Probabilistic characterization of weakly harmonic maps with respect to non-local Dirichlet forms

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.


Introduction
Harmonic maps are critical points of the Dirichlet energy defined on the space of maps between two manifolds.There are a lot of studies about harmonic maps from both analytic and geometric points of view.The regularity of harmonic maps is one of the most important problems.In general it is known that harmonic maps have singular points by the non-linearity of the Euler-Lagrange equation obtained from the variational problem for the Dirichlet energy.The theory of partial regularity for harmonic maps was obtained in [25,26] for energy minimizing maps, and in [4,11] for stationary harmonic maps.On the other hand, harmonic maps are related to the theory of probability and they also have been studied through stochastic processes.In fact, we can easily show by Itô's formula that smooth harmonic maps map Brownian motions on domain manifolds to martingales on target manifolds.In [23], it was shown that this probabilistic characterization of harmonic maps is valid even for weakly harmonic maps with respect to strongly local Dirichlet forms.Regularity of harmonic maps has been also considered form the view point of the theory of probability.The regularity of harmonic maps with values in spaces with a kind of convex geometry has been shown in [2,22].Recently, fractional harmonic maps, which are critical points of the fractional Dirichlet energy, are also studied.The study of fractional harmonic maps is initiated in [9,10] and the regularity theory for 1  2 -harmonic lines was obtained in those articles.The partial regularity for 1 2 -harmonic maps has been shown in [17].Recently, for α ∈ (0, 2), the partial regularity for α 2 -harmonic maps with values in spheres has been obtained in [16].In this paper, we first obtain a probabilistic characterization of harmonic maps with respect to regular Dirichlet forms which are not necessarily strongly local.This result is the extension of that of [23].Fractional harmonic mas are one of the most typical examples of such harmonic maps.To study harmonic maps for general regular Dirichlet forms by stochastic processes, we will apply the theory of martingales with jumps on a submanifold.Discontinuous martingales on manifolds are first introduced in [20] and further studied in [21].Recently, some properties concerning the convergence of discontinuous martingales have been studied in the author's previous paper [19] for the purpose of applications to harmonic maps.To state the characterization of harmonic maps by martingales on submanifold in a general situation, we briefly specify the setting.Let E be a locally compact and separable metric space, m a positive Radon measure with full support on E, and (E, F) a regular Dirichlet form on L 2 (E; m).For a special standard core C and an open set D ⊂ E, we denote by C D the family of functions in C whose supports are included in D. Let M be a Riemannian submanifold of a higher dimensional Euclidean space R d .For a map u = (u 1 , . . ., u d ) : E → M ⊂ R d such that u i ∈ F D loc satisfies some conditions, we can consider the following equation: where C D (R d ) = {ψ = (ψ 1 , . . ., ψ d ) : E → R d | ψ i ∈ C D for each i = 1, . . ., d}.
We can regard (1.1) as the Euler-Lagrange equation with respect to the energy functional E.
A map u : E → M satisfying (1.1) is called a weakly harmonic map on D. On the other hand, a regular Dirichlet form determines a symmetric Hunt process on E. We define a quasi-harmonic map as a map which maps the symmetric Hunt process to a martingale on M .The equivalence of weakly harmonic maps and quasi-harmonic maps was shown in [23] in the case where (E, F) is strongly local transient regular Dirichlet form and u is in the reflected Dirichlet space.On the other hand, harmonic functions in F D loc for general regular Dirichlet form were characterized by stochastic processes in [5].(In this paper, we mainly refer to Chapter 6 of [7].)By combining [5,23], and the theory of stochastic integrals along CAF's of zero energy developed in [6,15,18,29], we obtain the non-local version of the result of [23].To simplify the statement, we only state the case that a target manifold M is compact in Theorem 1.1 below, but we will show this kind of result under more general situations in Theorems 3.7 and 3.8 and those results in Section 3 include Proposition 4 of [23].
Theorem 1.1.Let M be a compact Riemannian submanifold of R d .We assume that a Borel measurable map u : E → M is in F D loc (M ) and quasi-continuous on D. Then u is weakly harmonic on D if and only if u is quasi-harmonic on D.
As a particular case of Theorem 1.1, we can obtain a discontinuous martingale on a submanifold by substituting an α-symmetric stable process into a map satisfying the Lagrange equation with respect to the fractional Laplacian.In this way, the fractional harmonic map discussed in [9,10] can be studied by using stochastic processes.
