Stanislav Molchanov

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We consider the Stochastic Partial Differential Equation u t = κ∆u + ξ(t, x)u, t ≥ 0, x ∈ Z d. The potential is assumed to be Gaussian white noise in time, stationary in space. We obtain the asymptotics of the almost sure Lyapunov exponent γ(κ) for the solution as κ → 0. Namely γ(κ) ∼ c 0 ln(1/κ) , where the constant c 0 is determined by the correlation(More)
We give a simple, transparent, and intuitive proof that all eigenvalues of the Anderson model in the region of localization are simple. The Anderson tight binding model is given by the random Hamiltonian H ω = −∆ + V ω on 2 (Z d), where ∆(x, y) = 1 if |x − y| = 1 and zero otherwise, and the random potential V ω = {V ω (x), x ∈ Z d } consists of independent(More)
We consider the parabolic Anderson problem ∂ t u = ∆u + ξ(x)u on R + × Z d with localized initial condition u(0, x) = δ 0 (x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) lies in the vicinity of, or beyond, the double-exponential distribution , we prove the following geometric characterisation of intermittency: with(More)
We study spectral properties of the discrete Laplacian H on the half space Z d+1 + = Z d Z + with a boundary condition (nn ;1) = tan(n +)(nn 0), where 2 0 1] d. We d e note by H 0 the Dirichlet Laplacian on Z d+1 +. Whenever is independent o ver ratio-nals (H) = R. Khoruzenko and Pastur KP] have s h o wn that for a set of 's of Lebesgue measure 1, the(More)
We consider an explicitly solvable model (formulated in the Rie-mannian geometry terms) for a stationary wave process in a specific thin domain Ωε with the Dirichlet boundary conditions on ∂Ωε. The transition from the solutions of the scattering problem on Ωε to the solutions of a problem on the limiting quantum graph Γ is studied. We calculate the(More)
We study scattering properties of the discrete Laplacian H on the half-space Z d+1 + = Z d Z + with the boundary condition (nn ;1) = tan(n +)(nn 0), where 2 0 1] d. We denote by H 0 the Dirichlet Laplacian on Z d+1 +. Khoruzenko and Pastur KP] have shown that if has typical Diophantine properties then the spectrum of H on R n (H 0) is pure point and that(More)
We s tudy spectral properties of the discrete Laplacian H ! on the half space Z 2 + = ZZ + with a random boundary condition (nn ;1) = V ! (n)(nn 0). Here, V ! (n) are independent and identically distributed random variables on a probability space ((F P). We show that outside the interval ;4 4] (the spectrum of the Dirichlet Laplacian) the spectrum of H ! is(More)
We study spectral properties of the discrete Laplacian H on the half-space Z d+1 + = Z d Z + with random boundary condition (nn ;1) = V (n)(nn 0)) the V (n) are independent random variables on a probability space ((F P) and is the coupling constant. It is known that if the V (n) have densities, then on the interval ;2(d + 1) 2(d + 1)] (= (H 0), the spectrum(More)
We describe a universal transition mechanism between annealed and quenched regimes in the context of reaction-diffusion in random media. We study the total population size for random walks which branch and annihilate on Z d , with time-independent random rates. The random walks are independent , continuous time, rate 2dκ, simple, symmetric, with κ ≥ 0. A(More)