Interplay of Analysis and Probability in Physics

It is widely recognised that stochastic effects need to be included in the modelling of many physical systems, while reciprocally the sciences provide a challenging potential area of application of stochastic processes. This creates an increasing need to combine analytic and stochastic techniques. The aim of this workshop was to address this need and contribute to the efforts to surmount the language barrier between analysts and probabilists by stimulating and encouraging exchange and joint research between the two communities. The focus of the workshop was on recent and currently emerging progress in the investigation of complex physical systems, using a combination of analytical and stochastic methods. Mathematics Subject Classification (2000): 60xx, 70xx, 74xx, 82xx. Introduction by the Organisers This workshop brought together researchers working in analysis and probability, stimulated an intensive exchange of ideas between the respective communities, and helped identify open problems at the boundaries and intersections of these areas. Various approaches to scale-bridging were the main theme of the first day. Particular problems included the derivation of the thermal conductivity of a diffusion process from an atomistic model (Stefano Olla), the development of a coarsegraining technique to calculate canonical ensemble averages efficiently (Frédéric Legoll), and a potential-theoretic approach to the analysis of the transition from the metastable state to the stable state for a lattice gas under the Kawasaki dynamics (Anton Bovier). Jürgen Gärtner and Jean-Dominique Deuschel discussed a range of models interacting with a random disorder. Spectral properties of large random matrices were in the focus of the morning session on Tuesday, in particular in the talks by Laszlo Erdős and Amir 3066 Oberwolfach Report 55/2008 Dembo. The afternoon was devoted to pinning problems, both from the probabilistic view points for polymer models (Giambattista Giacomin and Hubert Lacoin), and martensitic materials with imperfections (Patrick Dondl). Wednesday started with a review on recent progress of optimal transportation and gradient flows (Karl-Theordor Sturm). This was continued on Friday with a derivation of a particle model of the Ricci flow on a manifold (Robert Philipowski). Existence of Gibbs measures was another central theme of Wednesday. The existence and uniqueness of the Gibbs measure for point processes with a Hamiltonian depending on nearest-neighbour triples was investigated (Hans-Otto Georgii). A highlight of the conference was the analysis of lattice gradient models with nonconvex interaction energy. Stefan Adams explained, using rigorous renormalisation group theory, that the free energy is convex for sufficiently low temperatures. The talks of Thursday morning had scaling limits as one focus. This included a mean-field approach to clustering (Barbara Niethammer), the sharp interface limit of perturbed Allen-Cahn equations (Matthias Röger), and a detailed investigation of scaling limits of pinned random walks when the rate-function has a non-unique minimiser (Tadahisa Funaki). In the afternoon, Dirk Hundertmark presented the analysis of properties of soliton solutions for a nonlinear Schrödinger equation with relevance to dispersion management. On Friday, the topic of systems interacting with a random disorder was taken up again, covering ageing in the parabolic Anderson model (Marcel Ortgiese) and an interface model with general potentials (Takao Nishikawa). Besides the programme of altogether 25 talks of length between 30 and 45 minutes there was plenty of time for informal discussion and interaction between participants. This time was well-used with most discussions involving members of both the probability and analysis community. Beyond the mathematical discussion, several participants used the opportunity to discuss future collaborations in various international networks currently emerging in this area. There were 40 participants in total, 16 from Germany, 16 from other European countries (Switzerland, Italy, Spain, France and Britain), 5 from the US and 3 from Japan. All career stages were represented. The organisers thank the NSF for funding the participation of speakers from the US and Nadia Sidorova for collecting the extended abstracts.


Stabilization due to additive noise Dirk Blömker
Stabilization due to noise is a well-known phenomenon, and there are numerous publications over the last decades.But most examples are for multiplicative noise only.Stabilization can arise somewhat artificially by adding Itô-noise, due to the Itô-Stratonovich correction, as only multiplicative Stratonovich-noise is neutral for the linear stability.In other cases stabilization arises due to averaging over stable and unstable directions.A celebrated example is Kapiza's problem of the inverted pendulum [13].This averaging is also effective in case of deterministic rotation of the system [10].But there are very few examples due to additive noise.Very nice is the blow-up through a small tube [15].
We consider two very simple examples of stochastic partial differential equations (SPDEs) close to bifurcation.Using the natural separation of time scales, one derives effective stochastic differential equations (SDEs) for the amplitudes of the dominating pattern.Due to averaging, the noise not acting directly on the dominant pattern may appear as a stabilizing deterministic correction to the SDEs.