Next we will consider singularities of harmonic maps.Singularities of harmonic map heat flows with smooth initial data appearing at their explosion times were dealt with through stochastic processes in [27,28].In this paper, we will deal with singularities of harmonic maps in a weak sense including fractional harmonic maps based on Theorem 1.1.In this situation, the continuity of martingales obtained from harmonic maps at time zero can be thought as the continuity of harmonic maps along the paths of Markov processes.This kind of continuity corresponds to the continuity in the fine topology in classical potential theory.In [19], it was shown that the equivalent condition of the fine continuity of quasi-harmonic maps for Markov processes with jumps can be described by quadratic variations of martingales on manifolds.By combining Theorem 1.1 above and Proposition 4.11 of [19], we can obtain some equivalent conditions of the fine continuity of weakly harmonic maps with respect to general regular Dirichlet forms through martingales on manifolds.However, we note that the fine continuity is much weaker than the continuity in the topology of metric spaces in general.We will remark the equivalence does not hold for general weakly harmonic maps with respect to the Laplacian.One of the counterexamples is the everywhere-discontinuous weakly harmonic map constructed in [24].However, if we consider a class of weakly harmonic maps with respect to the Laplacian and the fractional Laplacian which satisfy some assumptions regarding tangent maps, we can easily show that the equivalence holds.The precise assumptions and statement will be given in Assumption 4.7 and Proposition 4.11, respectively.Typical examples satisfying these assumptions are energy minimizing maps.We give an outline of the paper.In Section 2, we first recall some facts regarding symmetric Markov processes and Dirichlet forms.Mainly we refer to [7,12].In addition, we recall the stochastic integral along continuous additive functionals of zero energy.In Section 3, we characterize harmonic maps with respect to regular Dirichlet forms by martingales on manifolds.In Section 4, we describe the connection between sinularities of weakly harmonic maps and those of martingales.Throughout this paper, given a topological space M , we denote by C 0 (M ) the set of all continuous functions on M with compact support.In the case that M is a manifold, we denote by C ∞ 0 (M ) the set of all C ∞ functions with compact support.For a, b ∈ R, we abbreviate max{a, b} and min{a, b} as a ∨ b and a ∧ b, respectively.For a stochastic process H and a stopping time τ , we write the stopped process as H τ defined by For two random variables X, Y , we write X ∼ Y if they have the same distribution.

Preliminaries on Markov processes
In this section, we recall the theory of Markov processes and Dirichlet forms in preparation for the proof of the main theorem.In particular, we focus on the stochastic integral along CAF's of zero energy introduced in [18] and further studied in [6,15,29].As for the general theory of Dirichlet forms, see [7,12] for details.Let E be a locally compact and separable metric space.We add a point ∆ to E and define E ∆ := E ∪ {∆}.Given f ∈ B(E), we extend the domain of f to E ∆ by f (∆) = 0. Let m be a positive Radon measure with full support on E. Let F be a dense subspace of L 2 (E; m) and E : F × F → R a non-negative definite symmetric quadratic form.We set for u, v ∈ F and α > 0. The pair (E, F) is called a Dirichlet form if the space (F, E 1 ) is a Hilbert space and it holds that (u ∨ 0) ∧ 1 ∈ F and For a regular Dirichlet form, the extended Dirichlet space F e is defined as the family of equivalence classes of Borel functions u : E → R with respect to the m-a.e.equality such that |u| < ∞ m-a.e. and there exists an E-Cauchy sequence {u k } ∞ k=1 in F such that lim k→∞ u k = u, m-a.e.For u ∈ F e , we can set where {u k } ∞ k=1 is an E-approximate sequence in F of u.For a regular Dirichlet form (E, F), there exists an m-symmetric Hunt process (Ω, {Z} t≥0 , {θ t } t≥0 , ζ, {P z } z∈E ∆ ) on E corresponding to (E, F), where θ t : Ω → Ω is the shift operator satisfying for all ω ∈ Ω, ζ is a life time of Z and P z is the distribution of {Z t } stating at z. Set For a σ-finite measure µ, we set In particular, if µ is a probability measure on E, P µ is a probability measure on where N µ is the family of all P µ -null