in(0,(d-2)/2)\) ensures the existence of moments of second order.

Reviewer: Dirk Blömker (Aachen)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
60G15	Gaussian processes
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Keywords:

stochastic partial differential equation; intermittency; extinction; density of support; singular measures