Scholarly article on topic 'Brownian motion and the formation of singularities in the heat flow for harmonic maps'

Brownian motion and the formation of singularities in the heat flow for harmonic maps Academic research paper on "Mathematics"

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Academic research paper on topic "Brownian motion and the formation of singularities in the heat flow for harmonic maps"

Probab. Theory Relat. Fields 105, 335-367 (1996)


Theory Related Fields © Springer-Verlag 1996

Brownian motion and the formation of singularities in the heat flow for harmonic maps

Anton Thalmaier

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany (e-mail:

Received: 3 March 1995 / In revised form: 12 February 1996

Summary. We develop a general framework for a stochastic interpretation of certain nonlinear PDEs on manifolds. The linear operation of taking expectations is replaced by the concept of "martingale means", namely the notion of deterministic starting points of martingales (with respect to the Levi-Civita connection) ending up at a prescribed state. We formulate a monotonicity condition for the Riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation. Such estimates lead to simple criteria for blow-ups in the nonlinear heat flow for harmonic maps with small initial energy.

Mathematics Subject Classification (1991 j:58G32, 58G11, 60H30

1 Introduction

The theory of harmonic maps connects nonlinear analysis, geometry and topology for Riemannian manifolds in a rather subtle way, see [9], [10], [11]. Harmonic maps M —> N provide a common generalization of the notion of geodesies (for M = S1 or 1) and harmonic functions (for N = M or №"). It is well-known that probability theory, namely the theory of Brownian motion, is related to the linear case of harmonic functions. Stochastic analysis provides tools to reduce (linear) partial differential equations (e.g., heat equation, Dirichlet problem) to solutions of ordinary stochastic differential equations, in such a way that the solution of the partial differential equation is given by taking expectations of a stochastic solution. Although stochastic analysis on Riemannian manifolds is well-developed [13], [14], [15], [20], an essential difficulty in applying stochastic methods to nonlinear geometric PDE problems comes from the fact that taking expectations of random variables is by definition a linear operation, ruling out,

for instance, straightforward generalizations to the harmonic map problem for general target manifolds.

There are established ways to define expectations (means) for random variables or probability distributions on manifolds, like the theory of Riemannian centres of mass [23], [30], or the notion of barycentres of measures, see e.g. [16]. Most applications of these concepts are related to random variables concentrated on domains which can be described geometrically in terms of convex geometry (see [25]). It appears difficult to adapt notions relying on convexity to situations where nontrivial topology and the global nature of the manifold is involved. The appropriate replacement of the linear expectation operators is given by a rather sophisticated nonlinear stochastic concept, namely the notion of (deterministic) starting points of martingales with the given random variable as terminal state. Martingales on manifolds depend on a linear connection, e.g., the Levi-Civita connection induced by the metric for Riemannian manifolds. With respect to this connection martingales provide a natural class of free motions relative to the given geometry.

This paper includes a discussion of the heat flow for harmonic maps in terms of martingales on Riemannian manifolds and investigates the development of singularities out of smooth initial data in finite time created by topological reasons. Our main intention is to clarify the probabilistic background of such phenomena and to establish stochastic methods in the field of global geometric evolution problems.

Throughout this paper, we assume that (M, g) is either a compact Riemannian manifold or (M,g) = (JSm, can) with the canonical Euclidean metric, and that all target manifolds N are compact. All manifolds are connected, and all maps are supposed to be smooth (C°°) if not stated otherwise explicitly. Solutions of the heat equation are always understood in the classical sense.

2 The heat flow for harmonic maps

Let (M, g) and (N,h) be smooth Riemannian manifolds and let/: M —> N be a smooth map. Associated with these data are the two fundamental forms of/.

(a) (First fundamental form of /). The pullback of the metric h under / gives a bilinear formf*h € F(T*M ® T*M) which is defined by

(f*h)x(u,v) = hf(x)(f*u,f*v) for u,v e TXM.

(b) (Second fundamental form of /). With respect to the Levi-Civita-connections on M and N one has Vdf e r(T*M ® T M ®f*TN) defined as covariant derivative of df =/„ € r(T*M ®f*TN).

By taking traces of the fundamental forms (with respect to the given metrics) we get

(i) \\df\\2 - tracef*h e C°°(M), the energy density of/, and

(ii) r(f) = trace W/ e r(f*TN), the tension field off.

Maps with vanishing tension field r(f) are called harmonic. Via the Euler-Lagrange equation they appear as stationary (critical) points of the energy functional

E(f) = !M\\df\\2dvo\

with respect to compactly supported variations of/. As in Hodge theory, where one seeks to realize de Rham cohomology classes by harmonic differential forms, a fundamental question is whether a given homotopy class of maps has a harmonic representative.

The basic existence problem is concerned with the deformation of a map f:M-^N into a harmonic map. A classical approach of determining harmonic representatives in a given homotopy class is the so called "deformation under the heat flow". Here one uses the heat equation

f Wt>u = H") on f°> x M (2 1)

I u\t=o =/

to establish a homotopy ut = u(t, •) between the initial map / = uq and a harmonic map, namely ua0 = lim, ut. Inherent to this method are several difficulties: one of them is that a priori only existence of local solutions to the heat equation is guaranteed, e.g., see [18], [22].

Theorem 2.1 (Short-term existence of solutions) Let M and N be compact. Then there exists T > 0 depending on the initial map f such that the heat equation has a unique smooth solution (t,x) >—» u(t,x)for (t,x) 6 [0, T[ x M.

Short-term existence of solutions also holds for not necessarily compact manifolds M, provided the energy density \\df\\2 of the initial map is bounded on M; then also \\du(t, • )||2 is bounded on M for any t <T. In their pioneering paper on the subject Eells-Sampson obtain the following global existence result, using strong curvature assumptions for N.

Theorem 2.2 [12] Let M and N be compact, and suppose the sectional curvature Riemw of N is non-positive. Then for any f G C°°(M ,N) the heat equation admits a unique, global, smooth solution u: [0, oo[ x M —s- N. As t —► oo, the maps u(t, •) converge smoothly to a harmonic map urx G C 00 (M, N) homotopic tof.

If the manifold M has a boundary cJM 0, then it is natural to consider the Dirichlet problem, namely whether or not a given (smooth) <p: dM N has an extension to a harmonic map u: M N with u\dM = (p. In the heat flow approach to this problem one works with the heat equation for a suitable extension/ of <fi to M, together with the Dirichlet boundary condition

The result of Eells-Sampson was extended to cover the case of manifolds with boundaries dM f 0 by Hamilton [18]. It is well-known that the curvature restriction Ricm'v < 0 can be weakened if initial and boundary data have small range [21].

The situation turns out to be much more complicated if the curvature assumption in Theorem 2.2 is dropped. In higher dimensions hardly any general global result is known for the homotopy problem unless Rlenvv < 0. There can be topological restrictions which prevent the heat flow from having any chance of converging or subconverging. For example, as is well-known [9], there exist no harmonic maps T2 —S2 with degree ±1. This implies that solutions of the heat equation for any initial data of degree ±1 cannot converge or subconverge to a harmonic map. A second even more fundamental problem is the question whether the heat flow exists for all i > 0 without curvature assumptions on N.

Local existence of solutions implies that there is a maximal interval [0, T[ where 0 < T < oo such that the solution exists on this interval but cannot be extended beyond. Whether the case T < oo, namely "blow-up in finite time", is possible to occur has been a challenging problem for a long time (see [10], p. 63). It is meanwhile known [4], [3] that blow-up in finite time is a natural phenomenon if the initial map mo =/ belongs to a nontrivial homotopy class and the initial energy E(uo) is sufficiently small.

Theorem 2.3 [4] Let M and N be compact, dim M > 3, and let be any nontrivial homotopy class in C(M ,N) such that

E& :=inf{E(f):/ e Jg? n C°°(M,N)} = 0.

Then there exists e > 0 such that iff G n C°°(M ,N) and E(f) < e then the solution of the heat equation with initial condition f blows up infinite time T. Moreover T = T(f) 0 as E(f) —► 0.

A result of White [44] guarantees = 0 for many nontrivial homotopy classes of maps M —> N. Namely, let M and N be compact Riemannian manifolds and u0 E C°°(M ,N), then

inf{£(f) :/ e C°°(M,N), f ~ M0} = 0

if and only if Mo is 2-homotopic to a constant map (i.e. 7t, (mo) = 0 for i < 2). This is equivalent to the condition that the restriction of uq to the 2-skeleton of some triangulation of M is homotopic to a constant map. Specifically, any map Mo € C°°(M,Ar) is homotopic to maps with arbitrary small energy, for instance, if

(a) 7Ti(M) = 0and7T2(M) = 0, or

(b) tti(N) = 0 and 7T2(/V) = 0, or

(c) tti(M) = 0 and tt2(AO = 0.

Note that if Ricm'v < 0 then 7t,(A') = 0 for i > 2, and any map M N that is 2-homotopic to a constant is already homotopic to a constant.