Swift-Hohenberg equation.In a series of papers [11,12], it was numerically and formally (using a center-manifold argument) justified that additive noise is capable of removing patterns in the one-dimensional Swift-Hohenberg equation.See also [9].The Swift-Hohenberg equation is an SPDE given by (SH) + σǫ∂ t β subject to periodic boundary conditions on [0, 2π] and β being a real-valued Brownian motion.The constants σ and ν measure the noise strength and the distance from bifurcation, respectively.The eigenfunctions of the differential operator are sin(kx) and cos(kx), k ∈ Z.The kernel span{sin, cos} is the space of the dominant pattern.Using the Ansatz with a fast Ornstein-Uhlenbeck (OU) process Z(t) = t 0 e −(t−τ ) dβ(τ ) and complexvalued amplitude A, we obtain the following amplitude equation [6]: It is interesting that the amplitude equation for the dominant behavior is deterministic, and noise leads to a stabilizing deterministic correction.For a precise statement and proof of the approximation result see [6], which treats a more general situation.Numerical approximation [9] shows that any moment of the uniform in space and time error grows logarithmically with the time-interval, while moments of the error for a fixed time seem to stay small for very long times.
Averaging with error bounds.In formal calculations for the derivation of the amplitude equation, the additional constant terms arise from square of noise in 3Aσ 2 (ǫ∂ T β) 2 , where β(T ) = ǫβ(T ǫ −2 ) is a rescaled Brownian motion on the slow time-scale T = ǫ 2 t.In the proofs, using the mild formulation (i.e., variation of constants), we consider the fast OU-process Crucial for the derivation of averaging with explicit error bounds is the following Lemma based on Itô's formula: Lemma [4,7] Let X be a stochastic process with bounded initial condition and differentials, i.e. dX = O(ǫ −r )dt + O(ǫ −r )d β and X(0) = O(ǫ −r ) for some r > 0.
Similar results hold true for other even powers of Z ǫ .For odd powers we have Burgers type equation.Stabilization effects were observed numerically in [1,14] for an equation of the following type; (B) ) subject to Dirichlet boundary conditions on [0, π] with dominant space span{sin}.The highly degenerate noise acts only on the 2 nd mode by β.Consider: with fast OU-process Z(t) = t 0 e −3(t−τ ) dβ(τ ).This is rigorously justified by apriori estimates.In [4] we obtain the following amplitude equation: in Stratonovich sense, with rescaled Brownian motion β(T ) = ǫβ(ǫ −2 T ).Obviously, 0 is stabilized for ν ∈ (0, σ 2 /88).For a precise statement of the approximation result and its proof in a significantly generalized situation see [4].Numerical justification in [9] verified the validity of the approximation for large times and moderate or even large ǫ.
Outlook -Open problems.We comment on a series of related results, generalizations and open problems.Interesting questions in regularity and scaling arise for example for Levy noise [5].
Averaging of martingals of the type T 0 XZ q ǫ dβ is necessary for (B) with highdimensional noise or for higher order corrections for (SH).The averaging is well known, but for error estimates in [4] we are based on Levy's characterization theorem, restricting the result to one-dimensional dominant modes.
Modulated patterns arise if the underlying domain is large or unbounded.Here we need to approximate by a modulated wave of the type A(ǫ 2 t, ǫx)e ix +c.c., where A solves a SPDE of Ginzburg-Landau type.See [4,8].The truly unbounded space with space-time white noise is still open.Solutions seem to be both spatially unbounded and not sufficiently regular for the tools available.
The results presented are limited to long transient time-scales.For the approximation of long-time behavior in terms of invariant measures for (SH) see [2].
Homogenization of random parabolic operators.Diffusion approximation Andrey L. Piatnitski (joint work with Marina Kleptsyna and Alexandre Popier) The talk focuses on homogenization problem for divergence form second order parabolic operators whose coefficients are rapidly oscillating functions of both spatial and temporal variables.The corresponding Cauchy problem takes the form (1) We assume that the coefficients of a(z, s) are periodic functions of spatial variables while their dependence of time is random stationary ergodic, α > 0.Moreover, the matrix a(z, s) is real symmetric, uniformly bounded and positive definite.
It was proved in [1], [2] that the solutions of the original problem converges almost surely to a deterministic limit, the limit function being a solution of homogenized equation with constant coefficients: (2) The question of interest is the asymptotic behaviour of the normalized difference of the original and homogenized solutions.
It turns out that the limits behaviour of the said normalize difference depends crucially on whether α < 2, or α = 2, or α > 2. In the talk we mostly dwell on the the self-similar case α = 2.
In order to formulate the diffusion approximation result we need an auxiliary function, so-called corrector.
Lemma (see [3]).The equation ∂ s χ(z, s) = div z a(z, s)[∇ z χ(z, s) + I] has a stationary in s and periodic in z solution.The solution is unique up to an additive (random) constant.