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

Dabrock N, Hofmanová M, Röger M (2021)
Probability Theory and Related Fields 179: 407-449.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Dabrock, Nils; Hofmanová, MartinaUniBi; Röger, Matthias
Abstract / Bemerkung
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-w,x,t(infinity) estimate for the gradient and an L-w, 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.
Erscheinungsjahr
2021
Zeitschriftentitel
Probability Theory and Related Fields
Band
179
Seite(n)
407-449
eISSN
1432-2064
Finanzierungs-Informationen
Open-Access-Publikationskosten wurden durch die Universität Bielefeld im Rahmen des DEAL-Vertrags gefördert.
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https://pub.uni-bielefeld.de/record/2937471

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Dabrock N, Hofmanová M, Röger M. Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs . Probability Theory and Related Fields . 2021;179:407-449.
Dabrock, N., Hofmanová, M., & Röger, M. (2021). Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs . Probability Theory and Related Fields , 179, 407-449. https://doi.org/10.1007/s00440-020-01012-6
Dabrock, N., Hofmanová, M., and Röger, M. (2021). Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs . Probability Theory and Related Fields 179, 407-449.
Dabrock, N., Hofmanová, M., & Röger, M., 2021. Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs . Probability Theory and Related Fields , 179, p 407-449.
N. Dabrock, M. Hofmanová, and M. Röger, “Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs ”, Probability Theory and Related Fields , vol. 179, 2021, pp. 407-449.
Dabrock, N., Hofmanová, M., Röger, M.: Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs . Probability Theory and Related Fields . 179, 407-449 (2021).
Dabrock, Nils, Hofmanová, Martina, and Röger, Matthias. “Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs ”. Probability Theory and Related Fields 179 (2021): 407-449.
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