A simple construction of the continuum parabolic Anderson model on \(\mathbf{R}^2\). (English) Zbl 1332.60094

Hairer, Martin; Labbé, Cyril

Electron. Commun. Probab. 20, Paper No. 43, 11 p. (2015).

A simple construction is presented for the solution to the continuum parabolic Anderson model (PAM) \[ \partial_t u_t= \Delta u_t + u_t\cdot \xi \] on \(\mathbb R^2\), where \(\xi\) is a white noise on \(\mathbb R^2\) independent of time. To overcome the difficulty that the divergence product \(u\cdot \xi\) is not well defined due to weaker Hölder regularities of the two arguments, and to construct the solution on the unbounded space rather than on a torus as appeared in some previous papers, a renormalization procedure as well as time-dependent weights for the spaces of distributions are proposed. The construction is, however, not valid for (PAM) in dimension 3 nor the generalized parabolic Anderson model.

Reviewer: Feng-Yu Wang (Swansea)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

continuous parabolic Anderson model; white noise; renormalization; time-dependent weights