Abstract
We prove that for almost every Brownian motion sample, the corresponding SLE\(_\kappa \) curves parameterized by capacity exist and change continuously in the supremum norm when \(\kappa \) varies in the interval \([0,\kappa _0)\), where \(\kappa _0=8(2-\sqrt{3})\approx 2.1\). We estimate the \(\kappa \)-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with \(\kappa \).
1 Introduction and main result
The Schramm–Loewner evolution with parameter \(\kappa > 0, \text{ SLE }_\kappa \), is a family of random conformally invariant growth processes that arise in a natural manner as scaling limits of certain discrete models from statistical physics. The construction of SLE uses the Loewner equation, a differential equation that provides a correspondence between a real-valued function—the Loewner driving term—and an evolving family of conformal maps called a Loewner chain. If the driving term is sufficiently regular the Loewner chain is generated by (or generates, depending on the point of view) a non self-crossing continuous curve which is obtained by tracking the image of the driving term under the evolution of conformal maps. For \(\kappa \) fixed and positive, \(\text{ SLE }_\kappa \) is defined by taking a standard one-dimensional Brownian motion \(B_t\) and using \(\sqrt{\kappa } B_t\) as driving term for the Loewner equation. Despite the fact that there are examples of driving terms strictly more regular than Brownian motion whose corresponding Loewner chains are not generated by continuous curves, it is known that for each fixed \(\kappa > 0\), the \(\text{ SLE }_\kappa \) Loewner chain almost surely is, see [10] and [9]. The \(\text{ SLE }_\kappa \) curves are random fractals and as \(\kappa \) varies their almost sure properties change. For example, when \(\kappa \) is between \(0\) and \(4\) the \(\text{ SLE }_\kappa \) path is simple, but when \(\kappa >4\) it has double points, and when \(\kappa \geqslant 8\) it is space-filling, see [10]. The Hausdorff dimension of the curve increases with \(\kappa \), see [2], while the Hölder regularity in the standard capacity parameterization derived from the Loewner equation decreases as \(\kappa \) increases to \(8\) and then the regularity increases again, see [4]. In all of these results, the exceptional event can depend on \(\kappa \) which is held fixed.
A natural question that seems to have occurred to several researchers, and is suggested by simulation (see [6]), is whether almost surely the \(\text{ SLE }_\kappa \) curves change continuously with \(\kappa \) if the Brownian motion sample is kept fixed. Note that a priori it is not even clear that there is an event of full measure on which the corresponding \(\text{ SLE }_\kappa \) Loewner chains are simultaneously generated by curves if \(\kappa \) is allowed to vary in an interval. An analogous question for the deterministic Loewner equation has been asked by Angel: If the Loewner chain corresponding to the driving term \(W_t\) is generated by a continuous curve \(\gamma \) and if \(\kappa <1\), is it true that the Loewner chain of \(\kappa W_t\) is generated by a continuous curve, too? This was answered in the negative by Lind et al. by constructing a non-random Hölder-\(1/2\) driving term \(\lambda _t\) with the property that the Loewner chain of \(\kappa \lambda _t\) is generated by a curve if and only if \(\kappa \ne \pm 1\), see Theorem 1.2 of [8]. More precisely, there exists a special \(T > 0\) such that the Loewner chain of \(\lambda _t\) is generated by a curve \(\gamma \) for \(t \in [0,T)\) but as \(t\) tends to \(T\) the curve spirals around a disc in the upper half plane and the limit of \(\gamma (t)\) as \(t \rightarrow T-\) does not exist. The function \(\lambda _t\) of this example is strictly more regular than Brownian motion. Indeed, it is well-known that for every \(\alpha < 1/2\) the Brownian motion sample path is almost surely Hölder-\(\alpha \), but it is almost surely not Hölder-1/2.
In this paper we will prove that \(\text{ SLE }_\kappa \) almost surely does not exhibit the pathological behavior described above, at least not for sufficiently small \(\kappa \). Let us state our main result in a slightly informal manner, see Sect. 4 for a precise statement of the full result; we prove more than is stated here. (In particular we will also estimate explicitly the relevant Hölder exponents.) In order to state the theorem, define \(\kappa _0 = 8(2-\sqrt{3}) \approx 2.1\).
Theorem 1.1
For almost every Brownian motion sample \(B_t\), the \(\text{ SLE }_\kappa \) Loewner chains driven by \((\sqrt{\kappa } B_t, \, t \in [0,1])\), where \(\kappa \in [0, \kappa _0)\), are simultaneously generated by curves that if parameterized by capacity change continuously with \(\kappa \) in the supremum norm.
See Sect. 3.1 for a sketch of the proof of Theorem 1.1 and Theorem 4.1 for a precise statement.
Let us make a few remarks. The restriction to \(t \in [0,1]\) is only a convenience and a similar result for \(t \in [0,\infty )\) holds if we consider instead continuity with respect to the topology of uniform convergence on compact subintervals. We emphasize that we prove that the curves change continuously with \(\kappa \) when the curves have a particular parameterization. This is a stronger topology than the one generated by the now standard metric used by Aizenman and Burchard in [1] which allows for increasing reparameterization of the curves. It can be checked that Theorem 1.1 also holds when \(\kappa > \kappa _\infty := 8(2+\sqrt{3})\) is allowed to vary. This may seem counterintuitive but can be viewed as a consequence of the fact that the regularity of the \(\text{ SLE }_\kappa \) curve in the capacity parameterization increases with \(\kappa \) when \(\kappa \geqslant 8\). (Intuitively, by duality, the boundary of the \(\text{ SLE }_\kappa \) hull becomes more and more like the real line when \(\kappa \) becomes large and so the Hölder regularity of the curve approaches the minimum of \(1/2\) and that of the driving term, which is the time-zero regularity of any chordal Loewner curve in the capacity parameterization.)
