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DDC: 510 × Subject: intermittency ×

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Now showing 1 - 3 of 3
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    Parabolic Anderson model with a finite number of moving catalysts
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Castell, Fabienne; Gün, Onur; Maillard, Gregory
    We consider the parabolic Anderson model (PAM) which is given by the equation partial u/partial t = kappaDelta u + xi u with ucolon, Z^dtimes [0,infty)to R, where kappa in [0,infty) is the diffusion constant, Delta is the discrete Laplacian, and xicolon,Z^dtimes [0,infty)toR is a space-time random environment. The solution of this equation describes the evolution of the density u of a ``reactant'' u under the influence of a ``catalyst'' xi.newlineindent In the present paper we focus on the case where xi is a system of n independent simple random walks each with step rate 2drho and starting from the origin. We study the emphannealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. xi and show that these exponents, as a function of the diffusion constant kappa and the rate constant rho, behave differently depending on the dimension d. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents,...
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    The Bouchaud-Anderson model with double-exponential potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Muirhead, Stephen; Pymar, Richard; Santos, Renato Soares dos
    The Bouchaud-Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour.
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    Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) König, Wolfgang; Perkowski, Nicolas; van Zuijlen, Willem
    We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log t, with the deterministic constant Χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.