1 Arch. Rational Mech. Anal. 144 (1998) c Springer-Verlag 1998 Stochastic Motion by Mean Curvature Nung Kwan Yip Communicated by R. V. Kohn Abstract We prove the existence of a continuously time-varying subset K(t) of R n such that its boundary K(t), which is a hypersurface, has normal velocity formally equal to the (weighted) mean curvature plus a random driving force. This is the first result in such generality combining curvature motion and stochastic perturbations. Our result holds for any C 2 convex surface energy. The K(t) can have topological changes. The randomness is introduced by means of stochastic flows of diffeomorphisms generated by Brownian vector fields which are white in time but smooth in space. We work in the context of geometric measure theory, using sets of finite perimeter to represent K(t). The evolution is obtained as a limit of a time-stepping scheme. Variational minimizations are employed to approximate the curvature motion. Stochastic calculus is used to prove global energy estimates, which in turn give a tightness statement of the approximating evolutions. 1. Introduction In this paper, we introduce stochastic perturbations into motion by mean curvature. This motion is but one example of the general notion of curvature driven flows for interfaces, which have been widely used in the modeling of solidification and coarsening processes [AC, Lan, KKL]. By an interface, we mean a hypersurface separating two domains in R n of different physical properties. The interfacial velocity is some prescribed function of curvature or other quantities defined in the bulk region. The motion law is such that the sum of the surface and bulk energies of the system is decreased in time. However, the physical environment is seldom deterministic there are perturbations coming from thermal fluctuations and impurities. Hence, it is important to consider the effects of randomness.
2 314 N. K. Yip Motion by mean curvature, in which the normal velocity of the interface simply equals its mean curvature, has attracted much attention. It involves interesting aspects of the topology of curves and surfaces and also poses challenging questions for partial differential equations. The motion can lead to singularities and topological changes of the interface. Several machineries have been developed to tackle these difficulties. [Bra] first used varifold theory to prove an existence result in arbitrary codimension. The technique so far only applies to the isotropic surface energy. [CGG] and [ES] established global unique viscosity solution to the level set formulation of the curvature motion. But there might be ambiguities of the interfacial locations due to the flattening of the level sets. The phase-field approach, in particular the Allen-Cahn equation [AC] in which the interface is diffused, was shown to produce motion by mean curvature in the sharp interfacial limit [BK] (spherical case), [MS, Che] (when a smooth solution of the interfacial motion exists). [ES] extended this result to the viscosity setting. [Ilm, Son] further showed that Brakke s varifold solution can be obtained from the phase-field approach. Recently, [ATW] used a variational method to prove a general existence result global in time. 1 The surfaces can have topological changes. The advantages of this approach compared with the previous ones are that it works directly in the sharp-interfacial regime and can handle very general (even non-smooth) anisotropic surface energy integrand. [ATW] called their evolution flat flow 2. The evolution is approximated by a time-stepping scheme which involves variational minimizations. We extend this approach to include stochastic noise. Questions about stochastic dynamics of interfaces have been widely discussed in the physics literature. The works touch upon lattice models, derivations of continuum (macroscopic) equations, the study of interfacial structures, scaling limits, etc. We refer to [BS, KrSp, Zan] for good introductions to these topics. [KO1] and [KO2] derived a stochastic version of curvature flows in the spirit of the phase-field equation. [Fun1] and [Fun2] studied random sharp-interfacial limits in the case of one spatial dimension and a convex curve in the plane. However, many questions remain unsettled in general situations Mathematical Approach In this paper, we tread stochastic motion by mean curvature in the sharpinterfacial regime by using techniques of geometric measure theory [Fed]. We establish a continuously time-varying subset K(t)} t of R n such that its boundary K(t), which is a hypersurface, has velocity formally equal to the (weighted) mean curvature plus some random vector. If K(t) undergoes pure motion by mean curvature without perturbations, its surface area decreases according to d dt H n 1 ( K(t)) = h(x) 2 dh n 1 x (1) x K(t) 1 [LS] also proved a similar result in the isotropic case. 2 The surface is shown to evolve continuously in time in terms of the flat norm for integral currents of geometric measure theory.
