The Parabolic Anderson Model

This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors. Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks.

[1]  R. Carmona,et al.  Stationary parabolic Anderson model and intermittency , 1995 .

[2]  J. Gartner,et al.  Time Correlations for the Parabolic Anderson Model , 2010, 1010.1510.

[3]  E. Bolthausen,et al.  Convergence of path measures arising from a mean field or polaron type interaction , 1993 .

[4]  R. Carmona,et al.  Sharp upper bound on the almost-sure exponential behavior of a stochastic parabolic partial differential equation , 1996 .

[5]  Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials , 2006, math/0606527.

[6]  J. Gärtner.  On Large Deviations from the Invariant Measure , 1977 .

[7]  W. König,et al.  The Universality Classes in the Parabolic Anderson Model , 2005, math/0504102.

[8]  W. Kirsch,et al.  Spectral Theory for Nonstationary Random Potentials , 2005 .

[9]  M. Aizenman,et al.  Localization at large disorder and at extreme energies: An elementary derivations , 1993 .

[10]  F. Hollander,et al.  4. Aspects of Trapping in Transport Processes , 1994 .

[11]  Tobias Povel,et al.  Confinement of Brownian motion among Poissonian obstacles in ℝd, d≥ 3 , 1999 .

[12]  F. Hollander,et al.  Intermittency on catalysts , 2007, 0706.1171.

[13]  Ryoki Fukushima.  From the Lifshitz tail to the quenched survival asymptotics in the trapping problem , 2009, 0905.4436.

[14]  Peter Morters,et al.  Ageing in the parabolic Anderson model , 2009, 0910.5613.

[15]  N E Manos,et al.  Stochastic Models , 1960, Encyclopedia of Social Network Analysis and Mining. 2nd Ed..

[16]  E. D. Giorgi.  Selected Papers , 2006 .

[17]  P. Antal.  Trapping problems for thr simple random walk , 1994 .

[18]  J. Gärtner,et al.  Almost sure asymptotics for the continuous parabolic Anderson model , 2000 .

[19]  S. Schmidt.  Das parabolische Anderson-Modell mit Be- und Entschleunigung , 2010 .

[20]  Screening Effect Due to Heavy Lower Tails in One-Dimensional Parabolic Anderson Model , 2000, math-ph/0007013.

[21]  J. Gartner,et al.  Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour , 2011, 1103.4848.

[22]  Adrian Schnitzler,et al.  Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst or Trap , 2010, 1010.1512.

[23]  N. Ueki,et al.  Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice , 2010, 1012.2508.

[24]  Transition from the annealed to the quenched asymptotics for a random walk on random obstacles , 2005, math/0501107.

[25]  A Scaling Limit Theorem for the Parabolic Anderson Model with Exponential Potential , 2010, 1009.4862.

[26]  J. Gartner,et al.  Intermittency in a catalytic random medium , 2004, math/0406266.

[27]  A. Sznitman.  Brownian motion, obstacles, and random media , 1998 .

[28]  D. Sokoloff,et al.  Intermittency, diffusion and generation in a nonstationary random medium , 2014 .

[29]  M. Cranston,et al.  Lyapunov Exponents for the Parabolic Anderson Model , 2002 .

[30]  Peter Muller,et al.  A Survey of Rigorous Results on Random Schrödinger Operators for Amorphous Solids , 2005 .

[31]  Transition asymptotics for reaction-diffusion in random media , 2005, math/0510519.

[32]  S. Molchanov.  Lectures on random media , 1994 .

[33]  P. Anderson.  Absence of Diffusion in Certain Random Lattices , 1958 .

[34]  F. Martinelli,et al.  Constructive proof of localization in the Anderson tight binding model , 1985 .

[35]  Alain-Sol Sznitman,et al.  On the confinement property of two‐dimensional Brownian motion among poissonian obstacles , 1991 .

[36]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[37]  S. Varadhan,et al.  On the number of distinct sites visited by a random walk , 1979 .

[38]  A. Klenke.  A Review on Spatial Catalytic Branching , 2009 .

[39]  Intermittency on catalysts : voter model , 2009, 0908.2907.

[40]  Rongfeng Sun,et al.  Survival Probability of a Random Walk Among a Poisson System of Moving Traps , 2010, 1010.3958.

[41]  George H. Weiss.  Contemporary Problems in Statistical Physics , 1987 .

[42]  Elliott H. Lieb.  Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation , 1977 .

[43]  Potential confinement property of the parabolic Anderson model , 2007, 0708.3207.