Heat flow is energy decreasing and, if the energy is sufficiently small, the deformation goes towards constant maps; blow-ups occur if this is impossible for topological reasons. The way analysis and topology combine in the heat flow to create singularities in finite time is far from being completely understood. In this paper we like to demonstrate that stochastic analysis provides natural tools to deal with such questions.

3 Brownian motion and harmonic maps

Let , P\(J%)tei+) be a given filtered probability space fulfilling the usual

conditions. An adapted stochastic process X with continuous paths on a Rie-mannian manifold (M,g) is a Brownian motion if for each <p € C°°(M) with compact support

<poXt-voX0-t [ (AMip)oXsds (f>0) (3.1)

defines a real-valued martingale; here Am is the Laplace-Beltrami operator on (M, g). We say X is BM(M, g), and write X = Xx if X§ = x a.s.

There is an intrinsic method of constructing Brownian motion on a Rieman-nian manifold (M,<?) of dimension m. Let W be a flat BM(1"). Denote by L\,..., Lm the canonical horizontal vector fields on the orthonormal base bundle O(M) over M, given by L,(k) = hu{uei) where h: tt*TM —> TO(M) is the horizontal lift induced by the Levi-Civita connection on M. Orthonormal frames u are read as isometries u: Wn —> T^U)M, and <?,■ stands for the /-th standard basis vector in ffim. Let uq be a ¿^-measurable random variable with values in O(M). Solving the 0(A/)-valued Stratonovich differential equation (SDE)

dU = £ Li (U) * dWl, U0 = u0, (3.2)

gives horizontal Brownian motion on 0(M), and the projection X = n o I J of the solution process U down to M defines a BM(M. g), started at x0 = ix o m0; see [14].

This geometric procedure of solving SDEs on the orthonormal base bundle O(M) can be extended to construct larger classes of random motions on M if we replace the Brownian motion W in (3.2) by a continuous semimartingale Z on ffim (started at 0). In this case, X = no U on M is called stochastic development of Z [15]. Thus, induced by the Levi-Civita connection on M, one can associate to each continuous semimartingale Z on 1™ a stochastic development X on M, together with a horizontal lift U of X on O(M), and therefore a notion of parallel transport along the paths of X via

//o,t = UtoU0~1: TXoM ^ TXlM. (3.3)

Stochastic development can be reversed in order to find the "anti-development" Z (with values in ffim) from the process X on M. More precisely, given a semimar-tingale X on M, we may construct U as horizontal lift of X on O(M) (starting at some initial frame uq above xq) and get the corresponding anti-development Z as a Stratonovich integral of the canonical connection 1-form'd on O(M) along the process U (see [14], [15], or [19]), namely as

where 1? G r(T*0(M) ® Mm) is the canonical 1-form of the connection, defined by i?u(e) = m_1(7r„e) for e G T„0(M). Thus, given one of the three processes Z, X, or U, up to a specification of the starting variables, the two others may be constructed. Note that the Riemannian quadratic variation [X,X] = Jg(dX,dX) of an M-valued semimartingale X depends only on the martingale part of its antidevelopment, namely [X,X] = [Z,Z] where [Z,Z] = [Z^Z1] + ... + [Zm,Zm] is the usual (Euclidean) quadratic variation of Z.

An adapted process X with values in the Riemannian manifold (M,g) is called a V-martingale (with respect to the Levi-Civita connection V) if X is the stochastic development of a continuous lRm-valued local martingale Z (see [15]). We only consider V-martingales with respect to Levi-Civita connections, thus we omit the specification of the connection in the sequel. The class of all martingales on (M,g) is denoted by Mart(M,g). Using the functional characterization of manifold-valued martingales, due to Darling [6], which relies on the richness of germs of convex functions, it is easy to see that the martingale property is a local property; hence there is a straightforward meaning of M-valued semimartingales being a martingale on stochastic intervals of the form [ct,t[ or ]a, r[.

BM(M, (¡) constructed by stochastic development lives on the canonical Wiener space that carries BM(Mm). The standard filtration on the m-dimensional Wiener space will be referred to as (m-dimensional) Brownian filtration in the sequel. All manifold-valued processes will be defined on some fixed m-dimensional Wiener space with its Brownian filtration, but with m not necessarily the dimension of the manifold. For instance, the image / o X of an M -valued Brownian motion X under a differentiable map f:M —< N is a semimartingale on N adapted to the m-dimensional Brownian filtration with m = dimM.

A consequence of the adaptedness to a Brownian filtration is that M -valued semimartingales start at deterministic points. Hence, in the correspondence between the continuous Rm-valued semimartingale Z, its M-valued stochastic development X, and the O(M)-valued horizontal lift U of X, there is only a choice for the initial frame L'() = u above Xo = x. Such a choice of u means selecting an isometry u: Mm T^U)M via z z'm . Under the identification of №m

and TXM via u G 0(M)x we may consider Z = uZ with values in TXM rather than Z with values in ®m. The horizontal lift U then still depends on u, but not the induced parallel transport //01 = Ut ° t/0_1 - Ut o u~l along X.

Maps between manifolds may be studied under the aspect how they transform certain classes of random motions, for instance Brownian motions, see [6]. The

action of a map / on BM(M, g) is described in terms of the first and second fundamental form of/.

Theorem 3.1 Let f: (M,g) —> (N,h) be a smooth map between Riemannian manifolds. Let X be a semimartingale on M with Xq = x, which comes by stochastic development from aflat semimartingale Z on TXM. The image process X =/ o X is a semimartingale on N with Xq -f{x), and hence determined by its anti-development Z in Tf^N. If ¡jt 0 = U0 o Ut~l and //( 0 = U0 o Ut~l denote the parallel transports along paths ofX, resp. X, we have

dZ = //~ df //0ji dZ + i //~0 Vdf(dX,dX). (3.5)

Specifically, ifX is a BM{M ,g), and correspondingly Z a BM(TXM), we get

dZ = //~0 df //0>( dZ + i//~0 r(f) oXdf, (3.6)

in this case the Riemannian quadratic variation [X,X] = Jh(dX, dX) reads as

d[X,X] = Zh(f*Uehf*Uei)dt=(\\df\\2oX)dt. (3.7)

The proof of Theorem 3.1 is based on the geometric Ito formula; see [15], or [19] pp. 442-4, for details. Equations (3.6) and (3.7) show that the martingale part in the anti-development of / o X measures the energy off along the path of X, while the drift part captures the tension field of/ along X. Specifically, a smooth map/: M N is harmonic if and only if it maps ¥>M(M, g) to MartOV. /?).

4 Expectations on manifolds

In this section we comment on various approaches to an intrinsic definition of expectations (means) of random variables on Riemannian manifolds; see [30], [23], [16], [25], [36], [1] and [2].

Definition 4.1 Let (N ,h) be a Riemannian manifold and £ an /V-valued random variable defined on a probability space {Q,^,P)\ further fix vo G N.

(i) (Barycentres) The point yo is said to be in the barycentre of £ if ^(yo) < E [ip o £] for all bounded convex functions [/ —> ffi such that U C N is open, convex, yo € U, and range (£) c U, a.e.

(ii) (Riemannian centres of mass; "Cartan means") Suppose that £ is L2-inte-grable in the sense that ]E[distTv(xo, ' ^ for some xq e N. The point yo is said to be a Cartan mean of f if yo is a local minimum of the function Q: N -» ffi+ given by Q(x) = E[disW(x, O2] •

(iii) (Martingale means) Let (^)t>o be a filtration on (JL.j^,P) fulfilling the usual conditions such that & = Then the point y0 is said to be a martingale mean of £ if there is a uniformly integrable martingale Y on

N, adapted to (¿%)t>o> starting at v'o and ending up at i.e., Y() = y0 and

Foo = a.s.

Each of the above concepts relies on a different aspect of the geometric setting, e.g., the notion of convexity, the distance function induced by the Riemannian metric, or the idea of a drift-free random motion. Only the martingale mean requires a filtration; it uses the starting point vq of a martingale F ending up at Fqo = £ as a substitute for the missing expectation "E [£]". (In addition, one may take Ys as replacement of the conditional expectation [£]"). Note that if -W) is trivial, e.g., in case of the Brownian filtration on Wiener space, adapted processes start at deterministic points. By definition, the martingale interpolating between yo and § is assumed to be uniformly integrable in the sense that its anti-development is an uniformly integrable martingale on M". Martingales on manifolds include continuous local martingales on Euclidean space; thus without the restriction to uniformly integrable martingales this notion would be too wide.

Example 4.2 Let (^fi.-W,P\(¿W)tea+) be a filtered probability space such that M is trivial and ^ = &. Take N = 1™, and let £ e Then

Fs = v'" .s > 0, defines a uniformly integrable martingale on M" with limit Y00 = f, a.s., and starting point Y0 = = I [£].

In Example 4.2 the martingale itself, but not its starting point, depends on the filtration — a situation that changes if we consider random variables £ with values in general Riemannian manifolds (N,h). Note that in the example yo = E[£] is also a Cartan mean and obviously in the barycentre of £ by Jensen's inequality.