We end this introduction with a question. From the point of view of probability theory what we do is to consider a specific coupling of \(\text{ SLE }_\kappa \) processes for different \(\kappa \) and prove almost sure existence of the curves and continuity as \(\kappa \) varies. As was realized by Schramm and Sheffield [11] it is possible to obtain \(\text{ SLE }_\kappa \) curves by a mechanism quite different from the usual one using the Loewner equation. Very roughly speaking, the construction considers certain “flow-lines” derived from the Gaussian free field (GFF) and by varying a parameter one gets \(\text{ SLE }_\kappa \) for different \(\kappa \), see [12] and the references therein. It seems natural to ask whether a similar continuity result as the one proved in this paper holds for GFF derived couplings of \(\text{ SLE }_\kappa \) for different values of \(\kappa \).
1.1 Overview of the paper
The organization of our paper is as follows. In Sect. 2 we discuss the deterministic (reverse-time) Loewner equation and derive Lemma 2.3 which estimates the perturbation of a Loewner chain in terms of a small supremum norm perturbation of its driving term. In Sect. 3 we start by giving the general set up of the proof of the main result along with a sketch its proof. We then give the necessary probabilistic estimates based on previously known moment bounds for the spatial derivative of the SLE map. (Some details of a modification of one of these derivative estimates are deferred to Appendix A.) The complete statement of our main result is given in Theorem 4.1 of Sect. 4, where the work of Sects. 2 and 3 is then combined to prove Theorems 1.1 and 4.1. We also prove Theorem 4.2, a quantitative version of Theorem 1.1.
2 Deterministic Loewner equation
Let \(W_t\) be a real-valued continuous function defined for \(t \in [0, \infty )\) and consider
This is the (chordal) Loewner partial differential equation and the function \(W_t\) is called the Loewner driving term. (As we will only work with the chordal version of the Loewner equation in this paper, we will usually omit the word “chordal”.) We will occasionally refer to (2.1) as the Loewner PDE. A solution \((f_t(z), \, t\geqslant 0, \, z \in \mathbb{H })\) exists whenever \(W_t\) is measurable and for each \(t \geqslant 0, f_t: \mathbb{H } \rightarrow H_t\) is a conformal map from the upper half-plane onto a simply connected domain \(H_t=\mathbb H {\setminus } K_t\), where \(K_t\) is a compact set. We call the family \((f_t)\) of conformal maps a Loewner chain and \((f_t, W_t)\) a Loewner pair. The family of image domains \((H_t)\) is continuously decreasing in the Carathéodory sense. We say that the Loewner chain \((f_t)\) is generated by a curve if there is a curve \(\gamma (t)\) (that is, a continuous function of \(t\) taking values in \(\overline{\mathbb{H }}\)) with the property that for every \(t \geqslant 0, H_t\) is the unbounded connected component of \(\mathbb H {\setminus } \gamma [0,t]\). Theorem 4.1 of [10] gives a convenient sufficient condition for \((f_t)\) to be generated by a curve:
Theorem 2.1
([10]) Let \(T>0\) and let \(W_t: [0, T] \rightarrow \mathbb R \) be continuous and \((f_t, W_t)\) the corresponding Loewner pair. Suppose that
exists for \(t \in [0,T]\) and is continuous. Then \((f_t, 0 \leqslant t \leqslant T)\) is generated by the curve \(\beta \).
It is sometimes convenient to write
There is another version of the Loewner equation that we shall use, namely the reverse-time Loewner ODE
If \((f_t)\) is the solution to the Loewner PDE (2.1) with driving term \(W_t\) and \((h_t)\) the solution to (2.3) with driving term \(W_{T-t}\), then it is not difficult to see that the conformal maps \(f_{T}(z)\) and \(h_{T}(z)\) are the same. Note that this identity holds only at the special time \(T\). In particular, the families \((h_{t})\) and \((f_{t})\) are in general not the same. It is often easier to work with (2.3) rather than directly with (2.1).
The standard Koebe distortion theorem for conformal maps gives a certain uniform control of the change of a conformal map evaluated at different points at distance comparable to their distance to the boundary. We will need similar estimates to control the change of a Loewner chain evaluated at different times and driven by “nearby” driving terms. The magnitude of the allowed perturbation depends on the distance to the boundary of \(\mathbb H \) and on the behavior of the spatial derivative of the conformal map. We first state the well-known estimates for the \(t\)-direction, see, e.g., [4] for proofs.
Lemma 2.2
There exists a constant \(0 < c < \infty \) such that the following holds. Suppose that \(f_t\) satisfies the chordal Loewner PDE (2.1) and that \(z=x+iy \in \mathbb H \). Then for \(0 \leqslant s \leqslant y^2\),
and
The next lemma considers a supremum norm perturbation of the driving term. The most important estimate is (2.6) which has appeared in a radial setting in [3]. It would be sufficient to prove our main result. The refinement (2.7) will be used to obtain better quantitative estimates on Hölder exponents using information about the derivative. We stress that we derive (2.6) with no assumptions on the driving terms other than the existence of a bound on their supremum norm distance.
