Stochastic mean curvature flow
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Date
2020
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Abstract
In this thesis we study a stochastically perturbed mean curvature flow (SMCF). In case of graphs, existence of weak solutions has already been established for one-dimensional and two-dimensional periodic surfaces with a spatially homogeneous perturbation. We extend this result by proving existence of solutions in arbitrary dimensions perturbed by noise which is white in time and colored in space. In addition, we work with a stronger notion of solution which corresponds to strong solutions in the PDE sense.
For this, we give a new interpretation of graphical SMCF as a degenerate variational stochastic partial differential equation (SPDE) with compact embedding. In order to infer existence of an approximating sequence, we extend the theory of variational SPDEs such that we can treat SMCF within this framework. In order to pass to the limit with the approximating sequence, we prove new a-priori bounds for graphical SMCF.
With this a-priori bounds, we can characterize the large-time behavior of solutions in case of spatially homogeneous noise. In particular, we will prove that solutions become asymptotically constant in space and behave like the driving noise in time. This strengthens a previously established one-dimensional large-time result by extending it to higher dimensions and proving stronger convergence.
Furthermore, we propose a numerical scheme for graphical SMCF which employs the variational interpretation we have analyzed before. Using this scheme, we present Monte-Carlo simulations visualizing the energy estimates we have used in the analytic part of this thesis.
Moreover, we discuss the regularity and uniqueness of solutions and give conditional results for both. We complement the previous results, especially the existence result, by investigating how they extend to SMCF with respect to an anisotropic notion of curvature.
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Stochastic mean curvature flow, Stochastic partial differential equations, Differential geometry