Time-reversal of multiple-force-point chordal SLE κ ( ρ )

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Introduction
The Schramm-Loewner Evolution (SLE κ ) with κ > 0 is an important family of random non-self-crossing curves introduced by Schramm [Sch00].They have been proved or conjectured to described a large class of two-dimensional lattice models at criticality.We refer the reader to [Law08,Sch11,Smi06] for basic properties of SLE and their relation to 2D lattice models.
The most basic version of SLE is the chordal SLE κ curve, which is a random curve between two boundary points of a simply connected domain characterized by conformal invariance and the domain Markov property.It was conjectured by Rohde and Schramm [RS05] that chordal SLE κ with κ ∈ (0, 8] satisfies reversibility.Namely, modulo a time reparametrization the time reversal of a chordal SLE κ curve is also a chordal SLE κ .The conjecture was first proved for κ ∈ (0, 4] by Zhan [Zha08b] using the so-called commutation coupling.The κ ∈ (4, 8) case was proved by Miller and Sheffield [MS16c] using the imaginary geometry theory.The chordal SLE 8 is the scaling limit of UST Peano curve with half free and half wired boundary conditions [LSW11] and therefore is also reversible.
Chordal SLE κ (ρ) curves are important variants of chordal SLE.They are still curves between two boundary points of a simply connected domain, but their laws depend on some additional marked points called force points.They were introduced by Lawler, Schramm and Werner [LSW03] in the theory of conformal restriction, and play a fundamental role in imaginary geometry as flow lines emanating from a boundary point [MS16a].In [MS16b,MS16c], it was proved that chordal SLE κ (ρ) for κ ∈ (0, 8) with at most two force points lying infinitesimally close to the starting point satisfy the reversibility.When there are force points away from the origin, the law of chordal SLE κ (ρ) is not reversible anymore.Recently, Zhan [Zha22] gave an explicit description of the law of the time reversal of SLE κ (ρ) when κ ∈ (0, 4] and all force points lie on the same side of the origin, and when κ ∈ (4, 8), all force points lie on the same side, and the curve is not boundary touching on this side.In the same paper, he conjectured that a similar result holds for general chordal SLE κ (ρ) with κ ∈ (0, 8) as long as the curve is non-boundary filling; see [Zha22,Conjecture 1.3].In this paper we prove his conjecture.
We now introduce a family of measures on curves describing the time reversal of chordal SLE κ (ρ).
Definition 1.1.Suppose x and ρ satisfy (1.1).We associate a power parameter α i,q ∈ R for each x i,q with α 0,L = α 0,R = 0. Define SLE κ (ρ; α) with force points x to be the measure on continuous curves in H from 0 to ∞ which is absolutely continuous with respect to SLE κ (ρ) with Radon-Nikodym derivative Let us recall some statements on the time reversal of chordal SLE κ (ρ) processes from existing literature.The first one is about the time reversal of SLE κ (ρ L ; ρ R ) processes, which is shown in [MS16b, Theorem 1.1] and [MS16c, Theorem 1.2].Let J : H → H be the map J(z) = −1/z.For a curve η, we write R(η) for its time reversal.
When all the force points lie on the same side of 0, the following theorem is shown in Theorem 1.2 and Section 3.2 of [Zha22] via the construction of the reversed curve.
In [Zha22], the time reversal of SLE κ (ρ R ) is described in terms of reversed intermediate SLE κ (ρ) (iSLE r κ (ρ)) process, which agrees with SLE κ (ρ L ; αL ) when normalized to be a probability measure.The ) with force points at 0 − and 0 + , as shown in [MS16b].iSLE r κ (ρ) process is described explicitly using a Loewner evolution based on Appell-Lauricella multiple hypergeometric function.The constant Z can be traced via [Zha22,(3.16),(3.19)]and [Zha22, Remark 3.6], and can be expressed by a hypergeometric function (in fact a product of the gamma functions) depending only on κ, ρ R but not on the location of the force points x R .
Our main result is the following.
We comment that the reversibility of SLE processes can also be inferred from the conformal welding of Liouville quantum gravity surfaces (see e.g.[DMS21, AHS20, ASY22]).For instance, by viewing the welding interface from the opposite direction, Theorem A is a direct consequence of [AHS20, Theorem 2.2].The time reversal of SLE κ (ρ − ; ρ + , ρ 1 ) with force points 0 − ; 0 + , 1 has also been discussed in [ASY22, Section 7.1] via the conformal welding of quantum triangles.We expect that this method can also be used to describe the time reversal of other types of SLE curves, such as radial SLE with force points and SLE on the annulus.