3 Stochastic Motion by Mean Curvature 315 where H n 1 denotes the Hausdorff (n 1)-dimensional measure and h is the mean curvature of K(t). This motion law can be extended to more general surface energy integrands Φ. The corresponding concept of curvature is then Φ-weighted mean curvature h Φ. Now the surface energy of K(t) decreases according to d Φ( K(t)) = h Φ 2 dh n 1. (2) dt K(t) We omit the word weighted for simplicity. In [ATW], such an evolution is recast as a negative gradient flow for the surface energy functional: d K(t) = Φ( K(t)). (3) dt An implicit time stepping scheme is used to solve this equation. During each time interval, the set K(t) is changed to a new shape which is a minimizer of an appropriate functional. This procedure approximates (3) in discrete time. We incorporate randomness into the above approach. Stochastic noises are introduced by means of a random flow of diffeomorphisms of the underlying space. This flow is generated by a Brownian vector field which is white in time but smooth in space. More precisely, we want K(t) to evolve according to the equation v n = h Φ + F, ˆn (4) where v n is the normal velocity of K(t), ˆn is the outward normal, h Φ is the Φ- weighted mean curvature 3 ; F is a white noise vector field defined on the whole background domain. (We only consider the normal component of F with respect to K(t) since the tangential part does not change the shape of K(t).) In essence, we have in mind that K(t) is evolving so as to reduce its surface energy Φ( K(t) but this motion is constantly perturbed by F which acts by deforming the space. One of the main difficulties in introducing white noise into such a geometric motion is how to combine the nonlinearity of the evolution and the statistical cancellation property of the noise. We achieve this by a time-splitting scheme. Within each time interval of t, we perform two operations. First, we change the set K(t) by minimizing a functional which is the same as the one in [ATW]. This approximates v n = h Φ. Then we transport the set by the flow of diffeomorphisms generated by F. We repeat this process for each interval. Using tools from stochastic calculus, we prove that the previous construction produces a tight sequence (as t ) of probability measures on an appropriate space of stochastic processes. Any weak limit of the measures concentrates on the space of continuous evolutions of K(t). The main theorem is stated in Section 2.7 after the introduction of some terminology and notations. An outline of the proof is given in Section 2.8. Some remarks are in order: The method we employ is quite general. It produces a sharp-interfacial evolution, allowing topological changes. In the deterministic case, the surface energy Φ can be any convex integrand. In our random case, we need it 3 The sign of h Φ is chosen so that the equation is well-posed: a sphere wants to shrink.
4 316 N. K. Yip to be in C 2 due to our use of Ito s formula from stochastic calculus which involves second derivatives. On the other hand, so far we can only show that our construction gives (4) formally. The motion law and regularity of our evolving set K(t) are not quite clear. For the unperturbed deterministic problem, it is shown in [ATW 7.4] that the variational approach gives the same evolution as the classical solution as long as the latter exists. It seems a challenging problem to prove similar results beyond the appearance of interfacial singularities Related Models As mentioned earlier, curvature driven flows have wide applications in modeling solidification processes. We refer to [Lan] and [KKL] for an introduction to these physical phenomena. A more general form of interfacial velocity is given by v n = µ(h Φ + ). (5) µ is the mobility function which measures the kinetics how fast the interface can react to driving forces. denotes bulk quantities which might depend on the temperature field, concentration of solutes, impurities, etc. We refer to [Gur] for a derivation of (5) from a thermodynamical point of view. [TCH] gives a review of several mathematical approaches to tackle such interfacial evolutions. In a more complete model, the bulk variables are subject to diffusion equations. This has been considered in [AW, Luc, Son]. Stochastic perturbation has also been incorporated in this case [Yip]. Many other phenomenological continuum equations have been developed to study similar growth processes, which are discussed in [BS, KrSp, Zan] and references therein. Typical equations considered in these works include f = A 1 2 f + A 2 ( f ) 2 + η, KPZ equation, (6) t f = A 1 4 f + A 2 2 ( f ) 2 + η, diffusion-dominated growth, (7) t where A 1 and A 2 are positive constants. η is commonly taken to be the space-time white noise: η(x, t), η(y, s) = δ(x y)δ(t s). Questions of particular interests concerning these equations include the interfacial structures and scaling exponents. In addition, [KO1] and [KO2] derived random interfacial dynamics starting from the time-dependent stochastic Ginzburg-Landau equation. Assuming that the interfacial curvature is small compared with the diffused interfacial thickness, they came up with the equation 4 t f = 1 + f 2 f div + η(x, t) (8) 1 + f 2 4 Here the interface is represented by the graph of f. The term 1 + f 2 accounts for the tilt of the interface.
5 Stochastic Motion by Mean Curvature 317 where the noise term satisfies the fluctuation-dissipation relation η(x, t), η(y, s) = C 1 + f 2 δ(x y)δ(t s). (9) This is similar to our equation (4) where we replace η by F, ( f, 1) with F being a vector field white in time but smooth in space. 2. Statement of Theorem and Outline of Proof We introduce here some notations for our theorem. In the whole paper, we work in a fixed domain O of R n with compact closure and nice boundary. (We can also regard O as an n-torus.) All random variables and stochastic processes (such as the Brownian motion and flows set forth later) are defined on a common probability space (, F,P)where F is a σ -field of and P is a probability measure on. E always means the expectation taken with respect to P Crystal Shape (K ) These are described by subsets of O with finite perimeters. K is called such a set if } K = sup div gdl n : g C 1 (O,Rn ), g 1 < (1) K K is called the perimeter of K. K is metrized by the L 1 norm: K L L 1 = K(x) L(x) dl n x = L n (K L). (11) x O (By abuse of notation, K can mean both the set K or its characteristic function χ K.) Each K K can also be considered as an n-dimensional integral current in the context of geometric measure theory [Fed]. It is denoted by [[K]]. [[K]] refers to its current boundary. The main properties we need for this kind of sets are compactness under L 1 of the collection K K : K M< } and the existence of a well defined notion of normal and boundary, namely, approximate normal (n K ) and reduced boundary ( K). These concepts are described in detail in [EG] and [Giu] Surface Energy (Φ) This notion is used to describe interfacial surface energy. A surface-senergy integrand Φ is a function from S n 1 to R +. It is usually extended to a map from R n to R + by positive homogeneity of degree 1: Φ(λv) = λφ(v) ( λ,v S n 1). Φ is called isotropic if Φ(v) = c v for some positive constant c. The Φ surface energy of K K is defined as Φ( K) = Φ(n K )dh n 1 (12) K where n K is the outward normal vector of K. 5 5 In this paper, K always denotes the reduced boundary of K.
