A support theorem for SLE curves

For all $\kappa>0$, we show that the support of SLE$_\kappa$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.


Overview
The support of a random variable X in a Polish space is the set of points x such that for any open neighborhood V of x, we have P(X ∈ V ) > 0. In this paper, the random variable X will be a random process, namely the SLE κ trace, and our goal is to describe its support.
Characterising the support of random processes such as Brownian motion and diffusions is an important research problem for stochastic (partial) differential equations, where it was initiated by Stroock and Varadhan [SV72] when they studied a strong maximum principle of a PDE operator. In [CF10] a support theorem was the key to a Hörmander/Malliavin theory for rough differential equations. The description of a support is also an important step to study the invariant measure of stochastic equations (see e.g. [TW18,CF18]). Other questions related to support theorems are large deviation estimates, or the continuity of solution maps of SDE and SPDE. SLE κ is an important random planar curve that shares many analogies with Brownian motion and other random processes. SLE κ is proven and conjectured to be the scaling limits or interface of many discrete models arisen from statistical physics (e.g. [LSW04, Smi01, SS05, SS09, CDCH + 14]). Instead of the Markov property, it satisfies a domain Markov property. Depending on the parameter κ, it has different regularities (similar to fractional Brownian motion). Moreover, SLE κ is defined through a family of deterministic ordinary differential equations called Loewner equations with the random input √ κB, where B is the one dimensional Brownian motion.
Motivated by the rich study of Brownian motion and other processes and by the similarities of SLE κ to them, it is natural to ask for a support theorem for SLE κ curves. Before stating the main result, let us give an overview of SLE κ .
We call the Loewner map with the input √ κB the Schramm-Loewner map. It is shown that this map is almost surely well-defined ([RS05, for κ = 8], [LSW04, for κ = 8]), that is, a.s. it gives rise to a curve. These random curves are called SLE κ curves, and (abusing notation) denoted γ κ (instead of γ √ κB ). The previous properties could remind us of stochastic differential equations (SDE). An SDE is also driven by a Brownian motion, and if one replaces the Brownian motion by smooth functions, then the SDE becomes an ODE and has a deterministic solution. Recall that the support of the solution to an SDE can be characterized by the solutions of the ODEs that arise by replacing the Brownian noise by Cameron-Martin paths (see e.g. [FV10,Chapter 19]). One could guess that the support of SLE can be described in the analogous way. We show in this paper that this is indeed true.
The main difficulty in proving such statements is that the Schramm-Loewner map (or, in the SDE case, solution map) is not continuous, and even only almost surely defined. If it were, the support theorem for SLE would immediately follow from the well-known support theorem for Brownian motion. Note also that the SLE κ curve is not a diffusion process, even though the Loewner equation with Brownian motion as an input can be seen as an SDE. Hence the method of proving support theorems for diffusion processes does not apply directly to SLE κ .
Consider the set D of all functions that have locally vanishing 1/2-Hölder constant. See Section 2 for the exact definition and properties of D. Our main theorem is the following. spaces to which the random process corresponds. SLE κ can be viewed as a subset of the plane, a continuous path, an α-Hölder function ( [Lin08], [JVL11]), a p-variation path, or an element of Besov spaces ( [FT17]). When we consider SLE κ only as compact subsets and measure distances by the Hausdorff metric, one can show a corresponding version of Theorem 1.1 by applying the method in [BJVK13, Lemma 8.2]. For Theorem 1.1 which is a characterization of the support of SLE κ in the sup-norm, one needs a non-trivial effort. We believe that a similar statement can be made for Hölder and p-variation spaces (see discussion in Section 6). (Similarly, there are different versions of the support theorem for Brownian motion and SDE; see for example [LQZ02], [BAGL94].) A direct consequence of Theorem 1.1 is that all the above statements are true in the strong topology of curves, which is weaker than the sup-norm topology. Consider the space of continuous paths α : [0, 1] → H modulo reparametrisation. Then the strong topology is defined by the metric where the infimum is taken over all increasing homeomorphisms from [0, 1] to [0, 1].
Corollary 1.2. Fix κ > 0. The support of SLE κ , in the strong topology, is the closure of which is equal to the closure of Theorem 1.1 consists of the two following results.