In Section 2.1, we recap the SLE κ (ρ) processes along with its coupling with the GFF as imaginary geometry flow lines in [MS16a].In Section 2.2, we establish a commutation relation for SLE κ (ρ; α) processes and recap the SLE resampling properties.Finally in Section 3, we prove Theorem 1.2.

Preliminaries
In this paper we work with non-probability measures and extend the terminology of ordinary probability to this setting.For a finite or σ-finite measure space (Ω, F, M ), we say X is a random variable if X is an F-measurable function with its law defined via the push-forward measure M X = X * M .In this case, we say X is sampled from M X and write M X [f ] for f (x)M X (dx).Weighting the law of X by f (X) corresponds to working with the measure d MX with Radon-Nikodym derivative d MX dM X = f , and conditioning on some event E ∈ F (with 0 < M [E] < ∞) refers to the probability measure Throughout this paper, for a continuous simple curve η from 0 to ∞ in H ∪ R, we shall refer to the subset of H\η consisted of connected components whose boundaries contain a subinterval of (−∞, 0) (resp.(0, ∞)) as the left (resp.right) part of H\η.For n ≥ 0, x = (x 0 , ..., x n ) ∈ R n+1 and a ∈ R, we write a + x for (a + x 0 , x 1 , ..., x n ) and ax for (ax 0 , ..., ax n ).The formal notation is used for weights and latter is for the locations of force points under dilation and SLE duality purposes (see Proposition 3.5).

SLE κ (ρ) process and the imaginary geometry
Fix κ > 0. We start with the SLE κ process on the upper half plane H. Let (B t ) t≥0 be the standard Brownian motion.The SLE κ is the probability measure on continuously growing curves (K t ) t≥0 in H, whose mapping out function (g t ) t≥0 (i.e., the unique conformal transformation from H\K t to H such that lim |z|→∞ |g t (z) − z| = 0) can be described by where W t = √ κB t is the Loewner driving function.For the force points and the weights ρ i,q ∈ R, the SLE κ (ρ) process is the probability measure on curves (K t ) t≥0 in H growing the same as ordinary SLE κ (i.e, satisfies (2.1)) except that the Loewner driving function (W t ) t≥0 are now characterized by (2.2) It has been proved in [MS16a] that the SLE κ (ρ) process a.s.exists, is unique and generates a continuous curve until the continuation threshold, the first time t such that W t = V j,q t with j i=0 ρ i,q ≤ −2 for some j and q ∈ {L, R}.Now we recap the definition of the Gaussian Free Field.Let D ⊊ C be a domain.We construct the GFF on D with Dirichlet boundary conditions as follows.Consider the space of smooth functions on D with finite Dirichlet energy and zero value near ∂D, and let H(D) be its closure with respect to the inner product (f, g) ∇ = D (∇f • ∇g) dx dy.Then the (zero boundary) GFF on D is defined by where (ξ n ) n≥1 is a collection of i.i.d.standard Gaussians and (f n ) n≥1 is an orthonormal basis of H(D).
The sum (2.3) a.s.converges to a random distribution independent of the choice of the basis (f n ) n≥1 .
For a function g defined on ∂D with harmonic extension f in D and a zero boundary GFF h, we say that h + f is a GFF on D with boundary condition specified by g.See [DMS21, Section 4.1.4]for more details.
Next we introduce the notion of GFF flow lines.We restrict ourselves to the range κ ∈ (0, 4).Heuristically, given a GFF h, η(t) is a flow line of angle θ if ) process coupled with the GFF h with illustrated boundary conditions as the (zero angle) flow line of h.The θ angle flow line of h then has the law as SLE κ (ρ To be more precise, let (K t ) t≥0 be the hull at time t of the SLE κ (ρ) process described by the Loewner flow (2.1) with (W t , V i,q t ) solving (2.2), and let F t be the filtration generated by (W t , V i,q t ).Let h 0 t be the bounded harmonic function on H with boundary values ), and λ(1

z). Let h be a zero boundary GFF on H and
Then as proved in [MS16a, Theorem 1.1], there exists a coupling between h and the SLE κ (ρ) process (K t ), such that for any F t -stopping time τ before the continuation threshold, K τ is a local set for h and the conditional law of h| H\Kτ given F τ is the same as the law of h τ + h • g τ .For κ < 4, the SLE κ (ρ) coupled with the GFF h as above is referred as a flow line of h from 0 to ∞, and we say an SLE κ (ρ) curve is a flow line of angle θ if it can be coupled with h + θχ in the above sense.For κ ′ > 4, the SLE κ ′ (ρ) curve coupled with a GFF −h as above is referred as a counterflow lines of h.