In the context of the martingale mean it appears natural to keep track of the "size" of the interpolating martingales. As mentioned above, the Riemannian quadratic variation of an N-valued martingale coincides with the quadratic variation of its anti-development in K". On the other hand, the size of a martingale on Euclidean space is measured by its quadratic variation process. Note that martingales Y with prescribed end state F00 automatically fulfill [Y -Y \ ^ < oo a.e., as a consequence of the martingale theorem.

For p > 1, a martingale Y with values in a Riemannian manifold (N,h) is called an HP-martingale if ffi([F,y]^2) < oo where [Y, Y] = f h(dY,dY) is the Riemannian quadratic variation of Y (see also [8]). A martingale mean y0 of an N -valued random variable £ is called an Hp -martingale mean if there is an Hp-martingale begun at y0 and converging to q. Then, the //''-norm

serves as a specification of the goodness of the mean value Fo = yo for the variable

Giving a brief comparison of the concepts in Definition 4.3, we remark that (at least on sufficiently small domains) both Cartan means and martingale means are compatible with barycentres (see [25]). For instance, let yo be a martingale mean for a random variable £ where F is the corresponding uniformly integrable

martingale with Y0 = y0 and Y^ - Then, for all (p € C°°(N), say such that \\d<p\\ is bounded, we have:

<p o Ys = ofi-iE3*[Js°° Vdif(dY,dY)] .

As a consequence, <p(yo) and E [(p o £] differ by the nonlinear correction term

|E [¡¿°Vd<p(dY,dY)]

depending on the complete martingale Y. Obviously, the correction is 0 if Vdip = 0 (i.e., ip affine), and <p o y0 < E [ip o £] if Vdip > 0 (i.e., tp convex). The obvious problem with barycentres comes from the fact that in general there are not enough convex functions to specify appropriate means for large range random variables. For example, on a compact Riemannian manifold (without boundary) there are no globally defined non-constant convex functions at all.

We like to stress that there is fundamental conceptual difference between Cartan means and martingale means which makes these notions incompatible in any general setting. For instance, fix an J^o-measurable random variable £ with values in N. For simplicity, suppose that range(0 C B C N where B is a sufficiently small regular geodesic ball such that dist^ = distg on B xB; here distB is the restricted metric on B defined by taking into account only curves connecting x and y within B. If y0 € B is a Cartan mean of then in particular, Vo is a critical point for

Q:N^ M, <2W = E[distw(x,02] ,

or, in other words, Efexp^1^)] = 0. Thus, for a Cartan mean yo of there is a martingale Y in TyoN, namely Ys = ¡exp"1^)], such that the /V-valued process Y = expyo Y starts at and terminates at i.e., Foo = In general, Y will not be a martingale unless expyo is totally geodesic. On the other hand, if yo is a martingale mean of then there is a martingale Y on N such that Y0 = y0 and Fqo = £ a.e. By definition, Y is the stochastic development of a martingale Y in TyoN = Mm. In both cases there is a martingale Y in TyoN, but dealing with the Cartan mean implies that Y is transported onto N via the exponential map (at the fixed point yo) to give a process Y on N connecting yo and £ (in general only a semimartingale), whereas the concept of the martingale mean uses the more complicated procedure of stochastic development to transport Y onto N: the frame in TyoN (used to identify TyoN and Mm) is carried along Y by stochastic parallel transport.

Throughout this paper, we restrict ourselves to H2-martingale means which is an appropriate class for the applications we have in mind. The term "martingale mean" will be used in this stronger sense. As pointed out, for //2-martingales Y,

Mr .KLo) = E[/0°° h(dY ,dY)\

measures the "size" of Y and specifies a "distance" of the mean Y(j = y0 to the variable In case uniqueness for Y fails, the (squared) H2-norm E([F, Y].x/)

allows one to compare different mean values and to specify a hierarchy of means to the random variable

In this paper the /V-valued random variables £ will be of the form £ -foXf for some t, or slightly more general, £ = / o X* for some stopping time r; here Xx is a BM(M,g) and/: M N a smooth map. The relevant filtration is the m-dimensional Brownian filtration on Wiener space where m = dim M. Since the class of martingales is invariant under transformations of the time scale, martingale means for random variables of the above type are covered by prescribing terminal values as t —oo. The Hp-norm of martingales is unaffected by a time change.

Example 4.4 For n > 2, let (S" ]. d'iP) be the (n — l)-dimensional standard sphere. The map

/: IT\{0} Sn~l,

is harmonic. Hence, for a BM(M") X = Xx started at x ^ 0, the angle process 6X:

Ox=f oXx: t > 0, defines a martingale on Sn~1. Its Riemannian quadratic variation is given by

d[ex,ex] = (n-i)(Rrr2dt, r = 11*11,

where Rr = ||XX|| is a Bessel process of dimension n started at r > 0. Thus, for x ^ 0 and any t > 0,

M\0x,ox]t) = wJ{gd[ex1exi) <oo,

and the point f(x) G Sn~l is a C/Y2-) martingale mean for the random variable £ = 6>f. But Edé?*,©*]«) oo, as * -> 0; hence, for x close to 0, jc f 0, the mean/(x) requires a martingale with large Hz-norm, if described by (<9*)o<s</-Small variations of Of, caused by varying x near 0, give rise to big changes in fix).

Note that, for x = 0 in Example 4.4, the angle process (0'J)S>O provides a martingale on (Sn~\ dû2) without a starting point. We get E(/£! d[0°, 6>0]) < oo for any 0 < £ < f, but E(/O'd[<9°,00]) = oo for t > 0. Thus <9° may be considered as martingale begun at t = —oo in its "intrinsic time".

5 The stochastic representation of the heat equation

We start discussing some nonlinear PDE problems on manifolds under the aspect of taking expectations on manifolds. Consider the following three types of problems.

(i) (Nonlinear heat equation). Let M and N be (compact) Riemannian manifolds and let/: M —> N be smooth. The map/ is deformed using the heat equation:

fM = ±r(w), u\t=0 = /.

(ii) {Nonlinear heat equation with a boundary condition). M is allowed to have a boundary dM f 0 and, as additional data, a smooth map dM —> N is given. The problem is solving the heat equation (5.1) as in (i), but together with the Dirichlet boundary condition u(t, ■)\dM = <p. (Note: J'\sm = '?)■

(iii) (Dirichlet problem). M and N are again (compact) Riemannian manifolds; M with boundary dM f 0, together with a smooth map <p\ dM —> N. The problem is to find u: M —> N harmonic with u \ ¡jm = <j>.

Note that in the flat case N = M" solutions are given in terms of Brownian motion by taking expectations, namely as

(i) u(t,x) = E[/ oXf] for the heat equation,

(ii) u(t,x) = E[/~ oXfATx] for the heat equation with a boundary condition,

(iii) u(x) = E [<f> o X*x] for the Dirichlet problem,

where Xx is BM(M, g) started atx, and tx = inf{,v > 0 : Xf £ dM} denotes the first hitting time of dM. These formulas are meaningless in the general situation; nevertheless there should be a stochastic interpretation involving appropriate expectations (means) of the N -valued random variables / °Xf,f °XfArX, or/' cX^, respectively. The key observation is given by the following theorem.

Theorem 5.1 For M compact, let u: [0,r[xM —> N be a smooth solution of the heat equation (5.1) and (t,x) G [0, T[ x M. Then the N-valuedprocess

is an H2-martingale on (N,h) with Fo - u(t.x) and Y, =foXf. Hence, u(t,x) is an H2-martingale mean of the N-valued random variable £ =/ o Xf.

Proof. We consider the jV-valued semimartingale Y as image of the 2+ x M-valued semimartingale Xs - (s.Xf ), 0 < .v < /, under the map (s,y) h u(t -s.y) and apply Theorem 3.1 to determine its anti-development Z in Tf^N. The fact that u solves the heat equation is equivalent to a vanishing drift component of Z; namely from formula (3.5), resp. (3.6), we get (modulo differentials of local martingales)

(//s 0 denotes parallel transport along Y). Hence, Y is an N-valued martingale, and obviously, Fo = u(t,x), resp., Y, = u(0,Xx) = f oXf. The Riemannian quadratic variation of F is given as

Ys = u(t — s,X*), 0 < s < t,

dZ = //sfi(~d,u + \r{u)){t - s,X*)ds =0

d[Y,Y] =h(dY,dY)= ||du(t-s, -)\\2(X*)ds .

From (5.4) we conclude immediately E([F, Y]t) = E(/¿h(dY,dY)) < oo, verifying that F is an H2-martingale. □

The conclusion of Theorem 5.1 also holds for not necessarily compact manifolds M if : du 112 is bounded on [0, t]xM. Moreover, Theorem 5.1 can immediately be adapted to the case dM f 0. For instance, if u: [0, T[ x M —> N is a solution of the heat equation with a boundary condition, then, for (f, x) e [0 ,T[xM,

Ys = u(t — s,X*Atx), 0<s<t,

is an ff2-martingale on (N,h) with Yo - u(t,x) and Yt =f oXfArX. Likewise, if u: M N solves the nonlinear Dirichlet problem,

Ys=u(X^), s> 0,

is an i/2-martingale, started at u (x) with limit u oX*x as .? oc.