Lemma 2.3
Let \(0 < T < \infty \). Suppose that for \(t \in [0 , T], f_t^{(1)}\) and \(f_t^{(2)}\) satisfy the chordal Loewner PDE (2.1) with \(W_t^{(1)}\) and \(W_t^{(2)}\), respectively, as driving terms. Suppose that
Then if \(z=x+iy \in \mathbb H \),
Moreover, for every \(t \in [0,T]\),
where \(I_{t,y}=\sqrt{4t+y^2}\).
Remark
Since the conformal maps are normalized at infinity there exists a constant \(c< \infty \) depending only on \(T\) such that for \(j=1,2, |(f^{(j)}_t)^{\prime }(z)| \leqslant c(y^{-1}+1)\) for all \(z=x+iy \in \mathbb H \) and all \(t \in [0,T]\). (This is a well-known property of conformal maps but can also be seen from the proof to follow.) Thus if \(y \leqslant 1\), say, and \(c_1<\infty \) and \(\beta _1 \geqslant -1\) are such that \(|(f_t^{(1)})^{\prime }(z)| \leqslant c_1 y^{-\beta _1}\), then (2.7) can be written
where \(c^{\prime } < \infty \) depends only on \(T, c_1, \beta _1\).
Proof of Lemma 2.3
We will start by proving (2.7). Let \(t \in (0,T]\) be fixed. For 0 \(\leqslant s \leqslant t\), write
Let \(z=x+iy\) be fixed and set
where \(h^{(j)}\) are assumed to solve (2.3) with \(\tilde{W}^{(j)}, \, j=1,2,\) respectively, as driving terms.
Define
and note that
Our goal will be to estimate \(|H(t)|\). We differentiate \(H(s)\) with respect to \(s\) and use (2.3) to obtain a linear differential equation
where
This differential equation can be integrated and with \(u(r) = {\exp }\{-\int _0^r \psi (s) \, ds\}\) we find
Hence, upon setting \(H(0)=0\),
Consequently,
and we see that we need to estimate the last factor in (2.10). We will first prove the bound corresponding to (2.7). Set
Note that (2.3) implies that for \(0 \leqslant s \leqslant t\),
and similarly for \(h^{(2)}\). In particular,
and similarly for \(f_t^{(2)}\). By the Cauchy–Schwarz inequality we have that
Here we used that \(y_s\) and \(v_s\) are always non-negative. We can write
It follows from the Loewner equation that \(y_t\) and \(v_t\) are both bounded above by \((4t + y^2)^{1/2}\), and we conclude using (2.11) that
We get (2.7) by combining the last estimate with (2.10) and noting that the Cauchy-Schwarz inequality implies that
It remains to prove (2.6). For this, note that
We can then estimate as in (2.12). Combined with (2.10), this proves (2.6) and concludes the proof.\(\square \)
3 Schramm–Loewner evolution and probabilistic estimates
Let \(B_t\) be standard Brownian motion. The Schramm–Loewner evolution \(\text{ SLE }_\kappa \) for \(\kappa \geqslant 0\) fixed is defined by taking \(W_t=\sqrt{\kappa }B_t\) as driving term in (2.1). We recall that for each \(\kappa \geqslant 0\), the \(\text{ SLE }_\kappa \) Loewner chain \((f_t^{(\kappa )})\) is almost surely generated by a curve, the (chordal) \(\text{ SLE }_\kappa \) path, \(\gamma ^{(\kappa )}\), see [10] and [9]. We also recall that the tip of the curve at time \(t\) is defined by taking the radial limit
where \(\hat{f}_t^{(\kappa )}(iy) := f_t^{(\kappa )}(\sqrt{\kappa }B_t+iy)\).
3.1 Set-up and strategy
The idea for the proof of Theorem 1.1 is simple and so before giving the details we shall first explain the main steps of the proof and give a few definitions. We write
and restrict attention to \((t,y, \kappa ) \in [0,1] \times [0,1] \times [0, \kappa _0)\). Our main goal is to show that for almost every Brownian motion sample \(B_t=B_t(\omega )\), the function \((t, \kappa ) \mapsto \gamma ^{(\kappa )}(t)\) defined by taking the radial limit \(\lim _{y \rightarrow 0+} F(t,y,\kappa )\) is well-defined and continuous for \((t,\kappa ) \in [0,1] \times [0, \kappa _0)\). (Recall the sufficient condition of Theorem 2.1.) This will clearly imply Theorem 1.1. Our strategy is similar to that of the proof of Theorem 5.1 of [10]. We partition the \((t,y,\kappa )\)-space in three-dimensional Whitney-type boxes \(S_{n; \, j, k}\) whose volumes decrease with the \(y\)-coordinate: Let
where \((n, j, k) \in \mathbb{N }^3\). (See Fig. 1 for a sketch.) The parameter \(q\) should for now be thought of as being (slightly larger than) \(1\). Let
be the corners of the boxes. The idea is to apply a one-point moment estimate and the Chebyshev inequality to control the magnitude of \(|F^{\prime }|\) at the corners \(p_{n; \, j, k}\) so that for suitable \(\beta < 1\) and \(j, k, n\),
where \(\rho \) is the decay rate in the moment estimate we use. (The decay rates of the probabilities in (3.3) depend on \(\kappa \) and \(\beta \) and we need to have \(\kappa < \kappa _0\) for the series in (3.3) to converge with \(\beta < 1\). In particular we need to be able to choose \(1< q <\rho -2\).) The Borel-Cantelli lemma then implies that there almost surely exists a random constant \(c<\infty \) such that \(\left| F^{\prime }(p_{n; \, j, k}) \right| \leqslant c 2^{n\beta }\) for all triples \((n, j, k)\) in the sum. With this derivative estimate we can then use the distortion-type bounds of Sect. 2 to show that the diameters of the box images decay like a power of the \(y\)-coordinate, that is,
where \(\delta > 0\) can be thought of as the smaller of \(q-1\) and \(1-\beta \). Once we have this it is easy to show that \(\lim _{y \rightarrow 0+} F(t,y,\kappa )\) exists. To prove continuity we estimate
by using (3.4) to sum the diameters of the box images along a “hyperbolic geodesic” in \((t,y,\kappa )\)-space connecting \((t_1,0,\kappa _1)\) with \((t_2,0,\kappa _2)\). The resulting Hölder exponents depend on the particular choices of parameters (\(\beta \) and \(q\)) and can be taken larger if we restrict attention to smaller \(\kappa \). To achieve the best exponents we will prove a local version of (3.4), which can then be used to patch together a global estimate with varying exponents.
