We investigate the nonstationary parabolic Anderson problem ∂u∂t=ϰLu(t,x)+ξt(x)u(t,x),u(0,x)≡1,(t,x)∈[0,∞)×Zd where ϰL denotes a nonlocal Laplacian and ξt(x) is correlated white-noise potential. The irregularity of solution linked to upper spectrum certain multiparticle Schrödinger operators that govern moment functions mp(t,x1,x2,⋯,xp)=⟨u(t,x1)u(t,x2)⋯u(t,xp)⟩. First, we establish weak form intermittency under broad assumptions on L positive-definite noise correlator B=B(x). then examine...

We study the non-stationary Anderson parabolic problem on lattice Zd, i.e., equation ∂u∂t=ϰAu(t,x)+ξt(x)u(t,x),u(0,x)≡1,(t,x)∈[0,∞)×Zd. Here A is a non-local difference operator, ξt(x), t ≥ 0, x ∈ family of correlated white noises and ϰ > 0 diffusion coefficient. The changes (large vs small) are responsible for qualitative phase transition in model. At first step analysis model reduced to solution stochastic differential (in standard Itô’s form) weighted Hilbert space l2(Zd, μ) with...

We consider a continuous-time branching random walk on Z in non-homogeneous environment. The process starts with single particle at initial time t=0. This can the lattice points or disappear intensity until it reaches certain point, which we call reproduction source. At source, split into two offspring jump out of evolves according to same law, independently each other and entire prehistory. aim this paper is study conditions for presence exponential growth average number particles every...

The paper contains the probabilistic analysis of Brownian motion on simplest quantum graph, spider: a system N-half axis connected only at graph's origin by (so-called Kirchhoff's) gluing conditions. limit theorems for diffusion such especially if $N \to \infty$ are significantly different from classical case = 2$ (full axis). Additional results concern properties spectral measure spider Laplacian and corresponding generalized Fourier transforms. continuation will contain study spectrum...

We consider a version of classical concentration inequality for sums independent, isotropic random vectors with mild restriction on the distribution radial part these vectors. The proof uses little Fourier analysis, Laplace asymptotic method and conditioning idea that traces its roots to some original works inequalities.

We study the limiting distribution of sum S N (t) = P i=1 e tX i as t → ∞, where (X ) are i.i.d.random variables.Attention to such exponential sums has been motivated by various problems in random media theory.Examples include quenched mean population size a colony branching processes with rates and partition function Derrida's Random Energy Model.In this paper, problem is considered under assumption that log-tail h(x) -log P{X > x} regularly varying at infinity index 1 < ̺ ∞.An appropriate...