Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials Export

@misc{citeulike:707070, abstract = {We study the parabolic Anderson problem, i.e., the heat equation with independent identically distributed random potential and localised initial condition. Our interest is in the long-term behaviour of the random total mass of the unique non-negative solution in the case that the distribution of the potential at one site is heavy tailed. For this, we study two paradigm cases of fields with infinite moment generating functions: the case of polynomial or Frechet tails, and the case of stretched exponential or Weibull tails. For potentials with either polynomial or stretched exponential right tails, we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.}, archivePrefix = {arXiv}, author = {van der Hofstad, Remco and Morters, Peter and Sidorova, Nadia}, citeulike-article-id = {707070}, citeulike-linkout-0 = {http://arxiv.org/abs/math.PR/0606527}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.PR/0606527}, day = {21}, eprint = {math.PR/0606527}, keywords = {anderson-localization}, month = {Jun}, posted-at = {2006-06-22 10:00:10}, priority = {2}, title = {Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials}, url = {http://arxiv.org/abs/math.PR/0606527}, year = {2006} }

TY - ELEC ID - citeulike:707070 L3 - citeulike-article-id:707070 N2 - We study the parabolic Anderson problem, i.e., the heat equation with independent identically distributed random potential and localised initial condition. Our interest is in the long-term behaviour of the random total mass of the unique non-negative solution in the case that the distribution of the potential at one site is heavy tailed. For this, we study two paradigm cases of fields with infinite moment generating functions: the case of polynomial or Frechet tails, and the case of stretched exponential or Weibull tails. For potentials with either polynomial or stretched exponential right tails, we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory. TI - Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials KW - anderson-localization AU - van der Hofstad, Remco AU - Morters, Peter AU - Sidorova, Nadia PY - 2006/Jun/21/ UR - http://arxiv.org/abs/math.PR/0606527 ER -