The first proposition implies that the set {γ λ : λ ∈ D} contains the support of SLE κ , while the second implies the other inclusion. Proposition 1.3 is similar to the Wong-Zakai Theorem [WZ65a,WZ65b], which is usually considered as an easier direction of support theorem. In principle, the Wong-Zakai Theorem says that if one regularizes or approximates the input (which is Brownian motion), then the output is also approximated. For SLE κ , κ = 8, this has been shown in [Tra15]. The result for κ = 8 follows from [LSW04].
For κ = 8, let γ 8 be a sample of the SLE 8 trace on the time interval [0, 1]. Then it is almost surely in the support of SLE 8 , i.e. for any ε-neighborhood B ε of γ 8 , SLE 8 is in B ε with positive probability. From [LSW04,Theorem 4.8] it follows that with positive probability, a segment of some UST Peano curveγ, mapped into H, is also in B ε . In particular, there is some sample ofγ such that γ 8 −γ ∞,[0,1] < ε.
Moreover, the UST Peano curve is constructed in [LSW04] as a simple, piecewise smooth curve. Therefore, by rounding off the edges and reparametrising by halfplane capacity (a precise argument is conducted in the proof of Proposition 6.4), we find a smooth Loewner curve γ (which in particular has a smooth driving function) with γ 8 − γ ∞,[0,1] < ε.
Proving Proposition 1.4 is the main part of this paper.

Strategy
First let us note a main difficulty in proving Proposition 1.4. Recall that in general, the Loewner map is not continuous, as the following example in [Law05,Page 116] shows.
Note that as sets, the traces γ (n) indeed come closer to the trace of the zero function, i.e. γ(t) = i2 √ t, but not as parametrized paths. Figure 1: The "Christmas tree".
The proof of Proposition 1.4 will be of the form: Naturally, one expects (A) to contain the condition that ξ − λ ∞ is small. But we need also something else that prevents a "Christmas tree" behaviour. The exact form of (A) will be formulated in Corollary 3.3. The condition is roughly as follows: where 0 = t 0 < t 1 < · · · < t n = 1 is a partition of [0, 1] depending on λ and ε, and the closeness of ξ to λ on [t k , t k+1 ] depends on how γ ξ t k behaves on [t k+1 , 1]. (γ ξ t k denotes the trace of the Loewner chain driven by ξ restricted to [t k , 1].) This structure of (A) allows us to make use of the independent increments of Brownian motion, which will imply that (A) is satisfied with positive probability. Now we explain roughly how we estimate the difference γ ξ − γ λ ∞,[0,1] in (1), and derive condition (A).
The right-hand sides of (2) and (3) have two things in common. They contain two terms. One is the difference between two conformal maps evaluated at the same point. The other term is the difference between the images of two points under the same map.
The first term of (3) can be estimated since the expectation of the moments of (f ξ t0 ) have been studied carefully; see [JVL11]. This is the strategy used in [Tra15]. However, upon investigating, one needs the expected moments of (f ξ t0 ) conditioned on ξ (which is a multiple of Brownian motion) close to a given λ, which is not known.
It turns out that the inequality (2) is approachable. To estimate the second term in the right-hand side of (2), we want the map f λ t0 to be uniformly continuous, uniformly in t 0 . This is true for sufficiently nice λ. This is where we impose the condition for λ in Theorem 1.1.
To control the distance between γ ξ t0 (t) and γ λ t0 (t), we just need to observe that when |t 2 − t 0 | is small, both points stay within a small box around 0; see Lemma 2.1.
For the first term of (2), we will apply (5) of Lemma 2.4. This lemma concerns the difference between two conformal maps driven by two driving functions. Roughly speaking, it tells us that where a b means a ≤ Cb for some fixed constant C > 0.
We get an estimate that can go arbitrarily bad if γ ξ t0 (t) gets close to the real line. Note that γ ξ t0 depends only on the increments of ξ from t 0 onwards. Since Brownian increments on disjoint time intervals are independent, we can "safely" require a smaller value for ξ − λ ∞,[0,t0] , depending on inf t∈[t1,t2] Im γ ξ t0 (t). The aforementioned argument works for SLE κ with κ ≤ 4 since a.s. inf t∈[t1,t2] Im γ ξ t0 > 0 given fixed t 0 < t 1 < t 2 . The situation becomes more complicated when κ > 4 since At the end, we show that it will not happen provided that ξ is close to λ, i.e.
This is another place where we will use the properties of functions in D.