We now turn to the details.
3.2 Probabilistic estimates
Before stating the basic moment estimate that we will use we need to introduce a few parameters. For \((\kappa , \beta ) \in (0, \infty ) \times (-1,1)\), let
It will also be useful to define
These notations (with the exception of \(\sigma \)) with corresponding moment estimates to follow have appeared in, e.g., [4] and earlier works by Lawler. We find these estimates more convenient to use and they give better Hölder exponents than those from [10], although for technical reasons we shall use a bound from the latter reference when we consider \(\kappa \) very close to and including \(0\). We remark that Lind [7] improved the estimates from [10] in a slightly different setup to essentially agree with the bounds we will use.
Remark
Roughly speaking, the functions in (3.5) are related in the following way: If \((\kappa , \beta )\) is fixed then as \(y \rightarrow 0\) the integral \(\mathbb E [|\hat{f}^{\prime }_t(iy)|^\lambda ]\) is supported on the event that \(|\hat{f}^{\prime }_t(iy)| \approx y^{-\beta }\) and this event has probability approximately equal to \(y^{\lambda \beta + \zeta }=y^\rho \), see [5].
Theorem 3.1
([4]) Suppose \((\kappa ,\beta ) \in (0,\infty ) \times (-1,1)\). There is \(c = c(\kappa ,\beta ) <\infty \) such that
for all \(n \in \mathbb{N }\), and \(j\) = 1, 2, ..., \(2^{2n}\), where \(\lambda \) and \(\zeta \) are defined by (3.5).
Using Theorem 3.1, the Chebyshev inequality implies that if \((\kappa , \beta ) \in (0, \infty ) \times (-1,1)\) then for all \(n \in \mathbb N , j=1,2,\ldots , 2^{2n}\),
where \(c = c(\kappa , \beta ) < \infty \). From (3.7), choosing parameters appropriately, we now get the almost sure control over the derivative at the corners \(p_{n; \, j, k}\) of the boxes \(S_{n; \, j, k}\) by summing and applying the Borel–Cantelli lemma. Notice that there are \(O(2^{n(q+2)})\) boxes at \(y\)-height \(2^{-n}\), where \(q\) determines the mesh of the partition in the \(\kappa \)-direction. The “optimal” choice of \(q\) depends on which interval of \(\kappa \) we consider. It turns out that we need to have \(q < \sigma \) for the Borel-Cantelli sums to converge; recall (3.3) or see (3.12) below. On the other hand, the decay rate claimed in (3.4) becomes
where
is the exponent from the distortion-type estimate (2.8). Thus we are led to consider \(\beta >\hat{\beta }_\kappa \) where \(\hat{\beta }_\kappa \) is a solution in \((0,1)\) to
where \(\sigma \) was defined in (3.6). If \(\beta \geqslant 0\), then if \(\kappa > 1\) we have \(\sigma = \rho -2\), while if \(\kappa \leqslant 1\), then \(\sigma = \lambda \beta \). We have not found a simple expression for \(\hat{\beta }_\kappa \) but we note the following properties which can be checked from (3.9) and (3.5).
Lemma 3.2
A solution in \((0,1)\) to the equation (3.9) exists if and only if \(\kappa \in [0,\kappa _0) \cup (\kappa _\infty , \infty )\), where \(\kappa _0=8(2-\sqrt{3})\) and \(\kappa _\infty =8(2+\sqrt{3})\). For each such \(\kappa \), call the solution \(\hat{\beta }_\kappa \). Then \(\hat{\beta }_\kappa \) increases continuously from \(0\) to \(1\) as \(\kappa \) increases from \(0\) to \(\kappa _0\) and decreases from \(1\) to \(0\) as \(\kappa \) increases from \(\kappa _\infty \) to \(\infty \). Moreover, if \(\beta > \hat{\beta }_\kappa \) then \(\sigma (\kappa , \beta )>\varphi (\beta )\).
Proof
We omit the details but note that the special values \(\kappa _0, \kappa _\infty \) can be found by solving \(\sigma = 1\).\(\square \)
Lemma 3.3
Let \(\kappa \in [0,\kappa _0)\). If \(\beta > \hat{\beta }_\kappa \) and \(\varphi (\beta ) < q < \sigma (\kappa , \beta )\), then there almost surely exists a (random) constant \(c = c(\kappa ,\beta ,q, \omega ) < \infty \) such that
for all \((n, j, k) \in \mathbb{N }^3\) with \(p_{n; \, j, k}\in [0,1] \times [0,1] \times [0,\kappa ]\).
