Regularity of SLE in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,\kappa )$$\end{document}(t,κ) and refined GRR estimates

Schramm–Loewner evolution (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {SLE}_\kappa $$\end{document}SLEκ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{\kappa }$$\end{document}κ times Brownian motion. This yields a (half-plane) valued random field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = \gamma (t, \kappa ; \omega )$$\end{document}γ=γ(t,κ;ω). (Hölder) regularity of in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\cdot ,\kappa ;\omega $$\end{document}γ(·,κ;ω), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883–924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3–4):413–433, 2014) showed a.s. Hölder continuity of this random field for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa < 8(2-\sqrt{3})$$\end{document}κ<8(2-3). In this paper, we improve their result to joint Hölder continuity up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa < 8/3$$\end{document}κ<8/3. Moreover, we show that the SLE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_\kappa $$\end{document}κ trace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\cdot ,\kappa )$$\end{document}γ(·,κ) (as a continuous path) is stochastically continuous in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}κ at all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa \ne 8$$\end{document}κ≠8. Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest.


Introduction
Schramm-Loewner evolution (SLE) is a random (non-self-crossing) path connecting two boundary points of a domain. To be more precise, it is a family of such random paths indexed by a parameter κ ≥ 0. It has been first introduced by [19] to describe several random models from statistical physics. Since then, many authors have intensely studied this random object. Many connections to discrete processes and other geometric objects have been made, and nowadays SLE is one of the key objects in modern probability theory.
The typical way of constructing SLE is via the Loewner differential equation (see Sect. 3) which provides a correspondence between real-valued functions ("driving functions") and certain growing families of sets ("hulls") in a planar domain. For many (in particular more regular) driving functions, the growing families of hulls (or their boundaries) are continuous curves called traces. For Brownian motion, it is a non-trivial fact that for fixed κ ≥ 0, the driving function √ κ B almost surely generates a continuous trace which we call SLE κ trace (see [16,18]).
There has been a series of papers investigating the analytic properties of SLE, such as (Hölder and p-variation) regularity of the trace [5,9,15,18]. See also [4,20] for some recent attempts to understand better the existence of SLE trace.
A natural question is whether the SLE κ trace obtained from this construction varies continuously in the parameter κ. Another natural question is whether with probability 1 the construction produces a continuous trace simultaneously for all κ ≥ 0. These questions have been studied in [10] where the authors showed that with probability 1, the SLE κ trace exists and is continuous in the range κ ∈ [0, 8 ( Stability of SLE trace was also recently studied in [12,Theorem 1.10]. They show the law of γ κ n ∈ C([0, 1], H) converges weakly to the law of γ κ in the topology of uniform convergence, whenever κ n → κ < 8. Of course, we get this as a trivial corollary of Theorem 1.1 in case of κ < 8/3. Our Theorem 1.2 (proved in Sect. 3.2) strengthens [12,Theorem 1.10] in three ways: (i) we allow for any κ = 8; (ii) we improve weak convergence to convergence in probability; (iii) we strengthen convergence in C( [0,1], H) with uniform topology to C p-var ([0, 1], H) with optimal (cf. [5]) p-variation parameter, i.e. any p > (1 + κ/8) ∧ 2. The analogous statement for α-Hölder topologies, α < 1 − κ 24+2κ−8 √ 8+κ ∧ 1 2 , is also true.
There are two major new ingredients to our proofs. First, we prove in Sect. 5 a refined moment estimate for SLE increments in κ, improving upon [10]. Using standard notation [14,18], for κ > 0, we denote by (g κ t ) t≥0 the forward SLE flow driven by √ κ B, j = 1, 2, and byf κ t = (g κ t ) −1 (· + √ κ B t ) the recentred inverse flow, also defined in Sect. 3 below. Write a b for a ≤ Cb, with suitable constant C < ∞. The improved estimate (Proposition 3.5) reads for 1 ≤ p < 1 + 8 κ . The interest in this estimate is when p is close to 1 + 8/κ. No such estimate can be extracted from [10], as we explain in some more detail in Remark 3.6 below.
Secondly, our way of exploiting moment estimates such as (1) is fundamentally different in comparison with the Whitney-type partition technique of "(t, y, κ)"-space [10] (already seen in [18] without κ), combined with a Borel-Cantelli argument. Our key tool here is a new higher-dimensional variant of the Garsia-Rodemich-Rumsey (GRR) inequality [7] which is useful in its own right, essentially whenever one deals with random fields with very "different"-in our case t and κ-variables. The GRR inequality has been a useful tool in stochastic analysis to pass from moment bounds for stochastic processes to almost sure estimates of their regularity.
We are going to prove the following refined GRR estimates in two dimensions, as required by our application, noting that extension to higher dimension follow the same argument. Lemma 1.3 Let G be a continuous function (defined on some rectangle) such that, for some integers J 1 , J 2 , Suppose that for all j, Then, under suitable conditions on the exponents, (2) .
Observe that the exponents q 1 j , q 2 j are allowed to vary, exactly as required for our application to SLE. We also note that the flexibility to have J 1 , J 2 > 1 is used in the proof of Theorem 1.2 but not 1.1.
One might ask whether one can further improve Theorem 1.1 to all κ ≥ 0. With the methods of this paper, it would require a better moment estimate in the style of (1) with larger exponent on the right-hand side. If such an estimate were to hold true with arbitrarily large exponent on the right-hand side (and any suitable exponent on the left-hand side), which is not clear to us, almost sure continuity of the random field in all (t, κ) with κ = 8 would follow.