sets in F µ ∞ .Denote the set of probability measures on E ∆ by P(E ∆ ) and let For a subset A ⊂ E ∆ , define the random time σ A by It is known that if A is a nearly Borel set, σ A and τ A are {F Z t }-stopping times.A subset A ⊂ E is said to be m-polar if there exists a nearly Borel set A 2 such that A ⊂ A 2 and The term q.e.stands for "except for an m-polar set."A positive Radon measure µ on E is said to be of finite energy integral if there exists C > 0 such that If µ is a positive Radon measure on E of finite energy integral, then for each α > 0, there exists a unique function U α µ is called an α-potential.We denote by S 0 the family of all positive Radon measures of finite energy integrals.We set a subset S 00 of S 0 as We use the abbreviation AF (resp.CAF) for additive functional (resp.continuous additive functional).Each PCAF A determines a measure µ A on E called Revuz measure which is characterized by where B + (E) is the set of all non-negative Borel functions on E. We denote the energy of an AF A by and the mutual energy of AF's A, B by We set Each additive functional in M is called a martingale additive functional (MAF).For details about MAF's, see [12,13].For u ∈ F e , set A [u] t := ũ(X t ) − ũ(X 0 ), where ũ is a quasi-continuous modification of u.Then A [u] is an additive functional of Z with finite energy.Moreover, by [12,13], there exists M [u] ∈ M and N [u] ∈ N c such that t , for all t ≥ 0, P z -a.s.q.e.z ∈ E and such a decomposition is unique.For u ∈ F e , the MAF M [u] can be decomposed into continuous part, jump part, and killing part as follows.Since M [u] is a P z -martingale for q.e.z ∈ E, the MAF M [u] can be written as the sum of its continuous martingale part M [u],c and purely discontinuous martingale part M [u],d and we can construct M [u],c and M [u],d as MAF's of Z.We further set where (N, H) is the Lévy system of Z, which is defined by a pair of a kernel N (z, dw) and a PCAF H of Z satisfying for any non-negative predictable process {Y s } and any k ∈ M and it holds that We set J(dzdw) := N (z, dw)dµ H (dz), k(dz) := N (z, ∆)dµ H (dz).
Since it holds that the above decomposition of M [u] yields the following Beurling-Deny decomposition: For ,c and diag(E) is a diagonal set of E × E. We denote the family of purely discontinuous MAF's of finite energy by Md .In [15], it was shown that there is a one-to-one correspondence between Md and the family of jump functions defined by For φ, ψ ∈ J , we denote φ ∼ ψ if φ = ψ N (z, dw)µ H (dz)-a.e. on E × E ∆ .Lemma 2.5 of [15] guarantees that for all φ ∈ J , there exists M ∈ Md such that and this is a one-to-one correspondence between J / ∼ and Md .In particular, for M ∈ Md , there exists K ∈ Md such that where φ is the jump function corresponding to M and φ(z, w) := φ(w, z) since φ + φ ∈ J .Now we can define the stochastic integral along CAF's of zero energy.We set By Lemma 3.2 of [18], we can define a linear operator γ : M → F and Γ : M → N * c as follows: For each M ∈ M, there exists a unique w ∈ F such that This w is denoted by γ(M ) and Γ is defined by Let M ∈ M and φ ∈ J its jump function.For g ∈ F e ∩ L 2 (E; µ M ), the stochastic integral g(Z) dΓ(M ) ∈ N c is defined by [15]).By [18], the CAF Γ(M ) can be characterized by lim where Proof.By derivation property, it holds that Moreover, in view of Lemma 3.1 of [18], we have Thus we obtain (2.5).
For an open set D ⊂ E, we set We will define the stochastic integral Then for g ∈ F b , it holds that where the right hand side is the Lebesgue-Stieltjes integral along A.
Proof.Take any l ∈ F D b .Then by (2.4), it holds that lim On the other hand, it holds that lim by Lemma 5.4.4 of [12], Theorem 2.2 of [18], and Therefore by Lemma 5.4.4 of [12] and Theorem 2.2 of [18], it holds that Thus we obtain the desired assertion.
For an open set D ⊂ E, we set Then (E D , F D ) is a regular Dirichlet form on L 2 (D; 1 D m).The symmetric Hunt process Z D corresponding to (E D , F D ) can be obtained as follows: We let for ω ∈ Ω, and ζ D (ω) := τ D (ω).Define the shift operator θ D t on Ω by where Then the inclusion adapted and satisfies the additivity ) Remark 2.4.This integral is independent of the choice of D 2 and φ D 2 by Lemma 3.4 of [15].Moreover, it is also independent of the choice of L ∈ M satisfying In fact, we have Γ(M − L) τ D 1 = 0 and consequently it holds that by Lemma 2.2.Thus we can deduce that the integral (2.6) is well-defined.
