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- ItemBrownian occupation measures, compactness and large deviations: Pair interaction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, ChiranjibContinuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.
- ItemThe Bouchaud-Anderson model with double-exponential potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Muirhead, Stephen; Pymar, Richard; Santos, Renato Soares dosThe 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.
- ItemAnalysis of the PSPG stabilization for the continuous-in-time discretization of the evolutionary stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) John, Volker; Novo, JuliaOptimal error estimates for the pressure stabilized Petrov-Galerkin (PSPG) method for the continuous-in-time discretization of the evolutionary Stokes equations are proved in the case of regular solutions. The main result is applicable to higher order finite elements. The error bounds for the pressure depend on the error of the pressure at the initial time. An approach is suggested for choosing the discrete initial velocity in such a way that this error is bounded. The "instability of the discrete pressure for small time steps", which is reported in the literature, is discussed on the basis of the analytical results. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.
- ItemPrecise asymptotics for the parabolic Anderson model with a moving catalyst or trap(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Schnitzler, Adrian; Wolff, TilmanWe consider the solution ucolon [0,infty) timesmathbbZ^drightarrow [0,infty) to the parabolic Anderson model, where the potential is given by (t,x)mapstogammadelta_Y_tleft(xright) with Y a simple symmetric random walk on mathbbZ^d. Depending on the parameter gammain[-infty,infty), the potential is interpreted as a randomly moving catalyst or trap. In the trap case, i.e., gamma<0, we look at the annealed time asymptotics in terms of the first moment of u. Given a localized initial condition, we derive the asymptotic rate of decay to zero in dimensions 1 and 2 up to equivalence and characterize the limit in dimensions 3 and higher in terms of the Green's function of a random walk. For a homogeneous initial condition we give a characterisation of the limit in dimension 1 and show that the moments remain constant for all time in dimensions 2 and higher. In the case of a moving catalyst (gamma>0), we consider the solution u from the perspective of the catalyst, i.e., the expression u(t,Y_t+x). Focusing on the cases where moments grow exponentially fast (that is, gamma sufficiently large), we describe the moment asymptotics of the expression above up to equivalence. Here, it is crucial to prove the existence of a principal eigenfunction of the corresponding Hamilton operator. While this is well-established for the first moment, we have found an extension to higher moments.
- ItemThe parabolic Anderson model on a Galton--Watson tree(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) den Hollander, Frank; König, Wolfgang; Soares dos Santos, RenatoWe study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.
- ItemEigenvalue order statistics for random Schrödinger operators with doubly-exponential tails(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Biskup, Marek; König, WolfgangWe consider random Schrödinger operators of the form Delta+zeta , where D is the lattice Laplacian on Zd and Delta is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for sigma with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where zeta takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.
- ItemThe parabolic Anderson model with acceleration and deceleration(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) König, Wolfgang; Schmidt, SylviaWe describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively similar to those from the constant-speed parabolic Anderson model in earlier work by various authors. Our proofs consist of adaptations and refinements of their methods, as well as a variational convergence method borrowed from finite elements theory.
- ItemLongtime 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, WillemWe 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.