A Garsia-Rodemich-Rumsey lemma with mixed exponents
In this section we prove a variant of the Garsia-Rodemich-Rumsey inequality and Kolmogorov's continuity theorem. The classical Kolmogorov's theorem goes by a "chaining" argument (see e.g. [13,Theorem 1.4.1] or [23,Appendix A.2]), but can also be obtained from the GRR inequality (see e.g. [21,Corollary 2.1.5]). In the case of proving Hölder continuity of processes, the GRR approach provides more powerful statements (cf. [6,Appendix A]). In particular, we obtain bounds on the Hölder constant of the process that are more informative and easier to manipulate, which will be useful in the proof of Theorem 4.1. (Although there are drawbacks of the GRR approach when generalising to more refined modulus of continuity, see the discussion in [23,Appendix A.4]. ) We discuss some of the extensive literature that deal with the generality of GRR and Kolmogorov's theorem. The reader may skip this discussion and continue straight with the results of this section.
There are some direct generalisations of GRR and Kolmogorov's theorem to higher dimensions, e.g. [ [1,3,8]. Moreover, there have been more systematic studies in a general setting under the titles metric entropy bounds and majorising measures. They derive bounds and path continuity of stochastic processes mainly from the structure of certain pseudometrics that the processes induce on the parameter space, such as d X (s, t) := (E|X (s) − X (t)| 2 ) 1/2 . A large amount of the theory is found in the book by Talagrand [23]. These results due to, among others, R. M. Dudley, N. Kôno, X. Fernique, M. Talagrand, and W. Bednorz. Their main purpose is to allow different structures of the parameter space and inhomogeneity of the stochastic process (see e.g. [2,11,23]).
We explain why the existing results do not cover the adaption that we are seeking in this section. The general idea for applying the theory of metric entropy bounds would be considering the metric d X (s, t) = (E|X (s) − X (t)| q ) 1/q for some q > 1.
Let us consider a random process defined on the parameter space T = [0, 1] 2 that satisfies where q 1 and q 2 might be different, say q 1 < q 2 . By Hölder's inequality, Write t = (t 1 , t 2 ), s = (s 1 , s 2 ). We may let where we can take q = q 1 (but not q = q 2 without knowing any bounds on higher moments of |X (s 1 , We explain now that we have already lost some sharpness when we estimated (3) using Hölder's inequality. Indeed, all the results [11,Theorem 3], [23, (13.141 Observe that we can take ϕ(x) = x q 1 at best. To apply any of these results, the condition turns out to be 1 α 1 + q 2 q 1 α 2 < 1. In fact, [23,Theorem 13.5.8] implies that we cannot expect anything better just from the assumption (4). More precisely, the theorem states that in general, when we assume only (4), in order to deduce any pathwise bounds for the process X , we need to havê with B denoting the ball with respect to the metric d, and μ e.g. the Lebesgue measure. In our setup this turns out to the condition 1 α 1 + q 2 q 1 α 2 < 1. We will show in Theorem 2.8 that by using the condition (2) instead of (4), we can relax this condition to 1 this is an improvement. We have not found this possibility in any of the existing references.
We now turn to our version of the Garsia-Rodemich-Rumsey inequality that allows us to make use of different exponents q 1 = q 2 . In addition to the scenario (2), we allow also the situation when e.g. |X (s 1 , Let (E, d) be a metric space. We can assume E to be isometrically embedded in some larger Banach space (by the Kuratowski embedding). To ease the notation, we write |x − y| = d(x, y) both for the distance in E and for the distance in R. For a Borel set A we denote by |A| its Lebesgue measure and In what follows, let I 1 and I 2 be two (either open or closed) non-trivial intervals of R.