Suppose that the process t → H t∧τ D 1 is a P z -local mimartingale for q.e.z ∈ E. Then for g ∈ F D loc the stochastic integral g(Z s− ) dJ τ D 1 is defined by Remark 2.6.The integral is well-defined, i.e. the right-hand side is independent of the choice of a function H ′ : [0, ∞) × Ω → R and M ′ ∈ M such that H ′τ D 1 is a P z -local martingale for q.e.z ∈ E and In fact, we have , where L = M ′ − M , and consequently, Then for all N ∈ N c , On the other hand, we can define the quadratic variation of Γ(L) τ D 1 as a P z -continuous local martingale for q.e.z ∈ E and it holds that Γ(L) τ D 1 = 0 P z -a.s. for m-a.e.z ∈ D 1 by Fatou's lemma.Thus Γ(L) τ D 1 is m-equivalent to zero as a CAF of Z D 1 .Therefore we can deduce that Γ(M ) t∧τ D 1 = Γ(M ′ ) t∧τ D 1 , t ≥ 0, P z -a.s.q.e.z ∈ D 1 .
For z ∈ E \ D 1 , it is obvious that Γ(M ) τ D 1 = Γ(M ′ ) τ D 1 = 0, P z -a.s.Thus the decomposition of J τ D 1 is unique and consequently the integral is well-defined.
3 Proof of Theorem 1.1 First we define martingales and harmonic maps with respect to regular Dirichlet forms.The theory of discontinuous martingales on manifolds was developed in [20].In [19], the author focused on discontinuous martingales on Riemannian submanifolds of higher dimensional Euclidean spaces as a special case considered in [20], and then extended them so that we allowed the killing of martingales.We begin with recalling semimartingales and martingales defined in [19].Let M be a Riemannian submanifold of R d .
) is an M -valued (P z , {F Z t })-martingale with an end point for q.e.z ∈ E. In Definition 3.3, we set 0 ∈ R d as the end point since we set the value of a function on E ∆ at ∆ as 0. If Z has no inside killing, u is quasi-harmonic on D if and only if for each relatively compact open set D 1 such that D 1 ⊂ D, {u(Z t∧τ D 1 )} t≥0 is an M -valued P z -martingale for q.e.z ∈ E. Next we will recall the following conditions for a function u ∈ F D loc which were considered in [5] and Chapter 6 of [7]: For any relatively compact open set D 1 , D 2 with ) and if we define a function f u by then f u ∈ F D 1 e , where φ D 2 is a function satisfying For a Borel measurable locally bounded function u ∈ F D loc satisfying (A), we can define E(u, ψ) for all ψ ∈ C 0 (D) ∩ F by The right-hand side of (3.3) is finite by Lemma 6.7.8 of [7].For a Borel measurable function u : E → R, we set e , where Lemma 3.4 follows by applying Lemma 6.7.6 of [7] to u + and u − .

locally bounded Borel measurable function satisfying (A) and (B). Then the process
is of P z -integrable variation for q.e.z ∈ E and the dual predictable projection of A t is We further set Then it holds that φu ∈ F D , h 1 ∈ F e , φu − h 1 ∈ F D 1 e , and

Proof. First we set
Then by Lévy system formula (2.2), it holds that for q.e.z ∈ E. Thus we obtain Thus A t is a process of integrable variation and its dual predictable projection is B t under P z for q.e.z ∈ E. In the second claim of Lemma 3.5, φu ∈ F D , h 1 ∈ F e , and φu − h 1 ∈ F D 1 e are obvious from the decomposition of F D e .Moreover, we have e by Lemma 3.4.The last claim also can be shown in the same way as the proof of Theorem 6.7.9 of [7] without the harmonicity of u.
Let C be a special standard core of (E, F) and D an open set of E. We set We define F D (R d ), F D (u * T M ) and F D e (u * T M ) in the same way.Definition 3.6.Let u ∈ F D loc (M ) be a locally bounded Borel measurable map satisfying (A) and (B).u is called a weakly harmonic map on D if We divide Theorem 1.1 into the following Theorems 3.7 and 3.8.Theorem 3.7.Let u : E → M be a Borel measurable map in F D loc (M ) which is locally bounded on D and satisfies (A) and (B).Suppose that u is weakly harmonic on D. Then u is quasi-harmonic on D. Theorem 3.8.Let u ∈ F D loc (M ) be quasi-harmonic on D, locally bounded on D and satisfy (A) and (B).Then u is weakly harmonic on D.
