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We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling.

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Abstract. We consider a conservative system of stochastic PDE's, namely a weakly coupled, one dimensional phase field model with additive noise. We study the fluctuations of the front proving that, in a suitable scaling limit, the front evolves according to a non-Markov process, solution of a linear stochastic equation with long memory drift. Part I. Introduction 1 General setting, model, and resultsThe term "sharp interface limit" denotes a scaling procedure aimed at the derivation of interfaces as geometric objects, e.g. surfaces of codimension one with bounded variation, that is, enough regular for the area measure to be well defined. Of course this makes only sense in the context of systems which undergo phase transitions and of states where different phases coexist. In the limit the other degrees of freedom are lost and we are left with the interface alone. Rigorous proofs are hard, yet a great variety of models has been successfully worked out. The mathematics involved is correspondingly rich, e.g. the theory of Γ-convergence (to study the sharp interface limit of Ginzburg-Landau like free energy functionals in relation with the equilibrium shape of the interface, as in the Wulff problem) and correspondingly the theory of Gibbsian large deviations (to study the same problems at the more microscopic level of statistical mechanics); singular limit in PDE's, like in the Allen-Cahn, Cahn-Hilliard and phase field equations, and correspondingly, at the microscopic level, hydrodynamic limits of spin or particle systems.This paper deals with fluctuations. Here again the questions are, first, whether in a sharp interface limit the system is described by a [fluctuating] interface with closed equations of motion and, secondly, the nature of such equations. The problem greatly simplifies in one space dimension where the limit interface is represented by a point which separates the two phases (one to its left and the other one *

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number,in terms of derivatives of powers of an elementary function that we call normalized left truncated exponential function. The unique explicit expression for the a k that appears to be known is that of Comtet in (Advanced Combinatorics, Reidel, 1974), which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet's expression yields an identity which is somehow unexpected if considering the combinatorial meaning of the terms. It suggests by analogy another possible formula for the coefficients, in terms of a normalized left truncated logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series. This in particular leads to the identification of a couple of simple implicit equations, which permits us to obtain also some recurrences related to the a k 's.

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the φ 4 1 -measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.2000 Mathematics Subject Classification. 82B24, 60K35, 81Q20. DOBRUSHIN STATES IN THE φ 4 1 MODEL 5

We consider a conservative system of stochastic PDE's, namely a one dimensional phase field model perturbed by an additive space-time white noise. We prove a global existence and uniqueness result in a space of continuous functions on Ê + × Ê. This result is obtained by extending previous results of Doering [3] on the stochastic Allen-Cahn equation.

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