6 318 N. K. Yip In this paper, we assume that Φ is in C 2 and is convex as a function from R n to R Φ-Weighted Mean Curvature h Φ The h Φ of a hypersurface K can be defined as a weighted sum of the mean curvatures of K or more generally as the first variation of the Φ-surface energy. A nice account of such a concept is given in [Tay]. Here we give the formula in the graph case. Suppose a surface in R n is represented by a graph: x n = f(x 1,...x n 1 ) and the surface energy integrand (assumed to be positively homogenous of degree 1) is given by Φ : (p 1,...p n ) R n R +. Then the Φ-weighted mean curvature of f is h Φ = n 1 i=1 x i ( ) Φ ( f, 1). (13) p i 2.4. Minimization Step Approximation of Motion by Mean Curvature Given t > and a K K, we replace K by a new set K which is a (Φ, t,k)-minimizer, i.e., K minimizes the functional Φ( L) + 1 Dist(x, K)dL n x (14) t x L K over all L K where L K = (L\K) (K\L) and Dist(, K) is the distance function from a point to the (topological or reduced) boundary of K. Such a change from K to K is an approximation of motion by mean curvature of K as explained in [ATW, 2.12] Perturbation Step Stochastic Flows We stochastically perturb the set K by deforming the domain O by using a Brownian flow. These and other related terminologies are described in Appendix B. We also refer to [KS] for basic concepts in probability theory such as random variables, stochastic processes, martingales, etc. Let (, F,P)be a probability space equipped with a filtration F t }t. Let F, Ft ; t< } be a Brownian Motion in the space of vector fields defined on O. The support of F is contained in O for all t. The local characteristics of F are denoted by (a(x,y,t),b(x,t)) x,y O,t, which belongs to the class B 2,δ ub (δ >) (Section B.2). Under this assumption, F can be written as F(x,t) = M(x,t) + t b(x, r) dr (15) where M(x,t) is a continuous C 2 (O,R n )-valued martingale with cross variation given by
7 Stochastic Motion by Mean Curvature 319 t M i (x, t), M j (x, t) = a ij (x,y,r)dr. (16) The Brownian flow ϕ s,t generated by F is the solution of the stochastic }t s differential equation t ϕ s,t (x) = x + F(ϕ s,r (x), dr), x O, s t 1. (17) s Under the stated assumptions of F, a unique solution for this equation exists. ϕs,t } t s is a 2-parameter family of C2 -diffeomorphisms of O. It equals the identity map outside O for all t s. We use ϕ s,t to perturb K: ϕ s,t K = (diffeomorphic) image of K under ϕ s,t. (18) This gives the effect that the boundary of K is transported by F Construction of Discretized Stochastic Motion by Mean Curvature Now we combine the previous ingredients and define our approximate evolutions. Let t = 1/N be the time discretization interval. We denote t i = i t. Starting from a fixed initial shape K at, we construct the discretized evolution K N (t) } as follows. For i N 1, t [,1] K N (t i + ) = a (Φ, t,k N (ti ))-minimizer, K N (t) = ϕ ti,tk N (t i + ) for t i t<t i+1, i.e., at each t i, we change the set by minimization so as to approximate motion by mean curvature; in between any two t i s, the set is perturbed by the Brownian flow ϕ. The K N (t) } thus defined is a piecewise continuous time-varying set with t [,1] discontinuities at the t i } s ( t i 1). Sometimes we will use K N i + to denote K N (t i + ). A similar remark holds for K N i.nowk N ( ) is a stochastic process taking values in K with its sample paths being right continuous with left-hand limits. Such a space is denoted by D ([, 1], K ) and is endowed with the Skorokhod topology 6. 6 For details of the Skorokhod topology, see [Bil]. Actually we can formulate our result without using this topology (see Theorem 1.1). We introduce this terminology here just because it is a very common space used in the study of piecewise continuous stochastic processes.