Proof
The result is clearly true if \(\kappa =0\), so let \(\kappa \in (0, \kappa _0)\) be fixed and choose \(\beta > \hat{\beta }_\kappa \) and \(\varphi (\beta ) < q < \sigma (\kappa , \beta )\). For \(n\) = 1, 2,..., let
be the event that \(\left| F^{\prime }(p_{n; \, j, k}) \right| \geqslant 2^{n \beta }\) for some \((j,k) \in \mathbb{N }^2\) with \(j 2^{-2n} \in [0,1]\) and \(k2^{-nq} \in [0,\kappa ]\). We have that
where the sum is over the above ranges of \(j\) and \(k\). Let \(0<\varepsilon < \kappa \) be fixed for the moment and assume that \(k2^{-nq} \in [\varepsilon , \kappa ]\). Then by the Chebyshev inequality and Theorem 3.1, for all \(n=1,2,\ldots \)
where \(c = c(\varepsilon , \kappa ,\beta ) < \infty , \zeta =\zeta (\kappa , \beta )\), and \(\lambda =\lambda (\kappa , \beta )\). (When performing the summation over \(j\) in (3.10) we have tacitly, if needed, estimated using a slightly smaller \(\beta \) to ensure that \(|\zeta /2 - 1|\) is bounded from below.)
We now consider the case when \(k2^{-nq} \in [0,\varepsilon ]\). We cannot directly quote Theorem 3.1 since a priori the multiplicative constant in that bound may blow up as \(k2^{-nq} \rightarrow 0\). Moreover, the setup used for the proof of Theorem 3.1 in [4] is such that it would require some work to verify that the constant can be taken to depend only on the largest \(\kappa < \kappa _0\) that we consider. Instead, for simplicity and as this is all we need, we will derive the needed estimate for small \(\kappa \) from Corollary 3.5 of [10], the proof of which can easily be seen to yield the required uniform constants. We have included a sketch of the appropriate modifications in Appendix A and by Lemma 5.1 we can find \(\varepsilon =\varepsilon (\beta , \sigma )>0\) (recall that \(\sigma =\sigma (\kappa , \beta )\)) such that if \(k2^{-nq} \in [0, \varepsilon ]\), then
where \(c_{\varepsilon } < \infty \) depends only on \(\varepsilon \).
Summing (3.10) and (3.11) over \(k\) shows that
where \(c=c(\kappa , \beta ) < \infty \). The last expression is summable over \(n\) and so the proof is complete by the Borel–Cantelli lemma.\(\square \)
We will now apply the uniform derivative estimate of the last lemma to show that the diameters of the \(F\)-images of the boxes decay like a power of their (minimal, say) \(y\)-coordinate. Since \({\sup }_{t \in [0,1]} |\sqrt{\kappa + \Delta \kappa }B_t-\sqrt{\kappa }B_t|\) is of order (a random constant times) \(\Delta \kappa \) for \(\kappa >0\) but only of order \(\sqrt{\Delta \kappa }\) at \(\kappa =0\) we must consider these two cases separately.
Lemma 3.4
Let \(\kappa \in [0,\kappa _0)\). There exist \(q > 0\) and \(\delta > 0\) and for every \(\varepsilon > 0\) almost surely a (random) constant \(c=c(\kappa , q, \varepsilon , \omega ) < \infty \) such that
for all \((n,j,k) \in \mathbb{N }^3\) with \(p_{n; \, j, k}\in [0,1] \times [0,1] \times [\varepsilon ,\kappa ]\).
Moreover, for each \(\varepsilon >0\) sufficiently small there exist \(q^{\prime } > 0\) and \(\delta ^{\prime } > 0\) and almost surely a constant \(c=c(q^{\prime }, \varepsilon , \omega ) < \infty \) such that (3.13) holds with \(\delta \) replaced by \(\delta ^{\prime }\) for all \((n,j,k) \in \mathbb{N }^3\) with \(p_{n; \, j, k}=p_{n; \, j, k}(q^{\prime }) \in [0,1] \times [0,1] \times [0,\varepsilon ]\).