Lemma 2.1
Let G ∈ C(I 1 × I 2 ) be a continuous function, with values in a metric space E, such that for all j, where q i j ≥ 1, Fix any a, b > 0. Then

Remark 2.2
The statement is already true when q i j > 0 (not necessarily ≥ 1) and can be shown by an argument similarly as in [ Let (x 1 , x 2 ), (y 1 , y 2 ) ∈ I 1 × I 2 . Using the above observation, we will approximate G(x 1 , x 2 ) and G(y 1 , y 2 ) by well-chosen sequences of sets.
We pick a sequence of rectangles I n 1 × I n 2 ⊆ I 1 × I 2 , n ≥ 0, with the following properties: In order for such a sequence of rectangles to exist, we must have Conversely, this condition guarantees the existence of such a sequence.

Remark 2.3
The dependence of the multiplicative constant C on |I 1 | and |I 2 | is specified in (11). This can be convenient when we want to apply the lemma to different domains. A more accurate version iŝ . Remark 2. 4 We could have added some more flexibility by allowing the exponents (q i j ), (β i j ) to vary with u 1 , u 2 , but again we will not need it for our result.
Remark 2. 5 We have a free choice of a, b ≥ 0 which affects the Hölder exponents γ (1) i j , γ (2) i j . In general, it is not simple to spell out the optimal choice of a, b and hence the optimal Hölder exponents. Usually we are interested in the overall exponents (i.e. min i, j γ (1) i j , min i, j γ (2) i j ), and we can solve to find the optimal choice for a, b.
For instance, in case β 1 j = β 1 and β 2 j = β 2 for all j, the best choice is resulting in But this is not necessarily the optimal choice.

Remark 2.6
Notice that the condition to apply the lemma does only depend on (β i j ), not (q i j ), but the resulting Hölder-exponents will.

Remark 2.7
The proof straightforwardly generalises to higher dimensions.
Using our version of the GRR lemma, we can show another version of the Kolmogorov continuity condition. Here we suppose I 1 , I 2 are bounded intervals.

Theorem 2.8 Let X be a random field on I
with measurable real-valued A i j that satisfy (2) and E[C q min ] < ∞ for q min = min i, j q i j .

Remark 2.9
In case α 1 j = α 1 and α 2 j = α 2 for all j, the expressions for the Hölder exponents γ (1) , γ (2) given above are sharp. In the general case, the exponents may be improved, following an optimisation described in Remark 2.5.