8 32 N. K. Yip 2.7. Theorem Tightness of Stochastic Motion by Mean Curvature Let Φ be a C 2 and convex surface-energy intergrand and let F be a Brownian vector field in the class B 2,δ ub (δ>) with support contained in O. Let N be the law of K N (t) } on D ([, 1], K ) induced by the construction described in t [,1] Section 2.6. Then N } N 1 is tight7. Furthermore, any weak limit of N satisfies the following statements. 1. (C([, 1], K )) = 1, i.e., is supported on the space of continuous evolutions of K. 2. Uniform Surface Energy Estimate. For any positive integer m, there is a constant C m such that ( Φ( K(t)) m : K C([, 1], K ) }) C m. (19) sup t [,1] 3. Weak Continuity Estimate. For any positive integer m and f C 2 (O ), there is a constant C(f,m) such that for s t 1, f(x)dl n x f(x)dl n 2m x d K K C([,1],K ) x K(t) x K(s) C(f,m) t s m. (2) 2.8. Outline of Proof As described in the previous main statement, we are establishing a compactness result. We will prove that our set evolution heuristically satisfies E K(t) K(s) 2m C m t s m in some weak sense. Then the Kolmogorov- Čentsov Theorem A.1, stated in Appendix A, says that K(t) varies continuously in time. In our discrete scheme, the set is changed by two procedures: minimization and stochastic perturbation. The estimate during the first step is essentially the same as in [ATW]. We show that formally (Corollary 3.2), change of set L 1 C t. The proof relies heavily on the regularity of the minimizers the boundary of any minimizer enjoys a lower density ratio bound (Proposition 3.1). For stochastic perturbations, we establish the following weaker form of the continuity statement: fdl n x fdl n x C(f) t K N (t+ t) K N (t) where f is any smooth function defined on O. 7 See the appendix for the definitions of tightness and weak limit of probability measures.
9 Stochastic Motion by Mean Curvature 321 Combining these two estimates, we have (see (31) and (32)) E K N (t) fdl n x K N (s) fdl n x 2m C(m, f ) t s m. This proves that the set evolves weakly continuously in time. Here we treat the sets as random measures. However, our sets are much better than arbitrary measures. They have a boundary notion which acts like a spatial distributional derivative. In this regard, we prove the uniform energy estimate (see (33)) } E sup Φ( K N (λ)) m C m, λ [,1] which implies that the sets have finite perimeters. With this extra ingredient, the weak continuity statement can be improved to strong L 1 continuity. The proofs of (31) (33) make use of the techniques from stochastic calculus, mainly Ito s formula and Martingale s inequality. For their statements we refer to [KS]. A particularly useful inequality is the Burkholder-Davis-Gundy Inequalities (BDG) [KS, ] which is used in several places in this paper. We state it here for later reference: Let M t be a continuous (local) martingale such that M =. Then for all m>, there are universal positive constants k m and K m (not depending on M) such that k m E ( M m ) ( τ E (Mτ )2m) K m E ( M m ) τ (21) where τ is any stopping time, M t is the quadratic variation of M and M t = sup s t M s. 3. Approximation of Motion by Mean Curvature In this section, we describe the estimates related to the minimization step. Recall the set-up in Section 2.4. Given a set K, we find a (Φ, t,k )-minimizer. In our actual application, K is the shape at time t i, and any minimizer K can be chosen to be the shape at t + i. The regularity of the (Φ, t,k )-minimizers is very important in establishing the continuity statement of the overall evolution. The starting point is a lower bound for the (n 1)-dimensional density ratio, from which we can control the volume change of the set. The existence of (Φ, t,k )-minimizers is easily deduced from the compactness property of integral currents or functions of bounded variations. Furthermore, any minimizer lies in the convex hull of K. The following is a collection of results from [AWT, 3.4, 5.3]. We set forth the following notations which are standard in geometric measure theory:
10 322 N. K. Yip B n (p, r) = x : p x r }, U n (p, r) = x : p x <r}, Φ = sup Φ(n), Φ = inf Φ(n), γ k (2 k n) is the isoperimetric constant, α(n) is the volume of unit n-ball in R n, β(n) is the Besicovitch-Federer covering constant for R n. Proposition 3.1 ((n 1)-Dimensional Density Bound for Minimizers [ATW, 3.4]). Let K be a (Φ, t,k )-minimizer. Then for all p spt [[K]], where θ = H n 1 ( K B n (p, r)) r n 1 θ for all <r t (22) ( ) 1 1/(n 2) ( ) } n 1 Φ nφ (n 1) n 1 inf 1, 2γ n 1 Φ 3D and D is the number from Proposition Proof. Denote T = [[K]]. Let p spt [[K]]. Define ρ(x) = x p. For all r>, consider T r = T x : ρ(x)<r}, m(r) = M(T r ) = H n 1 ( K B n (p, r)). } For almost every r>, the slice T,ρ,r = (T x : ρ(x) r ) = Tr is an integral (n 2)-current and M( T r ) = M T,ρ,r m (r) (since Lip ρ = 1). By the isoperimetric inequality, there is an integral (n 1)-current R supported in B n (p, r) such that R = T r = T,ρ,r and M(R) γ n 1 M( T r ) (n 1)/(n 2) γ n 1 m (r) (n 1)/(n 2). Consider the cone Q = [[p]] (R T r ).(Q is the n- dimensional current formed by joining p to all the points on R T r. For a precise definition, see [Fed, ].) Since (R T r ) =, we have Q = R T r and M(Q) r n M(R T r) r [ ] γ n 1 m (r) (n 1)/(n 2) + m(r). n Note that Φ(T + Q) Φ(T) = Φ(R+(T T r )) Φ(T r +(T T r )) = Φ(R) Φ(T r ). Since K is a (Φ, t,k )-minimizer, we have Φ( K) + 1 Dist(x, K )dl n x t K K Φ( (K + Q)) + 1 Dist(x, K )dl n x t L K where L is the set corresponding to the current K + Q. Since K L\K K K L, we deduce that } Dist(x, Φ(T r ) Φ(R) + L n K ) (K L) sup : x B n (p, r). t 8 This result implies that H n 1 (spt [[K]] [[K]]) =.
