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In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, where the $u$-field and the $\xi$-field are $ℝ$-valued, $\kappa \in [0,\infty )$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $\xi$-field plays the role of athat drives the equation. The initial condition $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$, split into two...

We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk...

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space–time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of...