Remark 2.10
The constants C can be replaced by (deterministic) functions that are integrable in (x 1 , x 2 ), without change of the proof. But one would need to formulate the condition more carefully, therefore we decided to not include it.
We point out that in case J 1 = J 2 = 1 and q 1 = q 2 , this agrees with the twodimensional version of the (inhomogeneous) Kolmogorov criterion [13, Theorem 1.4.1].
Proof Part 1. Suppose first that X is already continuous. In that case we can directly apply Lemma 2.1. The expectation of the integrals (6) and (7) are finite if β i j < α i j +1 for all i, j. By choosing β i j as large as possible, the conditions (β 1 −2)(β 2 −2)−1 > 0 and β 1 > 2, β 2 > 2 are satisfied if α −1 1 + α −1 2 < 1 and α 1 > 1, α 2 > 1. Since the (random) constants M i j in Lemma 2.1 are almost surely finite, X is Hölder continuous as quantified in (8), and the Hölder constants M 1/q i j i j have q i j -th moments since they are just the integrals (6). The formulas for the Hölder exponents follow from the analysis in Remark 2.5. Part 2. Now, suppose X is arbitrary. We need to construct a continuous version of X . It suffices to show that X is uniformly continuous on a dense set D ⊆ I 1 × I 2 . Indeed, we can then apply Doob's separability theorem to obtain a separable (and hence continuous) version of X , or alternatively constructX by settingX = X on D and extendX continuously to I 1 × I 2 . ThenX is a modification of X because they agree on a dense set D and are both stochastically continuous [as follows from (12) and (13)].
We use a standard argument that can be found e.g. in [22, pp. 8-9]. We can assume without loss of generality that X (x 1 , In particular, the conditions (12) and (13) imply that X (x 1 , x 2 ) is an integrable random variable with values in a separable Banach space for every (x 1 , x 2 ).

Fix any countable dense subset
We can pick an increasing sequence of finite σ -algebras G n such that G = σ n G n . By martingale convergence, we have Moreover, (12) implies . By Jensen's inequality and (13), we have In particular, X (n) is stochastically continuous, and since G n is finite, X (n) is almost surely continuous. Applying Lemma 2.1 yields i j are defined as the integrals (6) and (7) with A (n) i j . It follows that on D we have implying thatM i j < ∞, hence X is uniformly continuous on D.
One-dimensional variants of Lemma 2.1 and Theorem 2.8 can also be derived. Having shown the two-dimensional results Lemma 2.1 and Theorem 2.8, there is no need for an additional proof of their one-dimensional variants, since we can extend any one-parameter function G to a two-parameter function viaG(x 1 , x 2 ) := G(x 1 ). This immediately implies the following results.

Corollary 2.11 Let G be a continuous function on an interval I such that
Then For the sake of completeness we also state the one-dimensional version of Theorem 2.8.

Corollary 2.12 Let X be a stochastic process on a bounded interval I such that
for all x, y ∈ I , where A j , j = 1, . . . , J , are measurable and satisfy Then X has a continuous modificationX that satisfies, for any γ < min j

Further variations on the GRR theme
We give some additional results that are similar or come as consequence of Lemma 2.1. This demonstrates the flexibility and generality that our lemma provides. We do not aim for a complete survey of all implications of the lemma. We begin by proving the result of Lemma 2.1 under slightly weaker assumptions. The assumptions may seem a bit at random, but they will turn out to be what we need in the proof of Theorem 4.1.

Lemma 2.13
Consider the same conditions as in Lemma 2.1, but instead of (5), we assume the following weaker condition. Let r j > 1 and θ j > 0 such that for and all the points appearing in the sum are also in the domain I 1 .
Then the result of Lemma 2.1 still holds, with the constant C depending also on (r j ), (θ j ).
Proof We proceed similarly as in the proof of Lemma 2.1. We pick the sequence I n i a bit more carefully. Let d i > 0, R i > 1, i = 1, 2, be as in the proof of Lemma 2.1, and recall that we can freely pick R i ≥ 9. It is not hard to see that we can then pick a sequence of rectangles I n 1 × I n 2 in such a way that as n → ∞, and another analogous sequence of rectangles for (y 1 , y 2 ) that begins with the same I 0 1 × I 0 2 . The proof proceeds in the same way, but instead of the assumption (5), we apply (14) with some z 1 that we pick now.
Let n ∈ N. We pick z 1 := inf(I n 1 ∪ I n−1 1 ) if this point is in the left half of I 1 , and . Moreover, all the points z 1 + r k (u 1 − z 1 ) and ≥ θ j , as one can see in the proof. more than distance |I 1 |/2 ≥ 2c away (in the u 1 resp. v 1 direction) from the end of the interval I 1 .
We now have to bound Since we assumed which is the same bound as in the proof of Lemma 2.1. The rest of the proof is the same as in Lemma 2.1.
The following corollary is only used for Theorem 3.8.  (6) and (7)

Corollary 2.14 Consider the same conditions as in Lemma
is a control.