11 Stochastic Motion by Mean Curvature 323 Since L n (K L) M(Q), the last inequality gives Φ m(r) Φ γ n 1 m (r) (n 1)/(n 2) + r [ ] } Dist(x, γ n 1 m (r) (n 1)/(n 2) K ) + m(r) sup : x B n (p, r). n t By Proposition 3.3, Dist(x, K ) D t. Hence ( m(r) 1 rd ) n tφ m(r) Φ γ n 1 m (r) (n 1)/(n 2) Φ + rd [ ] n γ n 1 m (r) (n 1)/(n 2) + m(r), tφ Φ (γ )( n Φ ) rd n tφ m (r) (n 1)/(n 2). Now restrict r r = nφ t/3d and set C (n 1)/(n 2) = 2γ n 1 Φ /Φ. Then If r r t, then m(r) C (n 1)/(n 2) m (r) (n 1)/(n 2) ((n 1)m(r) 1/(n 1)) m(r) = m(r) (n 2)/(n 1) 1 C r n 1 m(r) ((n 1)C) n 1. m(r) r n 1 m(r ) r n 1 ( r The whole proposition follows if we set θ = r ) ( ) n 1 1 n 1 ( ) n 1 nφ. (n 1)C 3D ( ) 1 1/(n 2) ( ) } n 1 Φ nφ (n 1) n 1 inf 1,. 2γ n 1 Φ 3D Corollary 3.2 (Volume Difference Estimate). Let K be a set with a lower bound θ for the (n 1)-dimensional density ratio in the sense of (22). Let K be a (Φ, t,k )- minimizer. Then 9 L n (K K ) A(Φ, n) R θ H n 1 ( K ) + t R (Φ( K ) Φ( K)) for all R 1 2 t, where A(Φ, n) is a number depending only on Φ and n. 9 Note that we are making use of the lower density ratio bound of K, NOT K.
12 324 N. K. Yip Proof 1. By the fact that K is a (Φ, t,k )-minimizer, we have, using K as a comparison shape, that Dist(x, K )dl n t(φ( K ) Φ( K)). (23) K K Now, L n (K K ) = L n (K K Dist(x, K ) R}) + L n (24) (K K Dist(x, K ) R}). For the first term on the right of (24), we have by (23), that L n (K K Dist(x, K ) R}) t R (Φ( K ) Φ( K)). (25) For the second term on the right of (24), by Besicovitch-Federer Covering Theorem, we can cover K by balls of radius 2R such that they do not overlap more than β(n) times. Hence, L n (K K Dist(x, K ) R}) L n (B(p i, 2R)) = α(n) (2R) n B(p i,2r) = α(n)2 n R p i R n 1 B(p i,2r) α(n)2 n θ 1 R p i H n 1 ( K B(p i, 2R)) (by the lower density ratio bound for K ) α(n)β(n)2 n θ 1 RH n 1 ( K ). (26) The corollary follows by adding (25) and (26). Proposition 3.3 (ATW, 5.3). Let K be a (Φ, t,k )-minimizer. Then Dist( K, K ) D(Φ, n) t where D(Φ, n) depends only on Φ and the dimension. Proof. Suppose that there is a point p K such that B(p, R) K. (The proof for the case B(p, R) K c is similar.) As a comparison set, let K = K B(p, 2 1R). Then, Φ( K) + 1 Dist(x, K )dl n t K K Φ( K ) + 1 Dist(x, K )dl n t K K Dist(x, K )dl n t ( Φ( K ) Φ( K) ). (27) B(p,R/2)\K Note that Φ( K ) Φ( K) = Φ( B(p, 2 1R)) Φ( (B(p, 2 1 R) K)). 1 This proof follows [LS, 1.5]. It is much simplier than the original argument in [ATW, 4.2].