[44]  R. Carmona,et al.  Parabolic Anderson Problem and Intermittency , 1994 .

[45]  J. Gärtner,et al.  Correlation structure of intermittency in the parabolic Anderson model , 1999 .

[46]  V. Sidoravicius,et al.  Branching Random Walk with Catalysts , 2003 .

[47]  S. Varadhan,et al.  Asymptotics for the polaron , 1983 .

[48]  J. Gärtner,et al.  Annealed asymptotics for the parabolic Anderson model with a moving catalyst , 2006 .

[49]  J. Gärtner,et al.  Universality for the parabolic Anderson model , 2004 .

[50]  J. Gärtner,et al.  Parabolic problems for the Anderson model , 1998 .

[51]  E. Bolthausen,et al.  Moderate deviations for the volume of the Wiener sausage , 2001, math/0103238.

[52]  N. Ueki,et al.  Moment asymptotics for the parabolic Anderson problem with a perturbed lattice potential , 2010, 1012.2507.

[53]  G. Maillard,et al.  Parabolic Anderson model with voter catalysts: dichotomy in the behavior of Lyapunov exponents , 2010, 1010.4869.

[54]  M. Cranston,et al.  Lyapunov exponent for the parabolic Anderson model in Rd , 2006 .

[55]  Long-time tails in the parabolic Anderson model , 2000, math-ph/0004014.

[56]  Ya. B. Zel'Dovich,et al.  Intermittency in random media , 1987 .

[57]  Ryoki Fukushima.  Brownian survival and Lifshitz tail in perturbed lattice disorder , 2008, 0807.2486.

[58]  E. Lieb.  Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation , 1977 .

[59]  R. Hofstad,et al.  JOINT DENSITY FOR THE LOCAL TIMES OF CONTINUOUS-TIME MARKOV CHAINS: EXTENDED VERSION , 2005, math/0511169.

[60]  Fr'ed'eric Klopp,et al.  Spectral statistics for random Schr\"odinger operators in the localized regime , 2010, 1011.1832.

[61]  W. Konig,et al.  The Parabolic Anderson Model with Acceleration and Deceleration , 2010, 1010.3353.

[62]  A. Ruzmaikin,et al.  Lyapunov Exponents and Distributions of Magnetic Fields in Dynamo Models , 1994 .

[63]  E. Bolthausen.  Localization of a Two-Dimensional Random Walk with an Attractive Path Interaction , 1994 .

[64]  Werner Ebeling,et al.  Diffusion and reaction in random media and models of evolution processes , 1984 .

[65]  J. Gärtner,et al.  Intermittency on catalysts : symmetric exclusion , 2006, math/0605657.

[66]  W. Konig,et al.  Geometric characterization of intermittency in the parabolic Anderson model , 2005, math/0507585.

[67]  R. Carmona,et al.  Asymptotics for the almost sure lyapunov exponent for the solution of the parabolic Anderson problem , 2002, math/0206134.

[68]  G. I. Barenblatt,et al.  Chemical physics and hydrodynamics , 1992 .

[69]  G. Maillard,et al.  Parabolic Anderson model with a finite number of moving catalysts , 2010, 1010.4868.

[70]  Ryoki Fukushima.  Asymptotics for the Wiener Sausage among Poissonian Obstacles , 2007, 0709.1751.

[71]  J. Gärtner,et al.  Parabolic problems for the Anderson model , 1990 .

[72]  Peter Mörters.  The parabolic Anderson model with heavy-tailed potential , 2009 .

[73]  On a semilinear variational problem , 2011 .

[74]  J. Gärtner,et al.  Moment asymptotics for the continuous parabolic Anderson model , 2000 .

[75]  M. Wüthrich,et al.  Infinite volume asymptotics of the ground state energy in a scaled poissonian potential , 2000 .

[76]  P. Antal.  Enlargement of Obstacles for the Simple Random Walk , 1995 .

[77]  Infinite dimensional stochastic analysis , 1993 .

[78]  Srinivasa Varadhan,et al.  Asymptotics for the wiener sausage , 1975 .

[79]  Pierre Bernard,et al.  Lectures on probability theory , 1994 .

[80]  S. Molchanov,et al.  The Parabolic Anderson Model with Long Range Basic Hamiltonian and Weibull Type Random Potential , 2012 .

[81]  Werner Kirsch,et al.  An Invitation to Random Schr¨ odinger operators , 2007 .

[82]  F. Hollander,et al.  Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment , 2010, 1011.0541.