Continuity of SLE in Ä and t
In this section we show the main results Theorems 1.1 and 1.2. We adopt notations and prerequisite from [10]. For the convenience of the reader, we quickly recall some important notations. Let U : [0, 1] → R be continuous. The Loewner differential equation is the following initial value ODE For each z ∈ H, the ODE has a unique solution up to a time . One says that λ generates a curve γ if It is known [16,18] that for fixed κ ∈ [0, ∞), the driving function where B is a standard Brownian motion, almost surely generates a curve, which we will denote by γ (·, κ) or γ κ . But we do not know whether given a Brownian motion B, almost surely all driving functions √ κ B, κ ≥ 0, simultaneously generate a curve.
Furthermore, simulations suggest that for a fixed sample of B, the curve γ κ changes continuously in κ, but only partial proofs have been found so far. We remark that this question is not trivial to answer because in general, the trace does not depend continuously on its driver, as [ We will often use the following bounds for the moments of |f t (iy)| that have been shown by Johansson Viklund and Lawler [9]. In order to state them, we use the following notation. Let κ ≥ 0. Set for r < r c (κ).
With the scaling invariance of SLE, [9, Lemma 4.1] implies the following.
Moreover, C can be chosen independently of κ and r when κ is bounded away from 0 and ∞, and r is bounded away from −∞ and r c (κ). 3 Now, for a standard Brownian motion B, and an SLE κ flow driven by √ κ B, we writef κ t , γ κ , etc. We also use the following notation from [9].
v(t, κ, y) :=ˆy Observe that v(t, κ, ·) is decreasing in y and Therefore lim y 0f κ t (iy) exists if v(t, κ, y) < ∞ for some y > 0. For fixed t, κ, this happens almost surely because Lemma 3.1 implies So we can define as a random variable. Note that with this definition we can still estimate

Almost sure regularity of SLE in (t, Ä)
In this subsection, we prove our first main result. The theorem should be still true near κ ≈ 0 (Without any integrability statement for C, it is shown in [10].), but due to complications in applying Lemma 3.1 (cf. [10, Proof of Lemma 3.3]), we decided to omit it.
As in [5], we will estimate moments of the increments of γ , using Lemma 3.1. We need to be a little careful, though, when applying Lemma 3.1, that the exponents do depend on κ. Since we are going to apply that estimate a lot, let us agree on the following.
The first estimate is just [5, Lemma 3.2]. The second estimate follows from the following result (which we will prove in Sect. 5) and Fatou's lemma.
Below we write x + = x ∨ 0 for x ∈ R. Corollary 3.7 Under the same conditions as in Proposition 3.5 we have where C < ∞ depends on κ − , κ + , T , p, and ε.
Proof For a holomorphic function f : H → H, Cauchy Integral Formula tells us that where we let α be a circle of radius δ/2 around iδ. Consequently, For all w on the circle α we have w ∈ [δ/2, 3δ/2] and w ∈ [−δ/2, δ/2]. Therefore Proposition 3.5 implies By Minkowski's inequality, and the result follows since the length of α is πδ.
With Proposition 3.3, we can now apply Theorem 2.8 to construct a Hölder continuous version of the map γ = γ (t, κ), whose Hölder constants have some finite moments.
There is just one detail we still have to take into consideration. In order to apply Theorem 2.8, we have to use one common exponent λ on the entire range of κ where we want to apply the GRR lemma. Of course, we can choose new values for λ again when we consider a different range of κ.
Alternatively, we could formulate our GRR version to allow exponents to vary with the parameters. But this will not be necessary since we can break our desired interval for κ into subintervals.
Hence, there exists a set 2 with probability 1 such that for all ω ∈ 2 , we have γ (t, κ) =γ (t, κ) for all κ ∈ K and almost all t. Restricted to ω ∈ 3 = 1 ∩ 2 , the previous statement is true for all κ ∈ K and all t. We claim that on the set 3 of probability 1, the path t →γ (t, κ) is indeed the SLE κ trace driven by √ κ B. This can be shown in the same way as [16,Theorem 4.7].
We know that for fixed κ ≤ 4, the SLE κ trace is almost surely simple. It is natural to expect that there is a common set of probability 1 where all SLE κ traces, κ < 8/3, are simple. This is indeed true.