So far we have discussed SLE κ (ρ) processes on the upper half plane, and for general simply connected domains, the definition can be extended via conformal mappings.Namely, let x, y ∈ ∂D, x ⊂ ∂D be the force points and ψ : D → H be a conformal map with ψ(x) = 0, ψ(y) = ∞.Then a sample from the chordal SLE κ (ρ) process in D from x to y is obtained by first taking an curve η from SLE κ (ρ) with force points ψ(x) and then output ψ −1 (η).Observe that the term x i,q • (ψ i,q η ) ′ (x i,q ) in (1.2) is invariant under dilations of H, which implies that for a > 0 and an SLE κ (ρ; α) process η with force points x, the law of ψ • η is SLE κ (ρ; α) with force points ax.This implies that the notion of SLE κ (ρ; α) can also be extended to general simply connected domains by the same way.Moreover, if η is a flow line of some GFF h, then To simplify our language, we are going to extend the notion of SLE κ (ρ; α) processes to certain nonsimply connected domains.Let D ⊂ C be some domain and x, y ∈ ∂D, such that the boundary ∂D consist of two non-crossing simple curves η L D , η R D running from x to y which possibly intersect and bounce-off each other.Let with x 0,L = x − and x 0,R = x + , such that for i ≥ 1 and q ∈ {L, R}, none of the x i,q 's lies on η L D ∩ η R D .Further assume η L D visits x L in the order of x 0,L , ..., x k,L , and η R D visits x R in the order of x 0,R , ..., x ℓ,R .On each connected component D of D, let x D (resp.y D ) be the first (resp.last) point on ∂ D traced by η L D , and let i L D and j L D (resp.i R D and j R D ) be the largest and smallest integer such that Concatenate all the η D 's, and define the law of this curve from x to y in by SLE κ (ρ; α) in D with force points x.
We remark that our definition above is natural in the following sense.Temporarily assume α is zero.Let ψ L : C\η L D → C\(−∞, 0) and ψ R : C\η R D → C\(0, ∞) be the conformal maps sending x to 0 and y to ∞.
′ agrees with (2.5) on (0, ∞) with t = 0.In each connected component D construct the flow line η D of h from x D to y D , and the SLE κ (ρ) process in D can be understood as the concatenation of all the η D 's.For non-zero α we can further weight by the corresponding conformal derivatives.
The SLE κ (ρ) curve η satisfies the following Domain Markov property.Let τ be some stopping time for η.On the event that τ is less than the continuation threshold, the conditional law of η(t + τ ) t≥0 given η([0, τ ]) is an SLE κ (ρ) on H\K τ with force points x τ , where and if two force points x i,q and x j,q are equal, they could be merged into a single force point of weight ρ i,q + ρ j,q .

The coupling of the two flow lines
One important implication of the flow line coupling of SLE and the GFF is that, for two SLE κ (ρ) processes η 1 and η 2 coupled within the same imaginary geometry, one can easily read off the conditional laws of η 1 given η 2 and η 2 given η 1 .Suppose η 1 and η 2 are flow lines of h, then given η 1 , the conditional law of η 2 is the same as the law of the flow line (with some angle) of the GFF in H\η 1 with the flow line boundary conditions (see [MS16a, Figure 1.10] for more explanation) induced by η 1 , and vice versa for the law of η 1 given η 2 .Now we state the following commutation relation between SLE κ (ρ; α) processes.See Figure 4 for an illustration.Suppose (Ω, F) is a σ-finite measure space and X : Ω → A is a random variable with law µ.Also suppose (ν x ) x∈A is a family of σ-finite measures on (Ω, F).By first sampling X from µ and then Y from ν X , we refer to a sample (X, Y ) from the measure ν x (dy)µ(dx) on (Ω, F).