Proof
We will start with the first assertion. Let \(\varepsilon >0\) be given and assume that \(\varepsilon < \kappa \), since there is nothing to prove otherwise. Lemma 3.3 shows that if \(\beta > \hat{\beta }_\kappa \) and \(\varphi (\beta ) < q < \sigma (\kappa , \beta )\) then there almost surely exists a (random) constant \(c = c(\kappa ,\beta ,q, \omega ) < \infty \) such that
for all the box corners \(p_{n; \, j, k}\in [0,1] \times [0,1] \times [\varepsilon ,\kappa ]=:\mathcal S _\varepsilon \). (Note that (3.14) holds also for \(\varepsilon =0\).) Consider a fixed but arbitrary dyadic box \(S_{n; \, j, k}\subset \mathcal S _\varepsilon \). Let \(p \in S_{n; \, j, k}\). We will show that there exists \(\delta >0\) such that
Write
Since \(\left| \Delta t \right| \leqslant y^2\), Lemma 2.2 and Lemma 3.3 imply that
where \(c^{\prime } = c^{\prime }(\kappa ,\beta ,q, \omega ) < \infty \) almost surely. On the other hand,
by the Koebe distortion theorem and Lemma 3.3 combined with (2.4). Next, if \(\kappa ^{\prime }, \kappa ^{\prime } + \Delta \kappa \in [\varepsilon , \kappa ]\) then
where \(c^{\prime } = c^{\prime }(\varepsilon , \omega ) < \infty \) almost surely. The estimate (2.8) combined with Lemma 3.3, Koebe’s distortion theorem, and (2.4), show that
Consequently, by (3.16), (3.17), and (3.19) we get (3.15) with
which is clearly strictly positive. It remains to verify the case when \(p_{n; \, j, k}\in [0,1] \times [0,1] \times [0,\varepsilon ]\). For this, note that that all the estimates above except (3.18) and (3.19) hold in this case, too, with the assumption that \(\beta ^{\prime } > \hat{\beta }_\varepsilon \) and \(\varphi (\beta ^{\prime }) < q^{\prime } < \sigma (\varepsilon ,\beta ^{\prime })\). We replace (3.18) by
where \(c=c(\omega ) < \infty \) almost surely. This gives
Note that for \(\beta ^{\prime }>0\) fixed \(\sigma (\varepsilon , \beta ^{\prime })=O(1/\varepsilon )\) as \(\varepsilon \rightarrow 0\). Consequently, by taking \(\varepsilon >0\) sufficiently small (using also that \(\hat{\beta }_\varepsilon \) is increasing in \(\varepsilon \)) we have that \(\beta ^{\prime }>\hat{\beta }_\varepsilon \) and we can find \(q^{\prime }\) with
strictly positive. This concludes the proof.\(\square \)
We have seen in the (proofs of the) last two results that the anomalous behavior at \(t=0\) and \(\kappa =0\) can decrease the decay rate \(\delta \) of the box images in (3.13). In the next lemma we record a quantitative statement which restricts attention to \(t,\kappa \geqslant \varepsilon >0\) and therefore gives better exponents. In this case the sum (3.10) satisfies the stronger estimate (as opposed to when \(\varepsilon = 0\)) of being bounded above by an \(\varepsilon \)-dependent constant times \(2^{-n(\rho -2)} \leqslant 2^{-n\sigma }\). This implies that we can replace the requirement that \(\beta > \hat{\beta }_\kappa \) by \(\beta > \beta _\kappa \), where \(\beta _\kappa \) is the larger solution to
(Since \(\kappa \geqslant \varepsilon \) we do not need to use the estimate from Appendix A.) We note that this allows for a larger range of \(\beta \) and that \(\beta _\kappa \) may be negative.
Lemma 3.5
Let \(\varepsilon >0\) and let \(\kappa \in [\varepsilon ,\kappa _0)\). If \(\beta > \beta _{\kappa }\) and \(\varphi (\beta ) < q < \rho (\kappa , \beta )-2\) then there almost surely exists a (random) constant \(c=c(\varepsilon , \kappa , \beta , q, \omega )<\infty \) such that such that for all \((n,j,k) \in \mathbb{N }^3\) with \(p_{n; \, j, k}\in [\varepsilon ,1] \times [0,1] \times [\varepsilon ,\kappa ]\),
where
4 Hölder regularity and proof of Theorem 1.1
Let us now give a precise statement of Theorem 1.1. Let \((\Omega , \mathbb P )\) be a probability space supporting a standard linear Brownian motion \(B\). For \(\omega \in \Omega \) we write \((B_t(\omega ), \, t \geqslant 0)\) for the sample path of \(B\). Let \(\mathcal K \) be the space of continuous curves defined on \([0,1]\) taking values in the closed upper half plane \(\overline{\mathbb{H }}=\{z: \mathrm{Im } \,z \geqslant 0 \}\). We endow \(\mathcal K \) with the supremum norm so that it becomes a metric space.
Theorem 4.1
There exists an event \(\Omega _* \subset \Omega \) of probability \(1\) for which the following holds for every \(\omega \in \Omega _*\). The chordal \(\text{ SLE }_\kappa \) path \((\gamma ^{(\kappa )}(t, \omega ), \, t \in [0, 1])\) driven by \((\sqrt{\kappa }B_t(\omega ), \, t \in [0,1])\) and parameterized by capacity exists as an element of \(\mathcal K \) for every \(\kappa \in [0, \kappa _0)\), where \(\kappa _0=8(2-\sqrt{3})\). Moreover, \(\kappa \mapsto \gamma ^{(\kappa )}(\cdot , \omega )\) is continuous as a function from \([0, \kappa _0)\) to \(\mathcal{K }\).