6 Some non-uniqueness results

We now briefly describe some examples indicating that martingale means on manifolds are quite a delicate object, and which illustrate some of the issues arising in a global theory of expectations on manifolds. Here we mainly stick to the lack of uniqueness in the nonlinear Dirichlet problem.

Let M = Bi(0) c W" = {x e !m+1 : xm+1 = 0}, the closed unit ball in Mm, and N = S"', the standard m-dimensional sphere. We think of S'" as the unit sphere in Mm+1 and view M as unit ball in the equatorial hyperplane ffim+1 n {xm+1 = 0}. Thus dM = S"1-1 ^ Sm as equator with the inclusion i(x) = X.

Let X be BM(mm), started at X0 = 0, and r the first hitting time of dBx(0). The underlying filtration is the standard m -dimensional Brownian filtration. We take the S"-valued random variable £ = XT (actually with values in the equator S"' ~ 1 of S'"), and look for martingales Y on S'" with terminal value that is YT = £ a.s. Recall that this question is related to the Dirichlet problem of finding harmonic maps u: B\(0) —> Sm such that u equals the identity map on dBi(0) = Sm~l C Sm: for each such u the composition Y = u oX is a martingale with the desired property.

First, we look for martingales starting from the south (or north) pole and ending up at For this, let (i). r) be polar coordinates in B\({)), and write X = (&,R) such that 0 gives the angle of X on Sm and R the Euclidean distance of X to the origin. We restrict ourselves to "rotationally invariant" martingales Y on Sm of the form

Y = (0cos/z oR, sink oR) (6.1)

for some suitable C2-function /i: [0,1] —> M such that in addition Yo = south pole, i.e., h(0) = —7r/2, and YT = i.e., h( 1) = 0. The condition on h to make Y a martingale reads as

d2h m-ldh m~ I

— +- +__sm2ft=0. (6.2)

drz r dr

It is an elementary substitution to transform (6.2) to the equation for a damped pendulum which can be analyzed by standard methods (see also [40], pp. 92-93).

It follows that for m > 7 any solution h of (6.2) with h{0) = -7t/2 , h'(0) > 0 lies below the line h - 0, and is increasing and asymptotic to h = 0. This implies that there is no martingale of the type (6.1) with F0 = south pole and YT =

For 3 < m < 6, however, there are infinitely many different martingales starting from the south pole and ending up at Any solution h of (6.2) with h(0) = -7r/2, h'(0) > 0 is now asymptotic to the line h = 0 and crosses this line at infinitely many points 0 < r\ < r2 < r3 < ... with rt > oc; hence, as a consequence of the scale invariance of (6.2), for each i £ N

hi(r) = h(n r), 0 < r < 1,

gives a solution of (6.2) on [0,1] satisfying the boundary conditions /i, (0) = - Tr/2 and hi( 1) = 0. The corresponding martingales on S"

r = (0cos hi o R, sin hi oR)

possess Riemannian quadratic variation given by

d[Y', F'] = (frhi)2(R)dt + {m - 1) R~2 cos2 hi(R)dt.

Exploiting the fact that, by the scaling property of BM, for any r > 0 the processes ((1 jr)Rrit : t > 0) and (Rt : t > 0) are equivalent, we get

E([y!',i"']T) < E([Fi+1,y,+1]r) fori =1,2...

There are several features which should be stressed here. First, only the martingale Y1 of least H2-norm

||F1||//2 = (E[y1,F1]T)1/2

lives completely in the lower hemisphere. A remarkable feature is the failure of estimates on the size of the martingale in terms of the boundary data, as can be seen from

||l,i||^2 = E([lM',I"']r) ^oo as i —> oo .

Moreover, as i oo, note that Yt\T YtJ:T a.e. for each t > 0 where Y'x- = Uoo(X) and Wqq(x) = for x 4 0 in B\ (0). As explained in Example 4.4, the martingale Y°° = (0,0), defined on the interval ]0, r], is a martingale without a starting point. It corresponds to h = 0 in (6.1) and has the property that ¡¡ATd[Y°°, F°°] < oo for each £ > 0, but JJ d[Y°°, F°°] = oo. Nevertheless, if we consider the sequence of harmonic maps m, : B\ (0) —> Sm,

Ui(x)= (j^yCOS^(||x||), sinhi(\\x\\)) ,

it is true that the total energies £(«,) are uniformly bounded, i.e., Eiu,) < c < oo, while \\dm ||2(0) oo as i oo. This failure of interior estimates on harmonic

maps with bounded total energy is again in sharp contrast to the behavior of harmonic functions. We reconsider such phenomena in the subsequent sections.

Finally, if we also take into account martingales F on 5" with YT = not necessarily of the rotationally invariant form (6.1), starting points of such martingales may be quite arbitrary. For instance, if m = 3, then it is known [37] that for any xo <EBi(0) c IS3, there is a smooth harmonic map

u: 5i(0)\{*0} - S2

such that u\dBi(0) = icW^o)- Each such map u = uxo for xo^O composed with the Brownian motion X on M3 (started at Xo = 0) provides a martingale Y - u o X on S2 C S3 with F0 = m(0) and YT = For any 0 f x0 £ Bi(0), we thus get a martingale on S that actually lives on the equator of S3, ending up at the prescribed value By studying the construction of the maps u = uXQ [37], it is not very difficult to see that for appropriate choices of xo we can achieve that every point of S2 c S3 appears as initial point uXQ(0) of uXQoX; moreover uXQ oX is even an //2-martingalc. Translated into our terminology, this says that each point on the equator is a martingale mean of

We emphasize that the above results should not be interpreted as artifacts of an insufficient conceptual framework; they just reflect the topological nature of the problem and the nontriviality of the Dirichlet problem in this context.

7 Long-term behavior of the heat equation

As explained in section 5, associated to the nonlinear heat equation is the following reachability problem for V-martingales. Given m0 € C°°(M ,iV), a Brownian motion X* on (M ,g) such that Xq = x, and the random variable £ = uq a Xf for some x £ M, t > 0, the problem is to find an N-valued //2-martingale y = (7s.)0<s<f with Y, = £ a.e. The filtration is the m-dimensional Brownian filtration (m = dimM) with respect to which Xx is defined. The relevant question is how such a martingale (unambiguously defined as constant martingale for t = 0) changes as t increases.

There are results that guarantee existence and uniqueness of martingales with prescribed end states (see [25], [34], [35], [36], [7], [8]). They naturally require strong restrictions on £ for existence (either on the range of £ or on the norm of the derivative of £ considered as a smooth Wiener functional); uniqueness is usually only given within a certain class of martingales with sufficiently small Riemannian quadratic variation. The results of Kendall [25] give existence and uniqueness of martingale means for random variables with values in small domains, like regular geodesic balls; they exclude effects caused by global geometry; see also [26], [27], [28]. Results of Picard ([35], Theorems 2.2.1 and 3.1.1) cover random variables of the type £ = u0 o Xf (w0 smooth, M and N compact) at least for sufficiently small t and provide existence and uniqueness of martingale means in this case. Darling [7] constructs V-martingales on M" with

prescribed terminal value under local Lipschitz and convexity conditions on the connection V.

For small t, existence of martingales ending up at £ = «o ° X? at time t is guaranteed by Theorem 5.1 and the short-term solvability of the heat equation: There is always a smooth solution u: [0. T\ x M —> N of the heat equation with initial condition u(0, ■) = "(>, provided T > 0 is sufficiently small, and then

Ys -u(t - s,Xx), 0 <s<t,

is the martingale on N starting at u(t.x) with Y, = uQ oXf a.s. Its Riemannian quadratic variation is given by d [Y ,Y\ = \du(t — s. ■ )||2(X*)ds. We may choose T = T(uq) < oo such that [0, 7 [is the maximal interval where the solution exists.

Definition 7.1 Let u0 € C°°(M ,N) be a map between Riemannian manifolds (M ,g) and (N, h). For t <T and x e M, let u(t. x) eN be the martingale mean of Mo oXtx, as defined by the heat flow deformation of w0. We say that u(t, ■) blows up at time t = T if there is a point xo € M such that, for any e > 0,

limsup sup \\d u(t,x)\\2 = oo .

t/T x£Be(x0)

Thus, given the situation of Definition 7.1, we have one of the three alternatives:

(i) Blow-up in finite time, i.e., there exists T* = T*(uo) > 0 such that u(t, •) is regular for 0 < t < T* < oo, but blows up at t = T*(u0).

(ii) u(t, ■) is regular for all time, but blows up at T* = oo.

(iii) u(t, •) is regular for all time, and u(t. ■) subconverges to a map uX: = u(oo, •) which is necessarily C°° and harmonic; moreover ua0 is homotopic to MoLemma 7.2 Let u: [0, T[ x M —> N be a smooth solution of the heat equation

and let e(u) - denote the energy density of u. Then it is true that

(f -\AM)e(u) + \\Vdu\\2 < KMe(u) + KNe(u)2, (7.1)

where Km depends on the Ricci curvature of (M, g), and KN denotes an upper bound for the sectional curvature of(N,h).