We remark that the topological configuration of the two curves (η 1 , η 2 ) could be rather complicated, as they may intersect other, and both intersect the boundaries (−∞, 0) and (0, ∞), and we shall apply the definition of SLE κ (ρ; α) processes in non-simply connected domains as specified in the previous section.
Proof.If α i,q = 0 for all possible i and q, then as argued in [MS16a, Section 6], the pairs (η 1 , η 2 ) generated from the three ways can all be realized as sampling the angle (θ 1 , θ 2 ) = (0, − (ρ+2)λ χ ) flow lines of the GFF with boundary conditions as (2.5) and (2.6) and therefore the claim follows.Let P be the corresponding law of these two flow lines, which agrees with the law of (η 1 , η 2 ) constructed as in the third way before we do the weighting.
(2.8) Using a similar argument, one can show that if we let L be the law on the pairs (η 1 , η 2 ) constructed as in the second way of the statement, then d L dP is also the same as (2.8).Therefore the claim follows.Proposition 2.1 gives three equivalent ways to characterize the joint law of (η 1 , η 2 ).On the other hand, at least when α i,q = 0, the two conditional laws η 1 |η 2 and η 2 |η 1 as in Proposition 2.1 uniquely determines the joint law of (η 1 , η 2 ).
Proof.When η 1 a.s.does not intersect η 2 (i.e., ρ ≥ κ 2 − 2), the claim follows from the same argument as in [MS16b, Section 4].For the remaining case, we may first separate the starting and ending points of (η 1 , η 2 ) as in the first step of [MS16b, Proof of Theorem 4.1] and then apply the same argument in [MSW19, Appendix A].See also Appendix A for an alternative proof based on Markov chain irreducibility results in [MT09] and Lemma 3.1.
We are going to use the following variant of Lemma 2.2, which follows from exactly the same Markov chain remixing argument in [MS16b, Theorem 4.1] and [MSW19, Appendix A].
Lemma 2.3.Let κ ∈ (0, 4), ρ, ρ, x be the same as Proposition 2.1.For ε > 0, let D ε = ∪ q∈{L,R} ∪ i≥1 B(x i,q , ε).Fix ε sufficiently small such that 0 / ∈ D ε .Let P be the joint law of (η 1 , η 2 ) as described in Lemma 2.2, and P ε be the probability measure given by conditioning P on the event ) is a sample from some probability measure on curves in H running from 0 to ∞, such that the conditional law of η2 given η1 is the SLE κ (ρ; ρR ) in the right part of H\η 1 conditioned on not hitting D ε , and the conditional law of η1 given η2 is the SLE κ (ρ L ; ρ) in the left part of H\η 2 conditioned on not hitting D ε .Then the joint law of (η 1 , η2 ) is the same as P ε defined above.

Proof of Theorem 1.2
In this section, we prove our main result Theorem 1.2.We start with the κ ∈ (0, 4) case, where we first extend Theorem B to SLE κ (ρ; ρ R ) curves (i.e.adding a force point at 0 − in Theorem B) and then apply the SLE resampling properties (Lemma 2.2).For κ ∈ (4, 8) case, we shall use the SLE duality, and the κ = 4 case is covered in [WW17, Theorem 1.1.6].
Figure 5: An illustration of Lemma 3.1, where we show that flow lines of the GFF can stay arbitraily close to some given curve.Left: the curve γ intersects ∂D only at ±i, and we construct the curves γL , γR within the ε-neighborhood of γ.Let U γ be the region between γL and γR , hγ be some GFF in U γ with the same boundary conditions as h on ∂D ∩ ∂U γ and flow line boundary conditions on γL , γR such that the flow line ηγ of hγ from −i to i a.s.has positive distance to γL ∪ γR .Then by the GFF absolute continuity argument, the flow line η is contained in U γ with positive probability.Right: γ consists of 4 arcs in D, namely γ0 1 , ..., γ0 4 .Let γ1 = γ0 1 and construct γL 1 , γR 1 , U γ1 analogously.Let I 1 be the component of U γ1 ∩ D with ψ(σ 1,R γ ) on the boundary.By the same argument, the event where η hits I 1 at w 1 at some time τ 1 without hitting γL 1 ∪ γR 1 has nonzero probability.Then conditioned on η([0, τ 1 ]), we construct the curve γ2 from w 1 to ψ(ξ 1,R γ ) staying close to γ0 2 and iterate the same argument.