Organization of the paper
In Section 2, we gather some basic definitions and facts. In Section 3, we prove a lemma comparing two deterministic Loewner curves. Then we use it in Section 4 to prove Proposition 1.4 in the case κ ≤ 4. In Section 5, we prove a lemma that generalizes the proof of Proposition 1.4 to all κ > 0. In Section 6, we discuss further characterisations of the support and some open questions.

Acknowledgement
We would like to thank Steffen Rohde and Peter Friz for various discussions. The authors acknowledge the financial support from the European Research Council (ERC) through a Consolidator Grant #683164 (PI: Peter Friz). We also thank Peter Friz and Vlad Margarint for suggestions regarding the presentation of the paper. We thank the referees for helpful comments and inspiring questions.

Definitions and properties
Definition of the Loewner map. Let λ ∈ C([0, 1], R). Consider the family of Loewner equations with different initial values: The following lemma concerns how big the hull K t is. See [Won14, Lemma 3.2] for a (trivial) proof.
Lemma 2.1. Let (K t ) be the hulls generated by a driving function λ. Then for all z ∈ K t , If for every t ≥ 0, the limit exists and is continuous in The curve γ ⊂ H is called the Loewner curve driven by λ. We call a Loewner curve simple if it intersects neither itself nor R \ {γ(0)}. In that case, For each t ≥ 0, let .
The maps f t andf t are conformal on the upper half-plane H. The latter is a centred version of f t . To emphasize the dependence on λ, we also use notations γ λ , f λ t , and the likes.
The space D.
• There is a function δ(·, λ) : (0, ∞) → (0, ∞) such that A proof of this proposition can be found in the proof of [LMR10, Theorem 4.1]. There they have shown that λ generates a curve, and H\γ λ is a quasi-slit half-plane, therefore a John domain (see [Pom92,Section 5.2] for a definition). It follows from [Pom92,Corollary 5.3] that f t (as a conformal map from H to a John domain) is Hölder continuous on bounded sets, with Hölder constant and exponent depending on the John domain constant.
A particular example: When λ ≡ 0, then f λ t (z) = √ z 2 − 4t which is Höldercontinuous in z on any bounded set, uniformly in t.
The following will be our main ingredient to estimate the difference between conformal maps derived from the Loewner equation.
Lemma 2.4 ([JVRW14, Lemma 2.3]). Let f 1 t and f 2 t be two inverse Loewner maps with U 1 and U 2 , respectively, as driving terms. Then for t ≥ 0 and z = x + iy ∈ H Remark 2.5. The inequality (5) is the one that will be used. We do not use the full strength of Lemma 2.4. What we really need is an inequality of the form where Φ 1 , Φ 2 are two functions such that Φ 2 (y) > 0 and Φ 1 (0 + ) = 0. Therefore, one can replace (5) by an inequality in [Law05,Proposition 4.47] which says that 3 Comparing two Loewner curves. A deterministic estimate.
We will use the following lemma to compare the difference between two Loewner curves.
Remark 3.2. The lemma roughly says that where Φ is an increasing function with Φ(0 + ) = 0 that depends on the modulus of continuity of λ.
Note that c t0,t2 depends only on the increment ( We also see that when γ ξ behaves like the "Christmas tree", then c t0,t2 will be small. In order to prevent this behaviour, we can change ξ on the interval [0, t 0 ], making ξ − λ [0,t0] smaller while leaving c t0,t2 unchanged.
Proof of Lemma 3.1. Let λ and ξ satisfy the conditions of Lemma 3.1. Observe that . We follow (2) and estimate First we estimate the second term of the right-hand side.
Proof. Let t ∈ [0, 1]. In case t ≤ t 1 , applying Lemma 2.1 in the same way as in the proof of Lemma 3.1 implies If t ≥ t 1 , we find k ≥ 0 such that t ∈ [t k+1 , t k+2 ]. We apply Lemma 3.1 with the time points 0 ≤ t k < t k+1 < t k+2 . Observe Lemma 3.1 shows Remark 3.4. The list of conditions for Corollary 3.3 looks quite long. We describe roughly how we will find suitable variables such that the corollary can be applied. Suppose that λ and a are given. We will pickε and ∆t accordingly. Then, to choose ε 1 , ..., ε n and ξ, note that each c t k ,t k+2 depends only on the increments of ξ on the interval [t k , t k+2 ]. Therefore we can choose ε k depending on the increments of ξ on [t k , 1], and afterwards choose the increments of ξ on [t k−1 , t k ], then again choose ε k−1 , and so on.
4 Proof of the Support Theorem for κ ≤ 4 Let ξ(t) = √ κB t and let λ ∈ D. Let a > 0 be given. For simplicity, we first show Theorem 1.1 for κ ≤ 4. In this case γ ξ is a simple trace, as well as γ ξ t k for all t k ≥ 0. In particular, it will never touch the real line after time 0 and automatically guarantees the condition c t k ,t k+2 = inf t∈[t k+1 ,t k+2 ] Im γ ξ t k (t) > 0 of Corollary 3.3. The remaining task is to find a set of positive probability where all conditions of Corollary 3.3 are satisfied.
Suppose now that we have (arbitrary) random variables ε k ≤ε that are a.s. positive and measurable w.r.t. F t k ,1 (where F r,s denotes the sigma algebra generated by Brownian increments between time r and s). By inductively applying the independence of Brownian increments, we claim that To verify this claim, suppose that for some 1 ≤ k < n P(∀l ≥ k + 1 : (ξ − ξ(t l−1 )) − (λ − λ(t l−1 )) ∞,[t l−1 ,t l ] ≤ ε l ) > 0.

Proof of the Support Theorem for general κ
In case κ > 4, we can use the same proof as before, but the condition might be violated. Hence, our main task here is to add some condition that guarantees c t0,t2 (ω) > 0 for almost all ω.
This implies γ ξ t1 (t) ∈ I and consequently γ ξ t0 (t) ∈ ∂K ξ t0,t1 . Moreover, by a property of SLE proven by D. Zhan (see below), ∂K ξ t0,t1 intersects R only at its endpoints. Since γ ξ t1 (t) actually lies in the interior of I, then γ ξ t0 (t) lies in the interior of ∂K ξ t0,t1 which is contained in H. Finally, note that (9) is reasonable because by the assumption λ ∈ D and the freedom to chooseε, we can estimate where for I, at least in the extreme case ξ ≡ 0, it can be calculated that I = [−2 √ ∆t, 2 √ ∆t]. Now, we fill the above arguments with more rigorous details. First, we analyze f ξ t0,t1 as the inverse map of the Loewner flow driven by ξ(·) − ξ(t 0 ), t ∈ [t 0 , t 1 ]. In order to do so, we analyze the time-reversed Loewner equation. Recall that if (g t ) is the Loewner flow driven by ξ, then for any s 0 > 0 we can writef s0 = h s0 +ξ(s 0 ) where (h t ) is the solution of hits 0, and T (z) = ∞ for all z / ∈ R. Suppose thatf s0 : H → H\K s0 can be continuously extended to the boundary R. (This holds when the driver is in D, or a multiple of Brownian motion.) It is known that exists a closed interval I such that (10) Therefore we can analyze the interval I just by the time-reversed Loewner equation.
J. Lind has shown in [Lin05, Corollary 1] that if W has 1/2-Hölder constant less than 4, then T (x) is comparable to x 2 . A comparison argument will show that the result stays true if the driver is slightly modified. The next two results make it more precise.
Proof. Assume without loss of generality that T 1 (x + δ) = t, i.e. h 1 s (x + δ) exists for all s < t and only dies at time t.
We claim that for all s < t, ∂ s h 2 s (x) < ∂ s h 1 s (x + δ). (Note that this means |∂ s h 2 s (x)| > |∂ s h 1 s (x + δ)| since both are negative.) At s = 0 this is obviously true since . Now if the claim holds for all s < s 0 , then Consequently, . This shows that there cannot be a first time s 0 where the claim is violated. By the continuity of V j and h j t (x), and therefore also ∂ t h j t (x) in t, the claim is never violated at any time.
To finish the proof of the lemma, note that we have also shown above that for all s ∈ [0, t]. If T 1 (x + δ) ≤ t, this means that h 1 s (x + δ) − V 1 (s) = 0 for some s ≤ t, and consequently h 2 s (x) − V 2 (s) = 0 for some smaller s < t.
In the above argument, c and ∆t can be chosen such that the set {x ∈ R | f ξ ∆t (x) ∈ K ξ ∆t } contains more than the interval [−c √ ∆t, c √ ∆t]. Then this interval gets mapped to an inner segment of ∂K ξ ∆t , and not to its endpoints. In particular, γ ξ (t) is in the interior of ∂K ξ ∆t .
We remark that the assumption on ξ holds almost surely if ξ is a multiple of Brownian motion.
Proof of Proposition 1.4. The case κ ≤ 4 has already been shown in Section 4. The proof for κ > 4 is almost identical.

Further characterisations of the support and open questions
We note that the set is a deterministic set and does not depend on κ. One may ask for what specific λ (besides λ ∈ D) we have γ λ ∈ S?
First, it is worth mentioning that all curves in S are indeed Loewner curves, i.e. they satisfy the local growth property (which is not obvious since the closure is taken in the space C([0, 1]; H). We show this below in Proposition 6.3.
First recall that the half-plane capacity enjoys a uniform continuity property, described in [Kem17,Lemma 4.4]. We will apply it in the following way.
Here, for a compact set A ⊆ H, fill(A) denotes the complement of the unbounded connected component of H \ A, and A δ the δ-neighbourhood of A.
For a sequence of domains H n ⊆ H that contain a common neighbourhood of ∞, their kernel (with respect to ∞) is the largest domain H containing a neighbourhood of ∞ such that any compact K ⊆ H is contained in all but finitely many H n .
Lemma 6.2. Let H n ⊆ H be a sequence of simply connected domains that contain a common neighbourhood of ∞, and let H be their kernel (with respect to ∞). Let z ∈ H, 0 < r 1 < r 2 , and z 1 , z 2 ∈ B(z, r 1 ) ∩ H. If for all n the points z 1 and z 2 are in the same connected component of B(z, r 1 ) ∩ H n , then they are in the same connected component of B(z, r 2 ) ∩ H.
Proof. Since H is a domain, we can find a simple polygonal path α 1 in H from z 1 to z 2 . Note that such a path hits ∂B(z, r 1 ) only a finite number of times. Moreover, by a small perturbation we can choose α 1 to cross ∂B(z, r 1 ) at each such time. If α 1 does not cross ∂B(z, r 1 ) at all, we are done, so assume from now on that it does. Let U ⊆ H be an open neighbourhood of α 1 . The definition of kernel implies U ⊆ H n for all but finitely many n. Without loss of generality we restrict ourselves to that subsequence.
Suppose for the moment that (small neighbourhoods of) z 1 and z 2 lie in the same connected component of B(z, r 1 ) \ α 1 . This means that z 1 and z 2 can be connected by a simple path α 2 in B(z, r 1 ) that does not intersect α 1 except at its endpoints. In that case, α 1 ∪ α 2 is a simple loop and by the Jordan curve theorem separatesĈ into two components. Call the component that contains ∞ the "outside" component.
By construction, α 1 ∪ α 2 separates ∂B(z, r 1 ) into finitely many segments, alternatingly "inside" and "outside". Let A be an "inside" segment. Then there exists an open connected set U A ⊆ B(z, r 2 ) \ B(z, r 1 ) in the neighbourhood of A that is still "inside" α 1 ∪ α 2 . We claim that U A ⊆ H n for all n. This will imply that U := U ∪ A "inside" U A ⊆ H n for all n, and henceÛ ⊆ H. By alternatingly following segments of α 1 and U A , we see thatÛ connects z 1 and z 2 in B(z, r 2 ) ∩ H.
Let n ∈ N. By assumption, we can find a path α 3 in B(z, r 1 ) ∩ H n that connects z 1 to z 2 . Since α 2 ∪ α 3 ⊆ B(z, r 1 ), the winding numbers of α 1 ∪ α 2 and α 1 ∪ α 3 around U A are the same. Therefore U A is disconnected from ∞ (and hence also from R) by α 1 ∪ α 3 . Since α 1 ∪ α 3 ⊆ H n and H n is simply connected, we must have It remains to handle the case that (small neighbourhoods of) z 1 and z 2 lie in different components of B(z, r 1 ) \ α 1 . By construction, α 1 ∩ B(z, r 1 ) consists of finitely many segments. Pick the segmentα that bounds the component in which (a small neighbourhood of) z 1 lies, and letz 2 ∈α. Now z 1 andz 2 fulfil the conditions of the lemma again because any path in B(z, r 1 ) from z 1 to z 2 needs to crossα, and for each n one such path lies in H n (by the assumption on z 1 , z 2 ). Moreover, (small neighbourhoods of) z 1 andz 2 lie in the same component of B(z, r 1 ) \ α 1 . By the previous part of the proof, z 1 andz 2 are in the same connected component of B(z, r 2 ) ∩ H. Repeating this argument, the lemma in the general case follows by induction. Then the family (K t ) t∈[0,1] is parametrised by half-plane capacity and satisfies the local growth property.
Proof. Lemma 6.1 implies hcap(γ n [0, t]) → hcap K t for all t, so the parametrisation by half-plane is preserved. To show that (K t ) satisfies the local growth property, we will find for any ε > 0 some δ > 0 such that for all t there exists a crosscut of length less than ε in H \ K t that separates K t+δ \ K t from ∞. In the following, we call H n t := H \ γ n [0, t] and H t := H \ K t .
Now let t ∈ [0, 1] and z 1 , z 2 ∈ K t+δ \ K t . It suffices to consider z 1 , z 2 ∈ γ[t, t + δ] \ K t since this set bounds K t+δ \ K t . We claim that z 1 and z 2 are in the same connected component of H t \∂B(γ(t), 2ε). This will imply that there exists a segment of ∂B(γ(t), 2ε) ∩ H t that separates z 1 and z 2 from ∞ in H t (This can be seen e.g. by mapping H t to H). That segment is the desired crosscut.
By the choice of δ we have z 1 , z 2 ∈ H t ∩B(γ(t), ε), and we can find r > 0 such that B(z i , 2r) ⊆ H t ∩ B(γ(t), ε), i = 1, 2. Let n be large enough so that γ − γ n ∞ < r. In particular, B(z i , r) ⊆ H n t ∩ B(γ(t), ε), i = 1, 2. Note that the uniform convergence of γ n implies γ n [0, t] → K t in the sense of kernel convergence. Since γ n are simple, B(z 1 , r) and B(z 2 , r) are connected by γ n (]t, t + δ]) in H n t . Moreover, γ n (]t, t + δ]) ⊆ B(γ(t), ε + r) by the choice of δ and n. So by Lemma 6.2, B(z 1 , r) and B(z 2 , r) are in the same connected component of We turn back to the question of characterising S. Just from the definition of the support, for fixed κ = 8, we have a.s. γ κ ∈ S. Moreover, since piece-wise linear functions are in D, any γ λ that is approximated by a sequence of Loewner curves generated by piece-wise linear drivers is in S. In particular, [Tra15,Theorem 2.2] shows that if λ is weakly 1/2-Hölder and |(f λ t ) (iy)| ≤ Cy −β for some β < 1 and all t, y ∈ ]0, 1], then γ λ has such an approximation, hence is in S.
To see the last equality, suppose we have a simple curve γ ∈ C ∞ ((0, 1]; H) with γ(0) = 0. Then we can approximate it by a simple smooth curveγ ∈ C ∞ ((0, 1]; H) withγ(t) = i2 √ t on a very small time interval t ∈ [0, δ]. Thenγ is driven by a smooth driving function (see [EE01]), soγ ∈ S. (Strictly speaking, we also need to parametriseγ by half-plane capacity, but this will not change the approximation much, as the proof of Proposition 6.4 below shows.) We can say more.
• Can one strengthen the topology in Theorem 1.1? Note that the statement of Theorem 1.1 is the same regardless of κ. But as shown in [JVL11], for each κ there exists an optimal α * (κ) such that γ κ is α-Hölder continuous for α < α * (κ). Ideally, we would like to characterise the support of SLE κ in the α-Hölder space, or similarly, in the p-variation space where p > p * (κ) (see [FT17]). (Note that it is proved in [FS17] that γ λ is 1/2-Hölder continuous on [0, 1] for λ ∈ W 1,2 . Hence, almost surely the α-Hölder norm of (γ κ − γ λ ) is finite for some α > 0.) • We do not know how P( γ κ − γ λ ∞,[0,1] < ε) behaves as ε → 0 + . However as κ → 0, we believe that similarly to [Wan19] the following is true: • To what extent does the converse of Proposition 6.3 hold? Is every (not necessarily simple) curve, parametrised by half-plane capacity, that satisfies the local growth property in fact in S? (If not, is there a characterisation which curves are in S?) This would give us a full characterisation of S, generalising Proposition 6.4 to general curves in C([0, 1]; H). Update: In an ongoing work, the second author gives a positive answer to this question.