Lemma 7.2 is a well-known consequence of the Bochner formula for the energy density e(u) of u (see [9], section (6.8)). In particular, it shows

{§-t-\AM)e(u)<ce(u){\+e(u)) (7.2)

with a constant c = c(M,N) depending only on the geometry of M and N. Estimate (7.2) is the basic tool to establish small time solvability of the heat equation.

Moreover, estimate (7.1) easily allows one to exploit the role of negative curvature of the target manifold in the heat flow for harmonic maps; for simplicity, assume that both manifolds are compact.

Theorem 7.3 Let (M ,g) and (N ,h) be compact Riemannian manifolds, and suppose that the sectional curvature of N is non-positive, i.e. Riemv < 0. Furthermore, letu: [0, T[ x M —> N be a smooth solution of the heat equation. Then, for 0 < t < T and any x £ M,

E[\\du\\2(t -s,XZ)] < IE,[||du\\2(0,Xx)} eKu('-s), 0 < j < t, (7.3)

where the constant Km is given by (7.1) and Xx is BM(M,(/) with Xq = x. In particular, for s -0, we have the a priori estimate

\\du\\2(t,x) < E[\\du\\2(0,Xx)] eKMt, (7.4)

which excludes blow-up in finite time.

Proof. Let 0 < s < so < t. First, note that Km ~ 0 in (7.1), as a consequence of the curvature assumption Riem v < 0. Thus, by means of Ito's formula,


= E[e(u)(t - s0,X*)] + / - iAM)e(u)(t - p,Xxp)} dp

■J s

<E[e(u)(t-s0,X*Q)]+KM / E[e(u)(t - p,Xxp)]dp,

■J s

and hence, by Gronwall's Lemma,

E[||^||2(f < E[\\du\\\t -so,^)] eK^'s\

which completes the proof. □

Obviously (7.3) also gives a bound for the H2-norm of martingales starting at u(t,x) with end state uo oXx in terms of the initial energy E(uq).

Returning to the general case, the above discussion provides the following stochastic picture. Given u0 e C°°(M,N), for t < T* = T*(u0), there is the martingale Ys = u(t - s.X'), 0 < s < t, starting from u(t,x) and ending at m0 o Xf. If blow-up occurs at finite time T* < oo, as t / T*, the //2-norm of this martingale will increase to oo (for some x0). Nevertheless, for t > 7*, there may be new //2-maitingales with different starting point and terminal state m0 o Xf. Since, for instance, Brownian motion on (M, g) almost surely never hits subsets of codimension at least two, the stochastic description for t > T* is not affected by singularities of u(T*, •) in a subset of codimension at least two; for the martingale u(t-s,Xx), 0 < s <t, singularities in a polar set are insignificant because Xx will never see them (if not started in such a point). In particular, this will be the case if there is a smooth solution (t,x) ^ u(t,x) of the heat equation on a dense open subset of 1+ xM whose complement S has the property that

BM(M,<7) almost surely never hits any of the sections ({*} x M) fl £ (if not started there). Then, for any t > 0, the martingale u(t- s,X*) is well-defined for 0 < s < t, up to the possibly non-specified starting point u(t,x) if (t,x) <E S. Of course, at a singularity the homotopy class of the mapping u(t, ■) may change. In analogy to the classical expectation we use the suggestive notation

u(t,x) = ^iuooXf), t<T*,x£M,

for the martingale mean of u0 o Xf defined by the unique solution of the heat equation with initial condition u(0, - ) = uq\ further we write

^(«0°^), 0 < 5 < i ,

for the corresponding martingale Ys - u(i - ,s. X*), 0 < s < t. Note that the Markov property and the law of iterated expectations hold in this context. Indeed, if we write ut = S, (uq) for the solution u (i , ■ ) of the heat equation with initial condition u(0, •) = u0, then Ss (St(uo)) = Ss(ut) = Ss+t(uo) for s+t < T* = T*(u0) by the unique solvability of the heat equation; therefore in stochastic terms

ut O x; = g-(«o o Xj) \y=xi = ^(u0 O XfH), and

as long as s +1 < T*. Especially, we get = ^[^(uq oI/)|j=^].

Taking iterated expectations then gives, as in the usual theory,

(u0 o Xf)] = ^(«o ° Xf), for 0 < r < s < t < T*.

8 Blow-up in finite time - an example

The construction of martingales with prescribed terminal state leads to backward SDEs. It lies outside the intention of this paper to deal in general with the topic of finding such martingales using stochastic methods (see [7] for results in this direction); in the sequel we use martingale theory as a way to comprehend qualitative aspects of the nonlinear heat equation. In this section we discuss an example for finite-time blow-up; only in context of this example we briefly sketch the aspect of backward SDEs.

Let M be an m-dimensional differentiable manifold and r, n positive integers. Given a vector field Aq C r(TM) and a homomorphism A: M xW — TM of vector bundles over M, we consider the differential operator in Hormander form

with A, = A(■, Cj) e r(TM); in addition, we fix differentiable maps F: M x M" x Horn®'', IT) ->■ IT and/: M 1" .

Problem 1. Find a smooth function w: [0, t] x M —> }'n solving

f wu =Lu

1 u\s=0 =/■

J^-w = Lu + F(x,u,du • A) 5 (*)

Problem 2. Let X be a diffusion on M with generator L, constructed as solution of the Stratonovich SDE on M,

dX = A0(X)dt+A(X)*dW X0=x ,

where W is BM(Mr) on Wiener space; further let £ = f o X,. Find continuous adapted M"-, resp. Hom(Mr,M")-valued processes (Fs)o<s<i and (Cs)o<s<i such that

dY = C dW -F(X,Y,C)ds


Example 8.1 Let F = 0, and suppose £ G L2(P;M"). Then a solution (Y, C) of (**) is given by

rI=E3*K] = EK]+ f CdW, Jo

where the matrix process C is determined through Y via Ito's representation theorem.

Lemma 8.2 (i) Any solution of (*) gives a solution of (**) via Ys = u(t —s,Xf), Cs =du(t — s,X*)A(X*, •); in particular, Yq = u(t,x).

(ii) Under a global Lipschitz condition for F, and if in addition £ G L2(P;®ft), there exists a unique pair (Y,C) of square-integrable continuous adapted processes solving problem (**).

Proof. Part (i) is checked directly by means of Ito's formula. Part (ii) is covered by the work of Pardoux-Peng (see [32], [33]) on backward SDEs. □

Note that in PDEs connected to the harmonic map problem the function F depends in general quadratically on du, thus violating a global Lipschitz condition for F. Hence, the existence part of Pardoux-Peng cannot be used directly to determine martingale means.

The heat equation as a system of nonlinear parabolic PDEs is hard to deal with explicitly. With enough symmetries, however, the problem can be reduced to a scalar equation in only two variables.

We consider the following example, studied in [5]; see also [17]. Let M = Mm and N = Sm, m > 3, and let u0: lm ->• Sm be defined by

uq(x) = cosfto(r), sin/zo(r)) , r = ||x|| , (8.1)

where ho: [0, oo) —> M is differentiable, monotonic increasing, /?o(0) = — 7t/2, and ho(oo) = (2k + l)j with A; + . Let u(t, •) be the solution of the heat equation with u(0, ■ J = uq, in other words, u(t,x) = ??(uo°Xf), using the terminology of the last section. By symmetry considerations, we have

u(t,x) = (—cosh(t,r), sinh(t,r)j (8.2)

with an appropriate function h(t,r) of two variables. Hence, given the Sm-valued random variable u0 oXf for some fixed t > 0 (sufficiently small), the martingale Y with initial point Yy = u(t,x) and prescribed end state Y, = uq o Xf is of the form

Ys=u(t-s,X*)=(Gxscosh(t-s,Rrs),smh(t-s,Rrs)], 0<s<t, (8.3)

where Xx = (Ox ,R' ) denotes the decomposition of BM(Mm) into its angle and radial process. It suffices to determine the function h(t,r).

Remark 8.3 A process Y of the form (8.3) is a martingale on Sm with Y, = uqoXx if (8.2) solves the heat equation with u(0, ■) = uq which is equivalent to

dh 1 d2h m — 1 dh m — 1

dt 2 dr2 2r dr 4r2

sin 2h=0 where h(0, ■) = h0( ■). (8.4)

The Riemannian quadratic variation [Y ,Y\ of Y is given by / Q \ 2 fjj _ 1

d [Y, Y] ={—h(t-s,R))ds + —^cos 2h(t-s,R)ds. (8.5) \or J Rz

Now, let uo represent a nontrivial homotopy class in C(Mm,5m). Suppose that (8.2) is always extendable to a global smooth solution icl+xl™ —► Sm of the heat equation, u(t,x)= <?(uo °Xtx) for any (t, x) 6 M+ x Mm. The energy of M0 is given by

£(«„) = voKS-1) r [(g)2+ ^ cos2 A

rm~l dr. (8.6)

Note that, without changing the homotopy type of m0, the energy of the initial map can be made as small as we like. For instance, in any homotopy class we can find representatives with h0(t) = h0(oo) for t > so > 0 where £o may be chosen arbitrarily small. Taking such a representative means that uq is constant outside a small ball of radius e0 about the origin. On the other hand, BM(Mm) is transient for m > 3, which implies that the random variable u0 o Xx is almost constant for t large, namely equal to north or south pole, depending on k. Thus, we expect

lim u(t,x) = (0..... 0. 1) e Sm (north pole) if k is even, and


lim u(t,x) = (0,..., 0, -1) e Sm (south pole) if k is odd.

