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

Bertini, Lorenzo; Buttà, Paolo; Pisante, Adriano

doi:10.1007/s00205-017-1086-3

Cited by 18 publications

(22 citation statements)

References 45 publications

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“…In [16] also an existence result was proved, but only for short time intervals determined by a random variable that is not necessarily bounded from below. Other (formal) approximations of stochastically perturbed mean curvature flow equations have been studied, such as a time discrete scheme in [54] and stochastic Allen-Cahn equations in [7,8,25,48,53].…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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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“…Similar results have been obtained in the context of the stochastic Allen-Cahn equation. In [20,21] the authors study the same problem for d = 1 while in [23,5] it is extended to d = 2, 3. In particular, in [5] the limit considered is a joint sharp interface and small noise, but the starting point is at the mesoscopic scale, even though noise is also involved.…”

Section: Introductionmentioning

confidence: 99%

Action minimization and macroscopic interface motion under forced displacement

Birmpa

Tsagkarogiannis

2018

ESAIM: COCV

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We study an one dimensional model where an interface is the stationary solution of a mesoscopic non local evolution equation which has been derived by a microscopic stochastic spin system. Deviations from this evolution equation can be quantified by obtaining the large deviations cost functional from the underlying stochastic process. For such a functional, derived in a companion paper, we investigate the optimal way for a macroscopic interface to move from an initial to a final position distant by R within fixed time T . We find that for small values of R/T the interface moves with a constant speed, while for larger values there appear nucleations of the other phase ahead of the front.

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“…Finally, it cannot be excluded that the map t → Γ(t) has a Cantor part, which does not affect the cost functional constructed in [36]. A variational definition of S which takes into account also such Cantor part is provided in [6], its corresponding zero level set is given by the mean curvature flow according to the Brakke's formulation [9]. The rate functional S should describe the large deviations asymptotics of microscopic stochastic dynamics that leads to the motion by curvature of the interfaces.…”

Section: Introductionmentioning

confidence: 99%