e .Thus we can decompose u j (Z t ) as by Fukushima decomposition.Furthermore, since E(h j , ψ j ) = 0 for all ψ ∈ C D 1 (R d ) and j = 1, . . ., d by Lemma 3.5, it holds that where D f is the gradient of f as a function on R d .Then φD f • u ∈ F D (u * T M ).It suffices to show that u(Z) τ D 1 is a P z -semimartingale and is a P z -local martingale for q.e.z ∈ E. Since H t := h(Z t∧τ D 1 ) − h(Z 0 ) is an R d -valued P z -uniformly integrable martingale for q.e.z ∈ E, we can define the stochastic integral in the sense of Definition 2.5.Then it holds that lim holds and is a P z -local martingale for q.e.z ∈ E. Next we will show that u(Z t∧τ D1 ) is a semimartingale.Since it holds that and the second term of the right-hand side is a process of bounded variation, it suffices to show that f (φu(Z)) τ D 1 is a semimartingale.By Itô's formula for Dirichlet processes shown and it holds that is a P z -semimartingale for q.e.z ∈ E and so is N for each j = 1, . . ., d.Thus in the same way as in Remark 2.5, we can show that N is a CAF of locally bounded variation.
This enables us to see the stochastic integral along as the Stieltjes integral defined for P z -a.s.q.e.z ∈ E. Thus (3.7) and (3.10) mean that u is quasi-harmonic on D. This completes the proof.
as in the proof of Theorem 3.7.Then we can define the stochastic integral for each j = 1, . . ., d.Here the first two terms of the right-hand side are P z -local martingales.Moreover, since u is quasi-harmonic, the left-hand side is a P z -local martingale for q.e.z ∈ E. Thus we can deduce that s is a continuous local martingale additive functional for Z D 1 and consequently it equals zero in the same way as in Remark 2.6.Since φψ = ψ, it holds that Thus we obtain Thus u is a weakly harmonic map on D.
(ii) {u(Z t ) τ D 1 } t>0 can be extended to an M -valued P z -martingale with an end point indexed by t ≥ 0; Proof of Lemma 4.3.First we assume (i).Then we have lim ε→0 [X, X] ε t < ∞ and consequently, the process {X t } t≥0 is an M -valued martingale with the end point p by Theorem 1.2 of [19].Let A be the dual predictable projection of [X, X].Then we have Thus (ii) follows.
Next we assume (ii).We set a = sup 0≤t<∞ |∆X t |.Then a < ∞ since M is compact.We let the triple (B ε , C ε , ν ε ) be the characteristics of the d-dimensional semimartingale {X t } t≥ε with respect to the truncation function 1 {|x|<a} , that is, B t be the predictable locally bounded variation part of the canonical decomposition of special semimartingale {X t } t≥ε , and ν ε be the good version of the dual predictable projection of the random measure for 0 < t 1 ≤ t 2 , and J ∈ B(R d ).We note that for 0 < ε 1 < ε 2 , In particular, for 0 < s < t, the matrix is non-negative definite.Similarly, it holds that Thus we have Let f ∈ C ∞ (M ) be any smooth function and f ∈ C ∞ 0 (R d ) the extension of f given as (3.9).By Itô's formula, we have Moreover, for 0 < ε 1 < ε 2 , we have by (4.1) and (4.2),where Hess f = sup the right-hand sides of (4.3) and (4.4) converge to 0 as ε 2 → 0. Thus we can define U t , V t by We set H t := f (X t ) − U t − V t .Then {H t } t>0 is a local martingale with bounded jump since {X} t>0 is an M -valued martingale with an end point.Thus in the same way as the proof of Theorem 1.2 of [19], we can show that lim t→0 X t exists P-a.s.
To consider the continuity of weakly harmonic maps in the topology with respect to the metric on E and compare it with the fine continuity, hereafter we limit the scope of the metric space E to either of the following two cases as domains of harmonic maps: For a bounded smooth domain D ⊂ R m with smooth boundary and u ∈ H 1 loc (D), we set In this case, if we set where For a bounded smooth domain D ∈ R m with smooth boundary and u ∈ L 2 loc (R m ), we set See [16] for details about the space H α 2 (D).
The corresponding Markov processes Z are Brownian motion and α-stable process in Case 1 and Case 2, respectively.