Of course, uo cannot be homotopic to a constant map, if it represents a nontrivial homotopy class. Hence, the deformation u(t, ■) of uq must develop singularities.

The problem is to verify that this happens actually in finite time. Heuristically, it is then quite obvious that singularities appear at a finite number of points (tv, 0), since by symmetry, as t increases, u(t, 0) cannot move along great circles through the poles: the points u(t,0) have to "jump" between the poles.

In stochastic terms the problem can be described as follows. Let R denote an «-dimensional Bessel process Bes(m) (radial part of BM(ffira)) with generator

1 / dz m - 1 d \ 2\8?+ r dr)'

assume that R is defined on the one-dimensional Wiener space and adapted to its one-dimensional Brownian filtration For fixed t > 0, set £ = h0 o R't

with Rq = r. By Lemma 8.2 (i) (for t not too large) there are two adapted real processes (Ys,^)o<s<t and (Ci; Jf)0<s<, such that

J dYs = Cs dWs - F(R[, Ys)ds on [0, f] x Q, I Yt=£,

where F(Rrs,Ys) - (m - 1)(2J?;>~2sin^y,). Then h(t,r) = Y0. In other words, we get £ = Ys + CpdWp - f' F(Rrp, Yp)dp, which implies

ys = E* (£ + jf sin(2yp) dp), (8.7)

or equivalently,

(c + jf sin(2Fp) dp) -Yo + J^C, dWp .

Equation (8.7) represents Y as sum of the "linear conditional expectation" llv^1' [f ] plus an additional term taking into account the nonlinearity in the PDE (8.4). For r = 0, we get sin(2Fo) = 0 as a consequence of

f'm - 1

L ^ <00.

Therefore, h(t, 0) is necessarily an integer multiple of ty/2.

Formula (8.7) could be used for an explicit discussion of the heat flow for maps of the type (8.1). We are not going to pursue this example in further detail now; its main features will be covered by the general theory developed in the next section.

9 Monotonicity properties for manifold-valued martingales

Let (M,g) and (N,h) be Riemannian manifolds where N is compact. By Nash's theorem, we may assume that (N, h) is isometrically embedded into ? - for some if N. Let 7r: (THk^N 77V denote the orthogonal projection, which gives a

linear map ny\ TyN for each y e N. Writing/ = l of e C°°(M ,Re) for

the composition of a function/ G C°°(M.N) with the embedding /,: N <—? K^, we have

r</)(x) = 7T/W (AMf)(x), xeM. (9.1)

Thus under the identification r(f) = i,*T(f) we get Am] = rif) + v(f), where the second summand denotes the normal component of AMf ■ Comparison with the composition formula AM(t of) = i,,r(f) + trace(f*Vdt) shows iXf) = tracei/'* Vi//.). Hence v(f) depends quadratically on the differential df. In particular, the last decomposition implies that/: M —> N is harmonic if and only if (AMf)(x) JL Tf(x)N for each x & M.

From now on let M = Em (m > 3). We consider solutions of the heat equation

4j-t u = \t(u), u\t=0 = uq, (9.2)

where / = uq: W —> N is smooth and of finite energy E(f) = f \\df\\2 dx < oo. We also assume that e(u0) = \\df\\2 is bounded. In this case (7.2) provides an a priori estimate for e{u) on a small time interval which guarantees the existence of a solution to (9.2) locally, i.e. on a small time interval [0, t\ [ for some t\ > 0. The energy inequality

E(u(t,-))+[ [ \\^u\\2(s,x)dsdx < E(u0) (9.3)

Jo Jmm s

holds in this case for any t < t\. Moreover, \\du(t, -)[|2 is bounded for each t < t\. Via a fixed embedding l: N solutions of the heat equation take

values in 'ff\ and (9.2) reads as

§-tu - \Au + \v(u) = 0, u\t=0 = u0, (9.4)

where A is the usual Laplacian on Mm.

Now let u: [0, T[ x ffim N be a solution of (9.2) such that

E(uo) < oo, and e(u0) is bounded on Mm. (9.5)

The typical situation we have in mind is as in the example of section 8, where / = u0 has small energy E(f), but represents a nontrivial homotopy class in C°°($lm,N). To describe the evolution of u0 under the heat flow, we switch to the stochastic picture. As explained in section 5, for each (t,x) e [0, T\ x M™, there is an //2-martingale Y'-s = (Ys'>x)o<s<t with starting point Y('}'x = u(t.x) and terminal value Y,''x -f c Xr~ a.s. Let

[Y<'X,Y''X] = J hidY'^^dY'^)

denote the Riemannian quadratic variation of Y'-x. The next theorem shows that the Riemannian quadratic variations of these martingales share a specific monotonicity property which is basically a consequence of Brownian scaling. All subsequent results are essentially a consequence of this property.

Theorem 9.1 (Monotonicity Formula) Let u\ [0, J[ x Mm -> <V be a solution of the heat equation (9.2) such that (9.5) is fulfilled. Then, for each (t,x) G [0, T[ x Em and each a 6 ]0,1[,

$:i-h E /" h(dYc'x ,dYl'x) (9.6)

defines a non-decreasing function & on ]0, t].

We start the proof of Theorem 9.1 by giving a first lemma.

Lemma 9.2 Let u: [0, T[ x Mm N be as in Theorem 9.1 and (t,x) G [0, T[ x ®m. Then the following two statements are equivalent:

(i) For each a G ]0,1[ the function (9.6) is non-decreasing on ]0, i].

(ii) The function (f>(s) = il [e(u)(t — s.Z/)] is non-decreasing on ]0, /].

Proof. First note that h(dY^x,dYs''x) = \\du\\2(t - s,Xx)ds = e(u)(t - s,Xf)ds. Hence, under the assumption that cf>(s) = s E [e(u)(t — s,Xx)] =: s ip(s) is non-decreasing on ]0, f], we get for 0 < r < r' < t, with A := r'/r > 1,

[ ip(s)ds= f tp(s'/\)ds'/A= f (l/s')(s'/\)ip(s'/\)ds' J ar J ocr' J ar'

< [ (1 /s')s'<p(s')ds'= I (p(s')ds',

/ (XT'

which verifies condition (i). On the other hand, supposing that (i) is true, consider $(r) = E Jrarh(dYt-x,dY1'x) = far (p(s)ds for fixed a G ]0,1[. Then by assumption $>'(r) = ip(r) — onp(cer) > 0; hence ar <p(ar) < np(r) for any r G ]0,i]. Because a G ]0,1 [ is arbitrary this shows (ii). □

Proof (of Theorem 9.1) By Lemma 9.2 we have to show that

SKj^zjIE^Xt-s,^)] (9.7)

is non-decreasing on ]0, t]. We write s tp(s) = s E[e(M)(i,Xs0)] with

u(r,y) := u(t — r,x +y)

solving drii + ^t(m) = 0 on [0, t] x W, where dr denotes differentiation with respect to the time variable. The claim is a consequence of the following lemma. □

Lemma 9.3 Over an interval ]a, b [ C M+ let u: ]a, b [ x Mm N be a solution of

(.% + \t) U = 0 (9.8)

such that e(u) = ||t/fi||2 and \\dru |j are bounded on ]a, b [ x IRm.

Then, for a Brownian motion Xx on ffim with starting point x, the function

(i»:m^(i) = iE[e(i!)(j,x;)] (9.9)

is non-decreasing on ]a, b [.

Proof. We follow a similar calculation of Struwe [41], and show that

J (j)(s2) = s2 <p(s2)

is non-decreasing on ]y/a, \fb J. Of course, we may assume that x = 0. First note that, for fixed ,v > 0, the rescaled function

us(r,y) := ii(s2r,sy), (r,y) e ]a/s2,b/s2[ x lm

also solves (9.8).