13 Stochastic Motion by Mean Curvature 325 To simplify the notation, assume that the Wulff shape of Φ is a ball 11. Then for all r> and U R n, Φ( B(p, r)) 1 n 1 L n (B(p, r)) 1 n Φ( U) 1 n 1 L n (U) 1 n [ Φ( B(p, r)) Φ( U) Φ( B(p, r)) 1 ( L n (U) L n (B(p, r)) ) n 1 ] n. Assuming further that U B(p,r), we obtain [ Φ( B(p, r)) Φ( U) Φ( B(p, r)) 1 Φ( B(p, r)) L n (B(p, r)\u) L n (B(p, r)) ( 1 L n (B(p, r)\u) L n (B(p, r)) ) n 1 ] n where in the last inequality we have used 1 (1 x) n 1 n x for x 1. Now set U = B(p, 2 1 R) K in the above. Then 1 2 RL n (B(p, 2 1 R)\K) left-hand side of (27) right-hand side of (27) The extreme inequalities yield t Φ( (p, R/2)) L n (B(p, 1 2 R)\K) L n (B(p, 1 2 R)). 1 2 RL n (B(p, 2 1 n R)\K) t Φ( B(p,1 2 R))L (B(p, 2 1 R)\K) L n (B(p, 2 1 R)), 1 1 2R D(Φ, n) trn 1 R n, R D(Φ, n) t where D(Φ, n) depends only on Φ and the dimension. Proposition 3.4 (n-dimensional Density Bound for Minimizers). Let K be a (Φ, t,k )-minimizer. Then for all p K, L n (K B(p, r)), L n (K c B(p,r)) C(θ, n)r n for all <r t where C(θ,n) is a constant depending only on the lower bound θ for the (n 1)- dimensional density ratio bound (Proposition 3.1) and on the dimension. 11 The Wulff shape of Φ is the unique shape having the smallest Φ energy among all solids with unit volume. When Φ is isotropic, the Wulff shape is just the ball.
14 326 N. K. Yip Proof (by contradiction). Suppose that there exist a point p K and r t such that L n (K B(p,r )) Cr n (28) where C will be chosen below. (The proof for L n (K c B(p,r )) is similar.) Then L n (K B(p,r )) = r Thus there exists s with 1 2 r s r such that H n 1 (K B(p,s))ds Cr n. H n 1 (K B(p,s )) 2Cr n 1. (29) Considering the comparison set K = K\B(p,s ),wehave Φ( K) + 1 Dist(x, K )dl n t K K Φ( K ) + 1 Dist(x, K )dl n, t K K Φ( K) Φ( K ) + Dist(x, K )dl n. K K Now Φ( K) Φ( K ) = Φ( K B(p,s )) Φ(K B(p,s )). Invoking Propositions 3.1, 3.3, (29) and (28), we have θs n 1 2CΦ Cr n 1 + D t L n (K K ), t θ rn 1 2 n 1 2CΦ Cr n 1 + D Cr n t, θ 2 n CΦ t r 2 n 1. CD Choosing C small enough leads to a contradiction to the hypothesis that r t. In this paper, we just make use of the lower bound for the (n 1)-dimensional density ratio and the volume-difference estimate. However, any (Φ, t,k )-minimizer also enjoys other regularity properties: [[K]] is Bomberi (Φ,ω,δ)-minimal; spt [[K]] is Almgren (γ, δ)-restricted with respect to the empty set; spt K(t) is H n 1 almost everywhere a twice differentiable hypersurface (when Φ is smooth and elliptic). These are all stated in [ATW, Section 3].
15 Stochastic Motion by Mean Curvature Continuity and Energy Estimates We now prove the continuity and energy estimates which are crucial in showing the tightness of the probability measures induced by our time-discretization scheme. In the following, we use the notations introduced in Section 2.6. Let m be any positive integer, s t 1 and f be any C 2 function on O.WeuseC(f,m), C m to denote positive constants depending only on f, m and the size of O. Theorem 4.1. The processes K N (t) } constructed in Section 2.6 satisfy the N 1 following statements. Weak Hölder Continuity. For any f C 2 (O ), let Kf N (t) = x K N (t) f(x)dl n x. It can be decomposed as K N f (t) = SN f (t) + MN f (t). (3) (i) S N f has the estimate E Sf N (t) SN f (s) 2m C(f,m) t s m. (31) (ii) Mf N (t) is a piecewise constant function with jumps at the t i s. Moreover, for any t p,t q [, 1], E Mf N (t+ q ) MN f (t p ) 2m C(f,m) t q t p m. (32) Uniform Energy Bound. } E sup Φ( K N (λ)) m C m. (33) λ [,1] We start the proof by first defining the decomposition (3). For simplicity, we assume that t = t q +. Then set Sf N (t) = Kf N (t i ) Kf N (t+ i 1 ) + KN f (t+ ), (34) M N f (t) = <t i t <t i t K N f (t+ i ) K N f (t i ). (35) Essentially Sf N measures the changes of the sets due to the deformations by stochastic flows. Mf N describes the changes (or jumps) during the minimization steps to approximate motion by mean curvature. The proof of Theorem 4.1 relies on the use of Ito s formula to estimate various quantities. It is divided into four parts. Proof of (31) (Section 5). Proof of (33) (Section 6). Proof of (32) (Section 7).
