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Given a field ${B(x)}_{xinmathbf{Z}^d}$ of independent standard Brownian motions, indexed by $mathbf{Z}^d$, the generator of a suitable Markov process on $mathbf{Z}^d,,,mathcal{G},$ and sufficiently nice function $sigma:[0,infty)to[0,infty),$ we consider the influence of the parameter $lambda$ on the behavior of the system, begin{align*} rm{d} u_t(x) = & (mathcal{G}u_t)(x),rm{d} t + lambdasigma(u_t(x))rm{d} B_t(x) qquad[t>0, xinmathbf{Z}^d], &u_0(x)=c_0delta_0(x). end{align*} We show that for any $lambda>0$ in dimensions one and two the total mass $sum_{xinmathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for dimensions greater than two there is a phase transition point $lambda_cin(0,infty)$ such that for $lambda>lambda_c,, sum_{mathbf{Z}^d}u_t(x)to 0$ as $ttoinfty$ while for $lambda<lambda_c,,sum_{mathbf{Z}^d}u_t(x) otto 0$ as $ttoinfty.$
The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the
Diamond lattices are sequences of recursively-defined graphs that provide a network of directed pathways between two fixed root nodes, $A$ and $B$. The construction recipe for diamond graphs depends on a branching number $bin mathbb{N}$ and a segment
In this paper we study a stochastic version of an inviscid shell model of turbulence with multiplicative noise. The deterministic counterpart of this model is quite general and includes inviscid GOY and Sabra shell models of turbulence. We prove glob
We consider the annealed asymptotics for the survival probability of Brownian motion among randomly distributed traps. The configuration of the traps is given by independent displacements of the lattice points. We determine the long time asymptotics
We study the effects of dissipation on a disordered quantum phase transition with O$(N)$ order parameter symmetry by applying a strong-disorder renormalization group to the Landau-Ginzburg-Wilson field theory of the problem. We find that Ohmic dissip