On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment. (English) Zbl 1538.60161
Summary: With the recent developments on nonlinear SPDEs, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one-dimensional SPDE
\[
\partial_t \rho =\frac{1}{2}\Delta \Phi (\rho)-2\nabla (W^{\prime}\Phi (\rho)),
\]
where the spatial white noise \(W^{\prime}\) is replaced by a regularization \(W^{\prime}_{\varepsilon}\), as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in \(\varepsilon\)-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 60K37 | Processes in random environments |
| 60L50 | Rough partial differential equations |
Keywords:
SPDE; annealed; Brox diffusion; hydrodynamic; inhomogeneous; interacting particle system; quasilinear; quenched; regularization; Sinai random environment; zero-rangeReferences:
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