Stochastic Allen–Cahn equation with mobility

Bertini, Lorenzo; Buttà, Paolo; Pisante, Adriano

doi:10.1007/s00030-017-0477-3

Cited by 11 publications

(13 citation statements)

References 38 publications

(64 reference statements)

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“…where ı is a smooth positive function on R d with compact support. For γ > 0 the well-posedness and regularity properties of (1.4) in space dimension d ≤ 3 are discussed in [5]. We understand that for γ = 0 the process η γ is the space-time white noise.…”

Section: Introductionmentioning

confidence: 99%

“…From a technical viewpoint, our results will be obtained by suitably blending arguments from the analysis of the action functional, mostly imported from [40] (which relies on previous results, e.g., [26,32,42]), with basic tools of stochastic calculus and large deviations estimates for Markov processes. The restriction d ≤ 3 is inherited both from the analysis of the regularity properties of the stochastic equation (2.5) [5], and, as in [40], from the validity of the static result in [42].…”

Section: Introductionmentioning

confidence: 99%

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Stochastic Allen–Cahn Approximation of the Mean Curvature Flow: Large Deviations Upper Bound

Bertini

Buttà

Pisante

2017

Arch Rational Mech Anal

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Abstract. Consider the Allen-Cahn equation on the d-dimensional torus, d = 2, 3, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviations asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviations upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic Allen–Cahn Approximation of the Mean Curvature Flow: Large Deviations Upper Bound

Bertini

Buttà

Pisante

2017

Arch Rational Mech Anal

Self Cite

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“…Therefore, given a configuration σ 0 , we have 4) which is negligible if we choosek 6) so that η 1 << η 2 , as required in Sect. 7, formula (7.4).…”

Section: Too Many Jumps Are Negligiblementioning

confidence: 99%

“…A similar lower bound is obtained following the same reasoning. To have a negligible error in (7.3) we need the constraints (5.7), (5.33), (5.37) and 4) which is true from the choice made in (4.6). The next step is to replace f by the density H of the cost functional:…”

Section: Derivation Of the Cost Functionalmentioning

confidence: 99%

“…On the other hand, the large deviations have been studied in [14,16,17]. Furthermore, combining the above results, the large deviations asymptotics under diffusive rescaling of space and time are obtained in [5] [see also the companion paper [4]]: the authors consider coloured noise and take both the intensity and the spatial correlation length of the noise to zero while doing simultaneously the meso-to-macro limit. This double limit is similar in spirit with our work with the difference that our noise is microscopic and the "noise to zero" limit is replaced by a "micro-to-meso" limit.…”

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confidence: 99%

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Large Deviations for the Macroscopic Motion of an Interface

2017

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We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale. Furthermore, using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time, we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough.

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Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

Dabrock

Hofmanová

Röger

2020

Probab. Theory Relat. Fields

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We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $$L^{\infty }_{\omega ,x,t}$$ L ω , x , t ∞ estimate for the gradient and an $$L^{2}_{\omega ,x,t}$$ L ω , x , t 2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

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Stochastic Allen–Cahn equation with mobility

Cited by 11 publications

References 38 publications

Stochastic Allen–Cahn Approximation of the Mean Curvature Flow: Large Deviations Upper Bound

Stochastic Allen–Cahn Approximation of the Mean Curvature Flow: Large Deviations Upper Bound

Large Deviations for the Macroscopic Motion of an Interface

Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

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