We may choose some (non-random) constant ζ > 0 small such that this distance is at least ζ with probability greater than 1/2.Therefore it follows from the same argument of [MS17, Lemma 3.9] that, if we set U ζ γ := {z ∈ U γ ; dist(z, γL ∪ γR ) > ζ}, by the GFF absolute continuous property [MS16a, Proposition 3.4], the law of hγ is absolutely continuous w.r.t.h when restricted to the domain U ζ γ .Since flow lines are a.s.determined by and local sets of the GFF [MS16a, Theorem 1.2], and the flow line of hγ is contained within U ζ γ with probability at least 1/2, it follows that, with positive probability η is contained in U ζ γ and thus in the ε-neighborhood of γ, which finishes the case when γ ∩ R = {0}.
Proof.We sample an SLE κ (ρ R ) curve η 2 in H from 0 to ∞ with force points x R , and conditioned on η 2 , we sample an SLE κ (−ρ 0,L − 2; ρ 0,L ) process η 1 from 0 to ∞ in the left part of H\η 2 with force points 0 − and 0 + .Then it follows from the construction in Proposition 2.1 that conditioned on η 1 , η 2 is an SLE κ (ρ 0,L ; ρ R ) process in the right part of H\η 2 with force points (0 − ; x R ).
Corollary 3.1.Let ρ, x, x, ρ be as in Lemma 3.2, and γ be a curve in H from 0 to ∞ which is admissible w.r.t.(2 + ρ 0,L + ρ R , x R ).Let H R γ be the right part of H\γ.Sample an SLE κ (ρ) process η 0 in H R γ with force points x.Then there exists some constant Z(γ, ρ, x) such that the law of the time reversal of J(η 0 ) is equal to 1 Z(γ,ρ,x) times SLE κ (ρ; α) in J(H R γ ) with the force points x.
Proof.We apply Lemma 3.2 within (finitely many) connected components whose boundaries contain the force points x, and apply Theorem A for rest of the connected components (where there are no constants).Then the constant Z(γ, ρ, x) is now a (finite) product of the corresponding constants in each of the connected components.
The next lemma states that in some sense, the constant Z in Lemma 3.2 does not depend on the choice of x.This follows by comparing the two ways of viewing the marginal law of the η1 above: directly applying Lemma 3.2, and applying Proposition 2.1 to the pair (η 1 , η2 ).
Proof.The proof is organized as follows.We first construct a pair of reversed curves (η 1 , η2 ) by Proposition 2.1, and then apply Lemma 3.2 to get the conditional laws of η i := R(η i ) given η j for 1 ≤ i ̸ = j ≤ 2. Finally we apply Lemma 2.3 to identify the law of the forward curves (η 1 , η 2 ) with the usual SLE κ (ρ).
On the other hand, by Proposition 2.1, a sample (η 1 , η2 ) from Q ε can be produced by (i) sampling η1 from SLE κ (ρ; α) process conditioned on not hitting Dε (ii) weighting the law of η1 by Z 0 (η 1 ), where Z 0 (η 1 ) is the measure of an SLE κ (a − 2; ρR ; 0; αR ) process in the right component of H\η 1 with force points (0 − ; x R ) being disjoint from Dε and (iii) sampling an SLE κ (a − 2; ρR ; 0; αR ) process η2 in the right component of H\η 1 with force points (0 − ; x R ) conditioned on not hitting Dε .Meanwhile, by Lemma 3.2, Z 0 (η 1 ) is equal to Z(η 1 , (−a + ρ L ; a − 2), x R ) times the probability of an SLE κ (−a + ρ L ; a − 2) process in the left component of H\η 1 being disjoint from D ε .By Lemma 3.1, the latter probability is positive for any fixed η 1 , while by Lemma 3.3, the constant Z(η 1 , (−a + ρ L ; a − 2), x R ) is independent of η 1 .Therefore by comparing the marginal laws of η 1 and η1 , we conclude that under J(z) = −1/z, the time reversal of an SLE κ (ρ) process with force point x conditioned on not hitting D ε agree with the SLE κ (ρ; α) process with force point x conditioned on not hitting Dε .Since ε > 0 can be arbitrarily small, the claim therefore follows.