16 328 N. K. Yip Proof of the lower bound for the density ratio under stochastic perturbations (Section 8). To prove Theorem 4.1, we write the whole evolution as a stochastic integral, and patch together the estimates of each discretized interval by making use of the statistical cancellation property of the Brownian flow. Hence we need to make sure that the evolution is adapted to the filtration upon which the Brownian flow is defined. 12 To achieve this, it suffices to show the existence of a Borel measurable map Ɣ from K to K such that Ɣ(K) gives a (Φ, t,k)-minimizer. The technicality underlying this is treated in [SV, Chapter 12.1]. It is applied to our present situation in [Yip]. We do not repeat here. Next, we rewrite (86) here for later reference. d F α (x, t), F β (y, t) = a α,β (x, y), d γ F α (x, t), δ F β (y, t) = γ δ aα,β (x, y), d F α (x, t), γ F β (y, t) = γ aα,β (x, y) where the differentiation on a(, ) is with respect to the first variable and to the second variable. Note that we use, to denote the cross variation process between two semi-martingales. Recall also the relationship (17) between ϕ and F. 5. Proof of (31) Perturbations by Stochastic Flows Without loss of generality, assume that t = t q s = t p. Then, Sf N (t q) Sf N (t p) q = Kf N (t i ) Kf N (t+ i 1 ) i=p+1 ( q ) = f(x)dl n x f(x)dl n x i=p+1 x K N (ti ) x K N (t i 1 + ) ( q ) = f(x)dl n x f(x)dl n x i=p+1 x ϕ ti 1,t i (K N (t i 1 + )) x K N (t i 1 + ) ( q = f(ϕ ti 1 i=p+1 x K N (t i 1 + ),t i (x)) det(dϕ ti 1,t i (x)) dl n x ) f(x)dl n x. (36) x K N (t i 1 + ) 12 See [KS, Chapter 1] for the definition and the need of adaptedness.
17 Stochastic Motion by Mean Curvature 329 We apply Ito s Formula to rewrite the quantities in the parentheses and establish the following expression (see (4)), S N f (t q) S N f (t p) = tq t p A(K N (r), f, dr) + B(K N (r), f, dr) where A(K N (t), f, t) is a function of bounded variations and B(K N (t), f, t) is a semi-martingale. df (ϕ ti 1,t(x)) = = n α= n α= Ito s Formula for f(ϕ ti 1,t(x)), t i 1 t f x α (ϕ ti 1,t(x)) dϕ α t i 1,t (x) n β,γ=1 2 f (ϕ ti 1,t(x))d ϕt β x β x i 1,t(x), ϕ γ t i 1,t(x) γ f x α (ϕ ti 1,t(x)) df α (ϕ ti 1,t(x), dt) n β,γ=1 2 f x β x γ (ϕ ti 1,t(x))a β,γ (ϕ ti 1,t(x), ϕ ti 1,t(x), t) dt. (37) 5.2. Ito s Formula for det(dϕ ti 1,t(x)), t i 1 t where σ } is the collection of all permutations of (1, 2,,n)and sgn(σ ) is the sign of σ. After routine calculations 13, we obtain We write det(d(ϕ ti 1,t(x)) = σ sgn(σ ) ϕ1 t i 1,t (x) x σ(1) d [ det(dϕ ti 1,t(x)) ] ϕn t i 1,t (x) x σ (n) = div F(ϕ ti 1,t(x), dt) det(dϕ ti 1,t(x)) + 1 n ( ) 2 β γ β γ a β,γ (ϕ ti 1,t(x), ϕ ti 1,t(x), t) β,γ=1 det(dϕ ti 1,t(x)) dt. (38) 13 Similar calculation has been done in [Kun, 4.3.1].