Theorem 3.9 Let B be a standard Brownian motion. We have with probability 1 that for all κ < 8/3 the SLE κ trace driven by
√ κ B is simple.
Proof As shown in [18, Theorem 6.1], due to the independent stationary increments of Brownian motion, this is equivalent to saying that K κ t ∩ R = {0} for all t and κ, where K κ t = {z ∈ H | T κ z ≤ t} (the upper index denotes the dependence on κ). Let (g t (x)) t≥0 satisfy (15) with g 0 (x) = x and driving function U (t) = √ κ B t .
i.e. X is a Bessel process of dimension 1+ 4 κ . The statement K κ t ∩R = {0} is equivalent to saying that X s = 0 for all x = 0 and s ∈ [0, t]. This is a well-known property of Bessel processes, and stated in the lemma below.

Lemma 3.10 Let B be a standard Brownian motion and suppose that we have a family of stochastic processes X
Then we have with probability 1 that T κ,x = ∞ for all κ ≤ 4 and x > 0.
Proof For fixed κ ≤ 4, see e.g. [14,Proposition 1.21]. To get the result simultaneously for all κ, use the property that if κ <κ and x > 0, then X κ,x t > Xκ ,x t for all t > 0, which follows from Grönwall's inequality.

Stochastic continuity of SLE Ä in Ä
In the previous section, we have shown almost sure continuity of SLE κ in κ (in the range κ ∈ [0, 8/3[). Weaker forms of continuity are easier to prove, and hold on a larger range of κ. We will show here that stochastic continuity (also continuity in L q (P) sense for some q > 1 depending on κ) for all κ = 8 is an immediate consequence of our estimates. Below we write
Note that without sufficiently fast convergence of κ n → κ it is not clear whether we can pass from L q -convergence to almost sure convergence.
Proof Theorem 3.11 immediately implies the statement with · ∞ . To upgrade the result to Hölder and p-variation topologies, recall the following general fact which follows from the interpolation inequalities for Hölder and p-variation constants (see e.g. [6,Proposition 5.5]): Suppose X n , X are continuous stochastic processes such that for every ε > 0 there exists M > 0 such that P( X n p-var;[0,T ] > M) < ε for all n. If X n → X in probability with respect to the · ∞ topology, then also with respect to the p -variation topology for any p > p. The analogous statement holds for Hölder topologies with α < α ≤ 1.
In order to apply this fact, we can use [5, Theorem 5.2 and 6.1] which bound the moments of γ p-var and γ C α . The values for p and α have also been computed there.

Convergence results
Here we prove a stronger version of Theorem 3.2, namely uniform convergence (even convergence in Hölder sense) off κ t (iy) as y 0. For this result, we really use the full power of Lemma 2.1 (actually Lemma 2.13 as we will explain later). We point out that this is a stronger result than Theorem 1.1, and that our previous proofs of Theorem 1.1 and 1.2 do not rely on this section.
The same method as Theorem 4.1 can be used to show the existence and Hölder continuity of the SLE κ trace for fixed κ = 8, avoiding a Borel-Cantelli argument. The best way of formulating this result is the terminology in [5].
The following result is proved similarly to Theorem 4.1.

