Uniqueness of motion by mean curvature perturbed by stochastic noise

Abstract We present some uniqueness (non-fattening) results for the motion by mean curvature perturbed by stochastic noise. It is well known that for special initial data, the deterministic motion has multiple solutions, i.e., it develops interior. Our result for a particular evolution of curves in R 2 illustrates that stochastic perturbations can select a unique solution in a natural way. The noise we use is white in time and constant in space. The results are formulated both almost surely and in probability law.

[1]  P. Souganidis,et al.  Phase Transitions and Generalized Motion by Mean Curvature , 1992 .

[2]  J. Taylor,et al.  Overview No. 98 I—Geometric models of crystal growth , 1992 .

[3]  R. Wolpert.  Lévy Processes , 2000 .

[4]  Panagiotis E. Souganidis,et al.  Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations , 2000 .

[5]  Tom Ilmanen,et al.  Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature , 1993 .

[6]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[7]  G. Barles,et al.  A New Approach to Front Propagation Problems: Theory and Applications , 1998 .

[8]  P. Souganidis,et al.  Fully nonlinear stochastic partial differential equations: non-smooth equations and applications , 1998 .

[9]  L. Evans,et al.  Motion of level sets by mean curvature. II , 1992 .

[10]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[11]  G. Barles,et al.  Front propagation and phase field theory , 1993 .

[12]  Marius Buliga,et al.  Geometric Evolution Problems and Action-measures , 2022 .

[13]  Matteo Novaga,et al.  A stochastic selection principle in case of fattening for curvature flow , 2001 .

[14]  J. Taylor,et al.  II—mean curvature and weighted mean curvature , 1992 .

[15]  Luciano Tubaro,et al.  Fully nonlinear stochastic partial differential equations , 1996 .

[16]  Kenneth A. Brakke,et al.  The motion of a surface by its mean curvature , 2015 .

[17]  G. Bellettini,et al.  Two examples of fattening for the curvature flow with a driving force , 1994 .

[18]  D. Chopp,et al.  A Computed example of nonuniqueness of mean curvature flow in R3 , 1995 .

[19]  Yun-Gang Chen,et al.  Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations , 1989 .

[20]  Nung Kwan Yip,et al.  Stochastic Motion by Mean Curvature , 1998 .

[21]  H. Soner.  MOTION OF A SET BY THE CURVATURE OF ITS BOUNDARY , 1993 .

[22]  N. Yip.  Existence of Dendritic Crystal Growth with Stochastic Perturbations , 1998 .

[23]  Hitoshi Ishii,et al.  Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor , 1995 .

[24]  Panagiotis E. Souganidis,et al.  Front propagation: Theory and applications , 1997 .

[25]  Yonghoi Koo.  A FATTENING PRINCIPLE FOR FRONTS PROPAGATING BY MEAN CURVATURE PLUS A DRIVING FORCE , 1999 .

[26]  Panagiotis E. Souganidis,et al.  Fully nonlinear stochastic PDE with semilinear stochastic dependence , 2000 .

[27]  L. Ambrosio,et al.  Geometric evolution problems, distance function and viscosity solutions , 1997 .

[28]  Shun'ichi Goto,et al.  Generalized motion of hypersurfaces whose growth speed depends superlinearly on the curvature tensor , 1994, Differential and Integral Equations.