18 33 N. K. Yip 5.3. Combination of (37) and (38) d [ f(ϕ ti 1,t(x)) det(dϕ ti 1,t(x)) ] = f(ϕ ti 1,t(x))d [ det(dϕ ti 1,t(x)) ] + det(dϕ ti 1,t(x))df (ϕ ti 1,t(x)) + d f(ϕ ti 1,t(x)), det(dϕ ti 1,t(x)) = f(ϕ ti 1,t(x)) div F(ϕ ti 1,t(x), dt) det(dϕ ti 1,t(x)) + f(ϕ t i 1,t(x)) 2 n ( ) β γ β γ a β,γ (ϕ ti 1,t(x), ϕ ti 1,t(x), t) β,γ=1 det(dϕ ti 1,t(x)) dt n f + (ϕ ti 1,t(x))F α (ϕ ti 1,t(x), dt) x α α=1 + 1 n 2 f (ϕ ti 1,t(x))a β,γ (ϕ ti 1,t(x), ϕ ti 1,t(x), t) dt 2 x β x γ β,γ=1 det(dϕ ti 1,t(x)) n f + (ϕ ti 1,t(x))d F α (ϕ ti 1,t(x), t), div F(ϕ ti 1,t(x), t) } x α α=1 We can also write det(dϕ ti 1,t(x)). d F α (ϕ ti 1,t(x), t), div F(ϕ ti 1,t(x), t) = n δ aα,δ (ϕ ti 1,t(x), ϕ ti 1,t(x), t) dt. δ=1 The final formula we arrive at is f(ϕ tt 1,t(x)) det(dϕ tt 1,t(x)) dl n x K t = A i (f,x,r)dl n xdr+ t i 1 K where A i (f,x,r)denotes f(ϕ ti 1,r(x)) n β,γ=1 α,δ=1 β,γ=1 K t f(x)dl n x B i (f,x,dr)dl n x (39) t i 1 ( ) β γ β γ a β,γ (ϕ ti 1,r(x), ϕ ti 1,r(x)) n 2 f (ϕ ti 1,r(x))a β,γ (ϕ ti 1,r(x), ϕ ti 1,r(x)) x β x γ n f (ϕ ti 1,r(x)) δ x aα,δ (ϕ ti 1,r(x), ϕ ti 1,r(x)) α det(dϕ t i 1,r(x)) K
19 Stochastic Motion by Mean Curvature 331 and B i (f,x,dr)denotes f(ϕ ti 1,r(x)) div F(ϕ ti 1,r(x), dr) + In addition, we define n α=1 ti t } f (ϕ ti 1,r(x))F α (ϕ ti 1,r(x), dr) det(dϕ ti 1,r(x)). x α ti t A N (f,x,t)= i (f,x,r)dr, B i= t i 1 ta N (f,x,t)= B i (f,x,dr). i= t i 1 t Hence from (36), we get q ti Sf N (t q) Sf N (t p) = A i (f,x,r)dr + db i (f,x,r)dl n x K N i 1 + = i=p+1 tp t p t i 1 K N (r) da N (f,x,r)+ db N (f,x,r)dl n x. (4) Now A N (f,x,t)is a process of bounded variation and B N (f,x,t)is a semimartingale. Since the local characteristic (a, b) of F belongs to the class of B 1,δ ub (O ), we conclude from (113) that da N (f,x,t) C(f)dt, d B N (f, x, t), B (f,y,t) N C(f)dt where C(f) is a constant depending only on f and its derivatives up to second order. By the BDG Inequality (21), for any positive integer m,wehave tq E da N (f,x,r)+ db N (f,x,r)dl n x 2m C m E t p K N (r) tq + C m E t p K N (r) tq t p da N (f,x,r)dl n x K N (r) C m (f ) t q t p 2m + Cm (f )E tq t p C m (f ) t q t p m 2m db N (f,x,r)dl n x (x,y) (K N (r),k N (r)) d Hence the whole (31) is proved 14. 2m B N (f, x, r), B N (f,y,r) dl n xdl n y m (41) 14 There should also be a term for the bounded-variation part of the semi-martingale B N. But its estimate can be absorbed into that for A N.
20 332 N. K. Yip 6. Proof of (33) Uniform Boundary Estimates Statement (33) implies that almost surely the random measures associated with our sets have finite perimeters for all time. By the definition of minimizations, we always have Φ( K N i + ) Φ( K N i ).In between the minimizations, the set K N i + is deformed by the Brownian flow ϕ, i.e., for t i t<t i+1, Φ( K N (t)) = Φ(ϕ ti,t K N i + ). Combining the two steps (assuming that t = t q for simplicity), we obtain Φ( K N q + ) Φ( K N q ) Φ( K N q ) Φ( K N q 1 + ) + Φ( K N q 1 + ). q 1 Φ( K N + ) + Φ( K N i+1 ) Φ( K N i + ) i= q 1 = Φ( K N + ) + Φ(ϕ ti,t i+1 K N i + ) Φ( K N i + ). (42) Now we look at the term Φ(ϕ ti,t i+1 K N i + ) Φ( K N i + ) in detail. i= 6.1. Ito s Formula for Φ(ϕ s,t K), s t 1 Let K K. We borrow the notations and formulas from geometric measure theory, especially the change-of-variables formula for (n 1)-dimensional integration. We write Then Φ(ϕ s,t [[K]]) = [[K]] = t( K,1, σ ), ϕ s,t [[K]] = t(ϕ s,t K,1,σ t ). ϕ s,t K Φ(σ t )dh n 1 = K Φ( [ n 1 Dϕ s,t] σ)dh n 1. These notations can be found in [ATW, 3.1] and [Fed, Chapter 1]. We briefly describe them here. [[K]] is an (n 1)-integral current. σ denotes the approximate tangent plane of [[K]]. It is a simple unit (n 1)-vector in the Grassmann vector space n 1 Rn. n 1 Dϕ s,t is a linear map on n 1 Rn such that [ n 1 Dϕ s,t] (v1 vn 1 ) = (Dϕ s,t v 1 ) (Dϕ s,t v n 1 ), v 1,...,v n 1 R n. Let π t = [ Dϕ s,t ] σ. (It can also be treated as a vector in R n.) Since Φ is in C 2, by Ito s Formula, we have, dφ(π t ) = n i Φ(π t )dπt i i=1 n ij 2 Φ(π t)d πt i,πj t. ij