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

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.

### MSC:

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