Proof
Let \(\kappa \in [0,\kappa _0)\). We first show that almost surely, for all \((t,\kappa ^{\prime })\in [0,1]\times [0,\kappa ], \gamma ^{(\kappa ^{\prime })}(t) = \lim _{y \rightarrow 0+} F(t,y,\kappa ^{\prime })\) exists; \(F\) was defined in (3.1). Suppose that \(0 < y_1, y_2 < 2^{-N}\). The triangle inequality and Lemma 3.4 imply that there is an event \(\Omega _\kappa \) of probability \(1\) on which there exist a constant \(c<\infty \) and \(\delta >0\) such that for all \((t, \kappa ^{\prime }) \in [0,1]\times [0,\kappa ]\),
where for each \(n \in \mathbb{N }, Q_n=Q_n(t,\kappa ^{\prime }) \subset \{S_{n; \, j, k}\}\) is a Whitney-type box such that \((t, 2^{-n}, \kappa ^{\prime }) \in Q_n\). (Note that we when we apply Lemma 3.4 we consider separately the two cases when \(\kappa ^{\prime }\) is very small and when it is bounded away from \(0\) and the Whitney-type partition depends on which of the two cases we apply.) As \(N \rightarrow \infty \) the right-hand side of the last display converges to \(0\) and it follows that \(\gamma ^{(\kappa )}(t)=\lim _{y \rightarrow 0+} F(t,y, \kappa ^{\prime })\) exists for all \((t, \kappa ^{\prime }) \in [0,1] \times [0,\kappa ]\) on the event \(\Omega _\kappa \). Next, we wish to prove that on the event \(\Omega _\kappa , (t, \kappa ) \mapsto \gamma ^{(\kappa )}(t)\) is continuous for \((t,\kappa ) \in [0,1] \times [0, \kappa ]\). For this, let \(t_1, t_2 \in [0,1]\) be given. Define the “stopping time” \(N \in \mathbb{N }\) by
(We can assume that \(t_1 \ne t_2\).) Note that by the construction of the Whitney-type partition \(N=O(-{\log } |t_1-t_2|^{1/2})\). Using Lemma 3.4 we get, for sufficiently small \(y > 0\),
We have again tacitly, if needed, considered the two separate cases of Lemma 3.4 and we understand \(\delta \) as the smaller of the two exponents obtained. Notice also that the two points \((t_1, 2^{-N}, \kappa ^{\prime })\) and \((t_2, 2^{-N}, \kappa ^{\prime })\) may not be in the same level-\(N\) Whitney-type box, so we cannot, strictly speaking, apply Lemma 3.4 to estimate \(\left| F( t_1, 2^{-N}, \kappa ^{\prime }) - F( t_2, 2^{-N} , \kappa ^{\prime }) \right| \). However, it is clear that the points are contained in a translate of such a box or we can estimate directly as in (3.16). A similar remark applies when we estimate \(\left| F( t, 2^{-N}, \kappa _1) - F( t, 2^{-N} , \kappa _2) \right| \) below.
Now, if \(\kappa _1, \kappa _2 \in [0,\kappa ]\), we use the stopping time \(N=O(-{\log } |\kappa _1-\kappa _2|^{1/q})\) given by
instead of (4.1). We get
Letting \(y \rightarrow 0+\) we get that
holds on the event \(\Omega _\kappa \). We may take \(\Omega _*=\cap _n\Omega _{\kappa _0-1/n}\) and the proof is complete.\(\square \)
4.1 Quantitative estimates
We can see from the proof of Theorem 4.1 that we get Hölder continuity in both \(t\) and \(\kappa \) on any compact subinterval of \([0,\kappa _0)\). We will now state separately a quantitative version of the main result. For simplicity we shall only consider \(t, \kappa \geqslant \varepsilon > 0\), but all cases can be treated with similar arguments.
Recall the definition of \(\beta _\kappa \) at the end of Sect. 3.2. We define the following local Hölder exponents. Let
and then let
where the suprema are taken over \(\beta >\beta _\kappa \) and \(\varphi (\beta ) < q < \rho (\kappa , \beta )-2\). To find the value of \(\alpha _{\kappa }\), we take \(q = \rho (\kappa , \beta ) - 2\) and let \(\beta \) be the larger solution (for \(\beta \)) to
We have not found an explicit solution to this equation. However, if we replace \(\varphi (\beta )\) by the majorant \(\beta /4+3/4\) in (4.2), then we get a second order polynomial equation in \(\beta \) that we can solve to obtain the larger solution \(\beta _\kappa ^{\prime }\) below which gives an upper bound for the larger solution to (4.2) and so a lower bound for \(\alpha _\kappa \). We have that
which implies the estimate
As \(\kappa \) increases from \(0\) to \(\kappa _0 = 8(2-\sqrt{3})\), this lower bound decreases from \(1\) to \(0\).
In order to estimate \(\eta _\kappa \), we fix \(\kappa \in [0,\kappa _0)\) and note that for each fixed \(\beta \in (\beta _{\kappa },1)\), the function
is increasing for \(q \in (\varphi (\beta ), q_*]\) and is decreasing for \(q \in [q_*,\infty )\), where
Recall that we only may take \(q \in (\varphi (\beta ), \rho - 2)\) so the maximum occurs either at \(q = q_*\) or \(q = \rho - 2\). In fact,
We now plug in \(\beta =\beta ^{\prime }_\kappa \) from (4.3) and note that by the definition of \(\beta ^{\prime }_\kappa \) it holds that
Thus with this choice of \(\beta \) we see from (4.4) that
Again, as \(\kappa \) increases from \(0\) to \(\kappa _0 = 8(2-\sqrt{3})\), the lower bound decreases from \(1\) to \(0\).
Theorem 4.2
Let \(\varepsilon >0\) and \(\kappa < \kappa _0\) be given and let \(\alpha < \alpha _\kappa \) and \(\eta < \eta _\kappa \). There almost surely exists a (random) constant \(c=c(\varepsilon , \kappa , \alpha , \eta , \omega )<\infty \) such that for all \((t_j, \kappa _j) \in [\varepsilon ,1]\times [\varepsilon ,\kappa ], \, j=1,2,\)
Proof
This is proved in exactly the same way as Theorem 4.1 but using Lemma 3.5 instead of Lemma 3.4.\(\square \)
By an approximation argument we obtain the following corollary.