Since, by the scaling property of BM(2'", can),

EfUdfiJI^l.X!0)] =i2E[|M«||V,sXi0)] =52E[||rf«||V,^02)] ,

it suffices to check ^E\\\dus\\2(l,X®)\ > 0. Moreover, by the same reasoning as above, it is enough to consider j = 1, i.e., to establish

^EOldSj^l,*!0)] >0. (9.10)

From the definition of us follows jr-| =lus(r,y) = 2r dru(r,y)+du(r,y)-y. Notice that u(r, •) is a function from ffim to IR% and its differential du(r. •) is hence a function from I!m to Mm ® M^. Moreover, by the isometry of the embedding N we have \\du\\2 = (du,du), where (A,B) = f°r an

orthonormal basis (e\,...,em) of №m. Since P{Xf e dy} = g(y)dy, where g denotes standard normal distribution on Rm, we get:

£\s=1E[\\dus\\2(l,X°)} = i\s=lE[(dus, du,}(1,X°)] = 2E[(du(l,yX d(2dru(hy) + du(l,y)-y))\y=xo]

= 2 / (du(l,y),d(2dru(l,y) + du(l,y)-y))g(y)dy Jv

= 2 / (du(l,y), d[(2dru(l,y) + du(l,y)-y) g(y)])dy

- f (du(l,y),(gpidg)(y)® (2dru(l,y) + du(l,y)-y)). Jv

We substitute (gradg)(y) = -y g(y) into the second integral, and apply Green's formula to the first integral, exploiting Au = -2dru + v(u) and the fact that

2dru(l,y) + du(l,y)-y e THhy)N, ye ST.

Thus, we finally have

d 1 ^EOldfijIfd,*!0)] = £\s=1E[(dus,dus)(l,X°)]

/ (2drù(l,y),2drû(l,y) + dù(l,y)-y) eg(y)dy lw"

+ / (dû(l,y)-y,2drû(l,y) + dû(l,y)-y) eg(y)dy

= 2 \\2drù(l,y) + dû(l,y)-y\\2g(y)dy

= 2E||2drw(l,X1°) + dw(l,X10)-X°\\2 >0.

This completes the proof of Theorem 9.1. □

The proof of Lemma 9.3 allows us to extract explicit information about the growth of function (9.7) which in turn gives a quantitative version of the Monotonicity Formula.

Corollary 9.4 Let ]a, b [ C M+ and let 5: ]a,i>[xr —> N be a solution of

{dr + \T) 5=0 (9.11)

such that e(u) = ||dS||2 and \\dru\\ are bounded functions on ]a, b [ x №m. Further let <f>: ]a, b [ —► M be defined by

</>(s) = sE[fi(fi)(s,XJj:)]. (9.12)

Then for any s < t in ]a,b[ and x G Mm the following equality holds:

[< f \\2pdrù(p,Xx) + dù(p,Xx) ■ Xx\\2 }

№) + \ p———— \ dp = m.

Proof. Again it suffices to consider the case x = 0. Note that for solutions 5 of (9.11), where <p(s) = 4>(s\u) is given by (9.12), we get from the proof of Lemma 9.3

fs\s=l<t>{s2\u) = 2E\\2dru(l)X®) + du(\,X^) ■ X°f. On the other hand, we have

2pV(p2;S)= £|s=10(p2s2;5)= i\s=ls2E[\\dup\\2(s2,X^)} = i\s=1^(s2-,up),

where up(s,y) = u(p2s,py) is the corresponding rescaled solution of (9.11). The two equations combined give

<f>'(p\u) = E \\2pdru(pXP) + du(p,X°p) -X°\\2/p

which implies the assertion. □

Theorem 9.1 should be seen as a parabolic version of the monotonicity inequality developed by Schoen and Uhlenbeck (see [39], [38]) for energy minimizing harmonic maps: If/ G L\(fi,N) is energy minimizing on some domain

Q C Mm, i.e., if each point x of fi has a neighborhood U such that Eif) < E(f') for every f e L\{Q,N) satisfying/ =/' on Q\U, then

r2-mEr(f) < consts2'mEs{f) (9.13)

for 0 < r < s < dist(x, df2) and x e J?, where Er(f) = fBr(x) \\df \\2 is the energy off on the r-ball B,(x) about x. Note that in comparison to (9.13) the parabolic monotonicity formula 9.1 (expressed by the monotonicity of the function (9.7)) uses the heat kernel instead of Lebesgue measure on №m which seems to be a natural way to take into account the inhomogenity of space-time. We continue with a variation of the Monotonicity Formula.

Corollary 9.5 Letu: [0, T[ x Mm - > N be a smooth solution of the heat equation (9.2) such that (9.5) is fulfilled. For fixed (t,x) 6 [0, T[ x Mm and p> 0, consider the martingale Y''x'p on N,

Ys''x;p = u(p + t - s,Xx), p<s<p + t. (9.14)

Then the function

sHjE[e(K)(p + i- i,x;)] (9.15) is non-decreasing on [p, p +1], and for each a £ JO, 1[

rnli h(dY''x'p,dY''x''p) = E f [e(u)(p + t ~ s,Xx)]ds (9.16)

J ar J ar

is non-decreasing on [p/a, p +1\

Proof. Since u(r,y) :- u(p + t — r,y) solves (dr + jt)m = 0 on [p,p + t], the statement about (9.15) is an immediate consequence of Lemma 9.3. Again the statement about (9.16) follows by the same argument as in the proof of Lemma 9.3. □

Remark 9.6 Let u: [0, T[ x ffim — N be a smooth solution of the heat equation (9.2) such that (9.5) is fulfilled. The same argument as in 9.5 can be applied to

Yj'x'p = u(p + T - s,Xx), p < s < p + T ,

showing that

s ^ sE[e(u)(p + T - s,Xx)] is non-decreasing on ]p, T + p\. Especially,

exists for any p > 0, even if u blows up at x at time T.

10 Blow-ups in finite time and small energy

Crucial for this section is the observation that Theorem 9.1 allows one to use almost verbatim the machinery developed by Schoen and Uhlenbeck (cf. [38], Theorem 2.2) to establish a priori C1-bounds for solutions of the heat equation with small energy (see also [41]).

Theorem 10.1 There exists a constant £o = £o(m,N) > 0 depending only on m and N such that for any solution u: [0, T[ x ®m —>■ N of the heat equation satisfying condition (9.5) the following is true: If $(r) = E J^hidY'^ ,dY''x) < eq for some (t,x) G [0, T[ x ]Rm and some r such that 0 < r < t < T, then ||cfei||2(i,x) < C with a constant C = C (r, m,N, E(«o))-

Proof. Let u: [0.'/'[ x ~2m A' be an arbitrary solution of the heat equation. Fix (f0,x0) 6 [0,T[ x ffim, and consider <2>(r0) = E J^/2h(dY^xo,dY^) for some

0 < ff) < tfj. We show that there are (universal) constants £o = Zoim.N) > 0 and C = C (r0,m,N,E(u0)) such that if $(r0) < £o then \\du\\2(t0,x0) < C.

Denote by Ba(x) = {y £lm : j|y - x || < cr} the (open) Euclidean ball about

1 e 1* of radius a. Further, for a > 0 and (t,x) G M+ x ffim, consider the "parabolic ball"

P(a~, t,x) := [i - a2.J] x Ba(x)

at (t,x), where Ba(x) - {y E IRm : ||y - x|| < a} is the closed ball about x of radius a. Let 0 < r\ < 1 such that r2 = ro/8 with 6 > 4. First, we claim that for 0 < a < r\ and any point (f,x) E P(ry:to,xo) the following estimate holds:

— / [ e(u){t + a2-s,y)dsdy < c@(r0)+ ^-E(u0), (10.1) J a2 JBa(x) b

with constants c = c(m) and c* = c*(r0lm). Indeed, note that for (s,y) G [a2,2a2] x Ba(x) the Gaussian transition kernel may be estimated from below as

1 W2 / I|y -x||2\ ^ 1

^H^J exH--

where c\ = c\{m). In combination with Corollary 9.5 this estimate gives

— / / e{u)(t + a2 - s,y)dsdy < c\ / E e(u)(t + a2 - s ,Xx) ds Om Ja2 JBa(x) Ja2

<ci Ec(»)(i + a — s,Xx)ds .

Further, if we set r = /<> t. then 0 < r < r2, and hence for s G [r¡2.2r 2]

< - <c2 = c2(m),

Ps+T(X,:y) \ s uniformly in y. Thus, by using Theorem 9.1 once again, we get

e(u)(t + 0 — s,X*)ds < c2E / e(u)(t + a - s,X*)ds

e(u)(t0 + cr - s,X*)ds

ç2r{ ,4r{

...ds + ... ds

<2c2E / e(u)(to + a2 -s,X*)ds Jr0/2

= 2c2e/° e(u)(to-s,X^)ds.


Finally, for r0/2 — a1 < s < ro — a2, we can use the obvious estimate

\Ps+Axyy)-Ps(x0,y)\ <

I ^ m/2 ( I \m/2

1 \m/2 ,\\m/2 s

20 + a2)

) -exp(-

\\y -x0\ 2s

< c5 r2 = ce/S,

uniformly in y, with constants ct depending on rn and r0. From there, together with

fo-o2 f r-ra/2 pro

E/ e(u)(to — s,X*°) < E< / ...ds+ ... ds\< 2<P(r0), Jr0/2~<j2 Ur0/4 J rg/2 J

we get, as claimed,

Om Jo* JBa(x)

\ f [ e{u)(t + a2 - s,y)dsdy < 4c2 <2>(r0) + c2 c6 r0 E(u0)/6

Ja2 JB„(x)

= c(m)$(r0) + c*(m,r0)E(u0)/ë.

