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Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

Abstract

We study a stochastically perturbed mean curvature flow for graphs in \({\mathbb {R}}^3\) over the two-dimensional unit-cube subject to periodic boundary conditions. The stochastic perturbation is a one dimensional white noise acting uniformly in all points of the surface in normal direction. We establish the existence of a weak martingale solution. The proof is based on energy methods and therefore presents an alternative to the stochastic viscosity solution approach. To overcome difficulties induced by the degeneracy of the mean curvature operator and the multiplicative gradient noise present in the model we employ a three step approximation scheme together with refined stochastic compactness and martingale identification methods.

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Notes

  1. 1.

    This means for \(\epsilon \rightarrow 0\) the level sets the solutions \(f_t^\epsilon \) to (1.3) converge a.s. to some solution of mean curvature flow even in cases when \(f^0_t\) develops ’fattening’, i.e. has zero level sets of positive Lebesgue measure.

  2. 2.

    \({\mathcal {P}}\) denotes the predictable \(\sigma \)-algebra on \(\Omega \times [0,T]\) associated to \((\mathscr {F}_t)_{t\ge 0}\).

  3. 3.

    Here and in the sequel, we write \(L^2_x\) for \(L^2({\mathbb {T}}^N)\) and similarly for other spaces.

  4. 4.

    Here and in the sequel, \(A^*\) denotes the transpose of a matrix A.

  5. 5.

    If a topological space X is equipped with the weak topology we write (Xw).

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Correspondence to Martina Hofmanová.

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Hofmanová, M., Röger, M. & von Renesse, M. Weak solutions for a stochastic mean curvature flow of two-dimensional graphs. Probab. Theory Relat. Fields 168, 373–408 (2017). https://doi.org/10.1007/s00440-016-0713-5

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Keywords

  • Stochastic mean curvature flow
  • Weak solution
  • Martingale solution

Mathematics Subject Classification

  • 60H15
  • 53C44