Corollary 4.3
Let \(\varepsilon , \varepsilon ^{\prime } > 0\) be given. There almost surely exists a (random) constant \(c=c(\varepsilon , \varepsilon ^{\prime }, \omega ) < \infty \) such that for all \(t_1,t_2 \in [\varepsilon ,1]\) and \(\kappa , \kappa _1 \in [\varepsilon ,\kappa _0)\) with \(\kappa _1 \leqslant \kappa \),
References
Aizenman, M., Burchard, A.: Holder regularity and dimension bounds for random curves. Duke Math. J. 99(3), 419–453 (1999)
Beffara, V.: The dimension of the SLE curves. Ann. Prob. 36(4), 1421–1452 (2008)
Johansson Viklund, F.: Convergence rates for loop-erased random walk and other Loewner curves (2012)
Johansson Viklund, F., Lawler, G.F.: Optimal Hölder exponent for the SLE path. Duke Math. J. 159(3), 351–383 (2011)
Johansson Viklund, F., Lawler, G.F.: Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209, 265–322 (2012)
Kennedy, T.: Simulations available at http://math.arizona.edu/tgk/. Accessed 1 June 2012
Lind, J.: Hölder regularity of the SLE trace. Trans. AMS 360, 3557–3578 (2008)
Lind, J., Marshall, D.E., Rohde, S.: Collisions and spirals of Loewner traces. Duke Math. J. 154(3), 527–573 (2010)
Lawler, G.F., Schramm, O., Wendelin, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Prob. 32(1B), 939–995 (2004)
Rohde, S., Schramm, O.: Basic Properties of SLE. Ann. Math. 161, 883–924 (2005)
Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)
Miller, J., Sheffield, S.: Imaginary geometry I: Interacting SLEs (2012)
Acknowledgments
Fredrik Johansson Viklund acknowledges support from the Simons Foundation, Institut Mittag-Leffler, and the AXA Research Fund, and the hospitality of the Mathematics Department of University of Washington, Seattle. The research of Steffen Rohde and Carto Wong was partially supported by NSF Grants DMS-0800968 and DMS-1068105. Parts of the paper were written while the authors were visiting MSRI and we would like to thank MSRI for the support and hospitality. We would finally like to thank the referee for his/her careful reading and useful suggestions.
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Appendix A: Derivative estimate for small \(\kappa \)
Appendix A: Derivative estimate for small \(\kappa \)
In this appendix we sketch how to use Theorem 3.2 of [10] to obtain a derivative estimate with uniform constant for small \(\kappa \) that was used in Sect. 3. (The notation \(F^{\prime }(p_{n; j,k})\) is explained in Sect. 3.)
Lemma 5.1
Suppose that \(\beta > 0\) and \(\sigma >0\) are given. There exists \(\varepsilon >0\) depending only on \(\beta \) and \(\sigma \) such that for \(n=1,2,\ldots \),
where \(c_\varepsilon =e^{4/\varepsilon }\).
Proof
Let \(\varepsilon > 0\) be small and for the moment fixed. We shall apply [10, Theorem 3.5] to the \(\text{ SLE }_\eta \) reverse-flow, \(\eta \leqslant \varepsilon \), with the parameters \(\nu =1, b=2/\varepsilon \leqslant 4/\eta \) and \(a:= 2b + \eta b (1-b)/2\). Then in the notation of [10] (which differs from the one we have been using in the rest of the paper) we have \(a \geqslant 0\) and \(a-\lambda :=\eta b^2/2 - 2b<0\) (we define \(\lambda \) by this relation). Let \(h_t=h_t^{\eta }\) be the \(\text{ SLE }_\eta \) reverse flow and let \(u \mapsto t_z(u)\) be the random time-change defined by \(\mathrm{Im } \,{h_{t_z(u)}(z)} = e^u\). It is a well-known property of the reverse-flow that \(\left| \hat{f}^{\prime }_t(iy)\right| \) and \(\left| h^{\prime }_t(iy)\right| \) are equal in distribution, where \(\hat{f}_t\) is the solution to the Loewner PDE (2.1) driven by \(\sqrt{\eta } B_t\). Using the Loewner equation, one can check that \(|\partial _u {\log } |h^{\prime }_{t_z(u)}(z)|| \leqslant 1\) and therefore
Let \(y > 0\) and write \(t(u) = t_{iy}(u)\). Since \(u \geqslant {\log } y |h^{\prime }_{t_{iy}(u)}(iy)|\) we get
This estimate applies to the \(\text{ SLE }_\eta \) reverse-flow for all \(\eta \in (0, \varepsilon ]\). Clearly \(b(1-2b)<0\) if \(\varepsilon >0\) is sufficiently small and consequently,
and this bound is uniform in \(\eta \leqslant \varepsilon \). By choosing \(\varepsilon \) sufficiently small so that \(4/\varepsilon > (2+\sigma )/\beta -1\), this implies
as claimed.\(\square \)
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Johansson Viklund, F., Rohde, S. & Wong, C. On the continuity of \(\text{ SLE }_{\kappa }\) in \(\kappa \) . Probab. Theory Relat. Fields 159, 413–433 (2014). https://doi.org/10.1007/s00440-013-0506-z
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DOI: https://doi.org/10.1007/s00440-013-0506-z
Mathematics Subject Classification
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