Remark 4.5.In [24], T. Riviére constructed a weakly harmonic map u with respect to the Dirichlet energy from the 3-dimensional unit ball B 1 into the 2-dimensional unit sphere S 2 which is everywhere discontinuous.On the other hand, u has a quasi-continuous modification since the Dirichlet energy of u is finite.Thus there is a difference between the continuity in the Euclidean topology and that in the fine topology if we do not impose any restrictions for harmonic maps.
Definition 4.6.We assume the setting of either Case 1 or Case 2. Let u ∈ F D loc (M ) be a weakly harmonic map.Fix z 0 ∈ D and ρ > 0 such that B ρ (z 0 ) ⊂ D. Then ϕ : R d → M is called a tangent map of u at z 0 if for all R > 0, there exists a decreasing sequence {r where To see the equivalence of the continuity in Euclidean topology and that in fine topology in some cases, we impose some conditions on weakly harmonic maps.
Assumption 4.7.We assume the setting of either Case 1 or Case 2. Let u ∈ F D loc (M ) be a weakly harmonic map with E D (u).For each z 0 ∈ D and tangent map ϕ of u at z 0 , we assume that u satisfies the following two conditions.
(i) Under the notation in Definition 4.6, there exists a subsequence {r i j } such that E B R (u z 0 ,r i j − ϕ) → 0 as j → ∞.
(ii) u and ϕ have modifications which are continuous quasi-everywhere.In addition, u is continuous at z 0 if and only if any tangent map at z 0 is constant almost everywhere.
We remark that there are enough examples satisfying Assumption 4.7 as follows.) is a stable stationary harmonic map, then Assumption 4.7 is satisfied by [11,14].
Since Z * has the transition density, we have P * z (ϕ j (Z * t ) → 0 uniformly on any compact subset of (0, ∞) as j → ∞) = 1 for all z ∈ B R .
Since Z * has the same law as the part process of Z on [0, τ B R ), we have P z ϕ j (Z t∧τ B R ) τ B R − → 0 uniformly on any compact subset of (0, ∞) as j → ∞ = 1 for any z ∈ D and this holds for any fixed R > 0. On the other hand, for any bounded continuous function F on R d , E z 0 F (u z 0 ,r i (Z t ) τ B R − ) = E 0 F (u(z 0 + Z r α i t ) τα,r i − ) → F (u(z 0 )) by the scale invariance and bounded convergence theorem, where This yields that the law of ϕ(Z t ) τ B R − concentrates at u(z 0 ) for each t > 0. This implies that ϕ(Z) τ B R is constant along the path of Z B R almost surely under P 0 since it is a càdlàg process.Consequently, ϕ(Z) is constant P 0 -a.s.since we took an arbitrary R > 0. Take any continuous point z ∈ R d of ϕ.Then for any η > 0, there exists δ > 0 such that for all w ∈ B δ (z), |ϕ(z) − ϕ(w)| < η.On the other hand, since Z is irreducible, P 0 ({ω; Z(ω) ∩ B δ (z) = ∅}) > 0. Thus by taking t > 0 and ω such that the path t → ϕ(Z t (ω)) is constant and Z t (ω) ∈ B δ (z), we have Remark 4.14.At the moment we do not know if the same argument holds or not for all stationary harmonic maps since our proof of Proposition 4.11 relies on Assumption 4.7.

Lemma 2 . 2 .
Let M ∈ M and D a relatively compact open set of E. Suppose that there exists a CAF A of bounded variation of Z such that |µ A |(D) < ∞ and

Definition 2 . 3 .
adapted and a CAF of Z D .Let M ∈ M, D an open set of E and D 1 a relatively compact open set of D such that D 1 ⊂ D. The integral of locally bounded function g in

Lemma 3 . 5 .
Let D, D 1 , D 2 be open sets in E satisfying (3.1) and u ∈ F D loc a
[19] z -a.s.Proof.By Theorem 1.1, the map u is quasi-harmonic on D.Moreover, by the absolute continuity of the transition function, the triple ({u(Z t ) τ D 1 } t>0 , ζ, 0) is an M -valued P z -martingale indexed by t ∈ (0, ∞) for all z ∈ D 1 .(SeeLemma4.10 of[19]for details.) Thus the equivalence among (i) through (iii) in Proposition 4.2 can be shown by Theorem 1.2 of[19].The equivalence of (i) and (iv) can be verified by Lemma 4.3 below.Lemma 4.3.Let M be a compact submanifold of R d and ({X t } t>0 , ζ, p) an M -valued martingale with an end point indexed by t > 0. Then the following are equivalent: