Vision: Large deviation theory is a set of tools, techniques and methods to study scaling limits with tools from analysis (variational analysis, PDE) and probability theory (law of large numbers, ergodic theorems, central limit theorem) with many applications e.g. in stochastic PDEs, control theory, mathematical statistical mechanics, statistics, financial mathematic, percolation theory).
The focus is on sample path large deviations and scaling limits for weakly pinned random walks and integrated random walks. We study scaling limits and the corresponding large deviation principle of the integrated random walk perturbed by an attractive force toward the origin. In particular we analyse the critical situation that the rate function admits more than one minimiser leading to concentration of measure problems. The integrated random represents interface models (polymer.membrane models) with Laplacian interaction (bi-Laplace).
Reading material/proposed activity steps
1.) Learning basic principles of large deviation: The basic principles can be learned quite easily and quickly with the following book: Frank den Hollander: Large deviations, AMS (2000). Most relevant are chpater I-III - more specific reading hints to follow. The text is very basic, well written and easy to follow. To know these basic techniques is helpful whatever project you are going to do later in your thesis, let it be probability or applied analysis. In addition large deviations theory is part of the P2 - MASDOC probability module in term 2. Source: Library, or asking Stefan Adams for either pdf-file or template for producing copies.
2.) Random walk and integrated random walk with/without pinning:
Paper 1: T. Funaki and H. Sakagawa, Large deviations for $ \nabla\phi $ interface model and derivation of free boundary problems, in Stochastic Analysis on Large Scale Interacting Systems, 2002, pp. 173–211. Relevant are introduction and the $d=1$ case (Theorem 2.2 and its proof in § 6). (pdf-file)
Paper 2: E. Bolthausen, T. Funaki, T. Otobe, Concentration under scaling limits for weakly pinned Gaussian random walks, Probab. Theory Relat. Fields (2009) 143:441–480. This paper addresses the situation where the rate function has more than one minimiser (zero) - relevant are the $ d=1 $ cases only, in particular appendix B.
Paper 3: T. Funaki and T. Otobe, Scaling limits for weakly pinned random walks with two large deviation minimizers, J. Math. Soc. Japan Vol. 62, No. 3 (2010) pp. 1005–1041. This more recent work gives an alternative proof technique to paper 1 and its focus is on more than one minimiser (zero) of the rate function.
Paper 4: F. Caravenna and J.D. Deuschel, Pinning and wetting transition for (1 + 1)-dimensional fields with laplacian interaction, Annals of Probability, 36 (2008), pp. 2388–2433. This paper is the key resource for integrated random walk and Laplacian interaction (relevant are the introduction, the results on pinning free energy whereas the renewal approach can be skipped).
Paper 5: E. Bolthausen, T. Chiyonobu, and T. Funaki, Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates, ArXiv: 1406.7766v1 (preprint) (2014). This is the most recent one and adresses the challenging case of higher dimensions with more than one minimiser (zero) of the rate function.
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