Remark 4.4
The conditions for the exponents are the same as in [5]. In particular, the result applies to the (for SLE κ ) optimal p-variation and Hölder exponents.
Step 1 We would like to show that v andf (defined above) are Cauchy sequences in the aforementioned Hölder space as y 0. Therefore we will take differences |v(·, ·, y 1 ) − v(·, ·, y 2 )| and |f (iy 1 ) −f (iy 2 )|, and estimate their Hölder norms with our GRR lemma. Note that it is not a priori clear that v(t, κ, y) is continuous in (t, κ), but |v(t, κ, y 1 ) − v(t, κ, y 2 )| =´y 2 y 1 |(f κ t ) (iu)| du certainly is, so the GRR lemma can be applied to this function.
Consider the function The strategy will be to show that the condition of Lemma 2.1 is satisfied almost surely for G. As in the proof of Kolmogorov's continuity theorem, we do this by showing that the expectation of the integrals (6), (7) are finite (after defining suitable A 1 j , A 2 j ) and converge to 0 as y 0. In particular, they are almost surely finite, so Lemma 2.1 then implies that G is Hölder continuous, with Hölder constant bounded in terms of the integrals (6), (7).
We would like to infer that almost surely the functions v(·, ·, y), y > 0, form a Cauchy sequence in the Hölder space C α,η . But this is not immediately clear, therefore we will bound the integrals (6), (7) by expressions that are decreasing in y. We will also define A 1 j , A 2 j here.
• By the monotonicity of M A * , M 21 , M 22 in y we have that almost surely the functions v(·, ·, y) and (t, κ) →f κ t (iy) are Cauchy sequences in the Hölder space C α,η . This will show Theorem 4.1.
Step 2 We now explain that in fact, our definition of A 1 * does not always suffice, and we need to define A 1 j a bit differently in order to get the best estimates. The new definition of A 1 j will satisfy only the relaxed condition (14) [instead of (5)].
The reason is that, when |t −s| ≤ u 2 , |f t (iu)−f s (iu)| is estimated by an expression like |f s (iu)||B t − B s | which is of the order O(|t − s| 1/2 ). The same is true for the difference |f t (iu)−f s (iu)| [see (20) below]. When we carry out the moment estimate for our choice of A 1 * , then we will get

But recall from Proposition 3.3 that
which has allowed us to apply Lemma 2.1 with β 1 ≈ ζ +λ 2 + 1 in the proof of Theorem 3.2. When ζ > 0, this was better than just λ/2.
To fix this, we need to adjust our choice of A 1 j . In particular, we should not evaluate E|f t (iu)−f s (iu)| λ when u |t−s| 1/2 (here " " means "much larger"). As observed in [9], |f s (iu)| does not change much in time when u |t − s| 1/2 . More precisely, we have the following results. Lemma 4.5 Let (g t ) be a chordal Loewner chain driven by U , andf t (z) = g −1 t (z + U (t)). Then, if t, s ≥ 0 and z = x + iy ∈ H such that |t − s| ≤ C y 2 , we have where C < ∞ depends on C < ∞, and l < ∞ is a universal constant.
Proof The first two inequalities (18) and (19) follow from [9, Lemma 3.5 and 3.2]. The third inequality (20) follows from (19) by the Cauchy integral formula in the same way as in Corollary 3.7. Note that for z ∈ H and w on a circle of radius y/2 around z, we have |f s (w)| ≤ 12|f s (z)| by the Koebe distortion theorem.
We now redefine A 1 j . Let for s ≤ t, where the exponents 1/2 (−) < 1/2 denote some numbers that we can pick arbitrarily close to 1/2. (Of course,f t still depends on κ, but for convenience we do not write it for now.) Note that the integrands in A 12 and A 13 just make fancy bounds of according to (20). But now, in A 13 we are not integrating up to y any more. Thus, the condition (5) is not satisfied any more. But the relaxed condition (14) of Lemma 2.13 is still satisfied. Indeed, by (20), where by (18) Finally, with this definition of A 13 , we truly have E|A 13 (t, s; κ)| λ (−) = O(|t − s| (ζ +λ) (−) /2 ) and not just O(|t − s| λ/2 ); here λ (−) < λ is an exponent that can be chosen arbitrarily close to λ. Proposition 4.6 With the above notation and assumptions, if 1 < β 1 < ζ +λ 2 + 1, 1 < β 2 < p + 1, we have Proof These follow from direct computations making use of Lemma 3.1 and Corollary 3.7. They can be found in the appendix of the arXiv version of this paper.
The same analysis of λ and ζ as in the proof of Theorem 3.2 applies here. This finishes the proof of Theorem 4.1.

Proof of Proposition 3.5
The proof is based on the methods of [10,15].
Let t ≥ 0 and U ∈ C([0, t]; R). We study the chordal Loewner chain (g s ) s∈ [0,t] in H driven by U , i.e. the solution of (15). Let V (s) = U (t − s) − U (t), s ∈ [0, t], and consider the solution of the reverse flow The Loewner equation and therefore For r ∈ [0, t], denote by h r ,s the reverse Loewner flow driven by V (s) − V (r ), s ∈ [r , t]. More specifically, which implies from (21) that This implies also The following result is essentially [10, Lemma 2.3], stated in a more refined way.
Proof The proof of [10,Lemma 2.3] shows that The claim follows by estimating

Taking moments
Let κ,κ > 0, and let where B is a standard Brownian motion. In the following, C will always denote a finite deterministic constant that might change from line to line.
In order to make the idea precise, we will reparametrise the integral in order to match the setting in [15] and apply their results.

Reparametrisation
Let κ > 0. In [15], the flow with a = 2 κ is considered. To translate our notation, observe that If we letB s = √ κ B s/κ , then For notational simplicity, we will write just t instead of κt and B, h s , z s instead of B,h s ,z s .
In the next step, we will let the flow start at z 0 = i instead of iδ. Observe that Again, for notational simplicity we will stop writing the˜from now on. Now, let z 0 = i, and (cf. [15]) σ (s) = inf{r | y r = e ar } =ˆs 0 |z σ (r ) | 2 dr which is random and strictly increasing in s. Then This is the integral we will work with.
With (22) and Proposition 5.2 this will complete the proof of Proposition 3.5.
To proceed, we need to know more about the behaviour of the reverse SLE flow, which also incorporates the behaviour of σ . This has been studied in [15]. Their tool was to study the process J s defined by sinh J s = |z r | d B r is a standard Brownian motion and r c is defined in (17). The following results have been originally stated for an equivalent probability measure P * , depending on a parameter r , such that d J s = −q tanh J s ds + dW * s with q > 0 and a process W * that is a Brownian motion under P * . But setting the parameter r = 0, we have P * = P, q = r c , and W * = W . Therefore, under the measure P, the results apply with q = r c .
Note also that although the results were originally stated for a reverse SLE flow starting at z 0 = i, they can be written for flows starting at z 0 = x + i without change of the proof. One just uses [15,Lemma 7.1 (28) Recall that [9,15]  and A n = E exp(n) \E exp(n−1) for n ≥ 1, and A 0 = E 1 . Then (The constant C may change from line to line.) We proceed to estimating E where F is the filtration generated by B.
Note that y σ (s) = e as by the definition of σ . Moreover, on the set A n , the Brownian motion is easy to handle since by Hölder's inequality for any ε > 0. It remains to handle E |h σ (s),t/δ 2 (z σ (s) )| p/2 | F σ (s) .
The following result is well-known and follows from the Schwarz lemma and mapping the unit disc to the half-plane.
We would like to sum this expression in n.
Since ε > 0 can be chosen as small as we want, the condition to apply this is p > 2r c = 1 + 8 κ , and the exponent can be chosen to be greater than 2r c − p − ε for any ε > 0.
With this estimate for (24), the proof of Proposition 3.5 is complete.
Acknowledgements PKF and HT acknowledge funding from European Research Council through Consolidator Grant 683164. All authors would like to thank S. Rohde and A. Shekhar for stimulating discussions. Moreover, we thank the referees for their comments, in particular for pointing out the literature on metric entropy bounds and majorising measures, and for suggesting simplified arguments in the proofs of Lemma 2.1 and Theorem 2.8.
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.