We will exploit (10.1) for a small. First observe that there exists a0 G [0, r\[ such that

(r\ - cr0)2 sup e(u) = max (n - a)2 sup e(u). (10.2)

P(rj0;i0,x0) 0<a<ri P(cr;t0,x0)

Moreover, there is a point (t*,x*) G P(aoUo,xo) such that

eo := sup e(u) = e(u)(t*,x*). P(<T0''0<x0)

We may assume that e0 > 0. If we set p0 = {r\ - go)/2, then 0 < a0 + p0 < n, and hence from the choice of a0 and (t *. x *)

sup e(u) < sup e(u) < 4г0 • (Ю.З)

P(p0\t*,x*) P(o-0+p0-,tQ,x0)

Introduce pi := v'e« po, and rescale и as

v(t,x) := u(-—- +t*, +x* ) , (t,x) e P(pi;t0,x0),

V eo \/eo J

then v provides a solution of the nonlinear heat equation on P(p\; /о-Лэ) chosen such that e(w)(fo,xo) = 1. Moreover, by (10.3) we have

sup e(v) = — sup e(u)<4.

Р(Р1;г0л) e0 P(p0;t*,x*)

We want to show that pi < 1. First note that Lemma 7.2 implies


on P(|0i;io,*o), where the constant C\ depends only on N. Hence, if instead of e(v) we consider

e(t,x) := exp(Ci(i0 - 0)

on P(pi',t(s,xo), and if p\ > 1, we get from Moser's weak Harnack inequality for subsolutions of parabolic equations ([31], Theorem 3, p. 113) the estimate

1 = e(v)(t0,x0) <C2 f e(v)(s, у) dsdy , (10.4)


where the constant C2 depends on C\. Scaling back gives

/ e(v)(s,y)dsdy=e0m/2 e(u)(s,y)dsdy,

Jp(Ut0,x0) Jp (1/vioJV)

so that (10.4) leads to

l<C2e0m/2 ( f e(u)(s, y) dsdy

Jf-Xjeо JBl/v^(x*)

fljeQ /•

= C2eom/2 / / e(«)(i* + l/e0 -

< Сз ф(г0) + CA E(u0)/S = Сз e0 + c4 Е(щ)/6,

with uniform constants C3 = C3(m,N) and C4 = C4(r0,m,Ny, note that the inequality in the last line follows from (10.1), since l/л/ео < po < n as a consequence of the assumption p\ = т/ёоРо > 1. If we choose £o < 1/(2C3) and ё > 2 СаЕ(щ), we obtain a contradiction. Thus only p\ < 1 is possible, and hence, by the choice of ao,

max (n - a)2 sup e(u) = (rl — a0)2 sup e(u) < 4р$е0 = 4p2 < 4 .

0<o-<ri P(<r-,t0,xQ) P(a0;t0,x0)

Finally, choosing a - r\j2, hence (ri — a)2 = ro/46, we get for 6 sufficiently large

sup e(u) < 16 S/r0. (10.5)


The proof is complete. □

From (10.5) we read off more precise quantitative information how smallness of Riemannian quadratic variation translates into gradient bounds for solutions of the heat equation.

Corollary 10.2 Let u: [0, T[ x W —> N be a solution of the heat equation satisfying condition (9.5). Suppose that

$(r0) = E h (dY, dY) < e0 Jr0/2

for some (to, xo) £ [0, T[ x Mm such that 0 < ro < to <T, where e0 is determined by Theorem 10.1. Then, there exists 6 = 8(ro,m,N,E(uq)) such that

sup{e(M)(?0 ~s,x) : 0 <s < r0/<5, \\x - x0||2 < r0/6} < 4 S/r0. (10.6)

Moreover, S can be chosen independent of ro as long as ro exceeds some positive lower bound.

Theorem 10.1 can easily be adapted to obtain a global existence result for solutions of the heat equation.

Theorem 10.3 For any T > 0 there exists a constant £i = ¿~] (m. N. T) depending only on m, N such that any solution u\ [0, T[ x ffim —> TV of the heat equation with e(uo) bounded on Em and E(uo) < £i can be extended to a global (smooth) solution u: [0, oof x Mm —> N which converges to a constant harmonic map u^ as t —» oo.

Proof. Given (t,x) C [0, x Mm and 0 < r < t, we get by means of the Monotonicity Formula (see Corollary 9.5 with p = 0)

e[ h(dYt'x,dY,'x)= [ Ee(u)(t-s,Xx)ds = [ -iE[e(«)(i-ij;)]di Jr/2 Jr/2 Jr/2 s

< / E [e (u0){Xx)] / - ds = t E [e (u0)(Xx)] In 2

Jr/2 s

< c(t,m)E(u0) < e0 ,

for £\ < £0/c(t,m) where e0 = soUn.N) is the constant determined by Theorem 10.1. Note that the explicit form of the constant c(t,m) is tln2(2-K t)~m/2 and that m > 3 by assumption; hence the same constant e, that works for t also applies for t' > t; especially £\ may be chosen independent of t as long as t exceeds some positive value, say I > t{). However, as a consequence of Corollary 10.2, if E(uo) < £i = S\(m,N. i0), then there is a positive a = a(to,m,N,E(uq)) and a uniform a priori bound

\\du\\2(s, •) < C/T (10.7)

for T — a < s < T with a universal constant C depending only on to, m, N, E(uij). Now, given a smooth solution of the heat equation u: [0, 7'fx IP;'" —> N with e(uo) bounded and E(u0) < e\ = ei(m,N, t0), let u: [0, T'[ xW1 N be its maximal extension in the sense that u can not be extended beyond the time interval [0, T'[ as a smooth solution of the heat equation. Hence, if T < oo, then u(t, •) will blow up at t = 7''. But by the above reasoning

[|du\\2(s, ■) < C/T'

for T' — a < s < T', which shows that blow up at time T' is impossible. As a consequence, we must have T' = oo, which means that u is extendable to a global solution u: [0, oof x W" N. The fact that u(t, •) —» u^ = const as t —> oo can easily be derived from the uniform global decay

||dM||2(f, - )<C/t

for t>T. □

Note that in Theorem 10.3 the initial map wo is automatically homotopic to a constant map. On the other hand, in case of homotopically nontrivial maps liq. Theorem 10.3 immediately leads to blow-up results in finite time.

Corollary 10.4 Let T > 0, and let f | = s i (in. /V. T) be determined by Theorem 10.3. Then, for homotopically nontrivial smooth initial data uq: Mm —» N such that e(uo) € L°° and E(uq) < £i, solutions of the heat equation

= u\t=0 = u0 (10.8)

blow up before time T. In fact, blow-up time T* = T*(uq) approaches 0 as E(uq) decreases to 0.

Proof. Otherwise, by Theorem 10.3, there would exist a global solution to (10.8) inducing a homotopy uq ~ u^ = const, in contradiction to the assumption that uq is homotopically nontrivial. □

11 Conclusion

Deformation by heat flow reduces the total energy of the initial map uq: M —> N, while the average energy along paths of Brownian motion (measured by the H2-norm of the martingales with prescribed terminal state uq oXf) may blow up. We discussed the case M =W" with the flat Brownian motion; analogous results can be given in the general situation by exploiting the fact that locally about each point any Riemannian metric is approximately Euclidean (see [42]).

Specifically, if u: [0. iof x M ^ N is a smooth solution of the heat equation, we have the N -valued martingale Ys ~ u(to —s.Xf), 0 < 5 < to, and the function

4>(r) - E h(dY, dY) is non-decreasing on ]0, iol- Hence, for x G M, there are two possibilities:

(i) lim#(r) > 0, which means that blow-up occurs at (/q. x); in this case the

r—> 0

process (Fs)o<-s<i0 is a martingale without starting point.

(ii) lim <P(r) = 0, which implies that the heat flow u can be extended around


(to,x); moreover Yq = u(to,x).

There are many important related questions. Of course, the above discussion stresses the desirability of constructing appropriate martingales with purely probabilistic methods and without relying on solutions the heat equation. For instance, let/: M —> N be a smooth map between compact Riemannian manifolds (M,g) and (N,h). Is there always an H2-martingale Ys (defined at least for 0 < s <t) with terminal state f oXf, even for t > T*, if T' is the first blow-up time in the heat flow with initial map /? What is a straightforward stochastic way to construct such martingales in general situations? (We deal with these questions in [41], [42]).

Note that the heat equation is no longer well-defined as classical PDE beyond the first singularity; one has to switch to the framework of distributional solutions. Nevertheless, questions about martingale means of random variables of the type £ =/ o X? make sense for arbitrary values of t.

Finding starting points of martingales with a prescribed terminal state puts the heat equation in a quite general setting and provides "solutions" to the heat equation in a weak but canonical sense.

Acknowledgement. The author is indebted to Marc Amaudon for pointing out an error in the first version of this paper.


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