Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Theorem 1.1. There exists, up to multiplication by a constant, a unique Riemannian metric on J ∞which is invariant under the left and right action of SU(1,1). This metric is defined by an orthonormal basis as given in (1.2) below.

Proposition 2.1. Given u ∈L2(S1), the vector field U1 vanishes at z = 0 and is C1 in D = {z: |z| < 1}.

Theorem 3.3. There exist a constant c, independent of ζ and y > 0, such that in terms of ¯∂= ∂θ + i∂y the following estimate holds: E ¯∂Zx,t(ζ) 2 ⩽ct. (3.4)

Lemma 3.4. The covariance of Zx,t is a C5 function on P × P . Denoting (ζ,ζ ′) := a(ζ,ζ) + b(ζ,ζ) + a(ζ ′,ζ ′) + b(ζ ′,ζ ′) −2a(ζ,ζ ′) −2b(ζ,ζ ′), then (ζ,ζ ′) ⩽c|ζ −ζ ′|2, (ζ,ζ ′) ∈Pδ × Pδ; (3.11) (ζ,ζ ′) ⩽c∥ζ −ζ ′∥2 log− ∥ζ −ζ ′∥  + 1  , (ζ,ζ ′) ∈P × P. (3.12)

Lemma 3.5. There exist strictly positive numerical constants c1, c2 such that α(ζ1,ζ2)  ⩽c1 exp  −c2d(ζ1,ζ2)  , β(ζ1,ζ2)  ⩽c1 exp  −c2d(ζ1,ζ2)  , γ (ζ1,ζ2)  ⩽c1 exp

Theorem 3.6. We have ∥Dy1,y2∥L∞⩽∥Dy1,y2∥A ≃ρ(y1,y2) := inf(y1,y2) sup(y1,y2); ∥Dy1,y2∥2 L2 ≃inf(y1,y2) × ρ(y1,y2). (3.17)

Theorem 3.7. We have Dy1,y2(θ)  ⩽c inf  ρ, y1y2 θ2  and  S1 Dy1,y2(θ) dθ ⩽2√y1y2. (3.18)

Theorem 3.8. There exists a numerical constant c such that D(ζ1,ζ2)  ⩽c exp  −dP (ζ1,ζ2)  . (3.20)

Proposition 4.1. Define the vector field A(ζ) = v(ζ) + iw(ζ), ζ = θ + iy, where v = 0, w(y,θ) = 6  n>1 1 n2 −1  6 K′′(yn) + K(yn)  K′(yn). (4.2) The following Stratonovich and Itô equations are equivalent:

Lemma 4.2. The trajectories of the diffusions associated to L have infinite lifetime and stay in P up to their lifetime.

Theorem 4.3. Eqs. (4.1)–(4.3) define a unique stochastic flow  Ψx,t of C1 diffeomorphisms of D1; moreover, lim ρ→1  Ψx,t  ρ exp(iθ)  = gx,t(θ) uniformly in θ ∈S1.

Theorem 5.1 (Smooth welding theorem). Denote Dρ := {z: |z| < ρ} and let f ρ x,t(z) = F ρ x,t ◦ Ψx,t −1(z), z ∈ Ψx,t(Dρ); gρ x,t(z) = F ρ x,t(z), z /∈ Ψx,t(Dρ). Then f ρ

Theorem 5.2. For each r < 1, the limit lim ρ→1 f ρ x,t(z) =: ϕx,t(z) exists uniformly in z ∈Dr and defines a univalent function ϕx,t on D1.

Lemma 5.3. Denote δρ(F ρ x,t) = lim ε→0+ 1 ε (F ρ x,t)−1 ◦F ρ+ε x,t −Id. (5.8) Then δρ(F ρ x,t)(z) = 1 π

Corollary 5.4. As ρ →1, F ρ x,t converges uniformly with all its derivatives in a neighbour- hood of z = 0. For the univalent function f ρ x,t, all coefficients of its Taylor expansion at the origin converge.

Corollary 5.5. We have δρ(F ρ x,t)  ⩽2 |z| ρ(ρ −|z|) (5.12) and δρ+ε(F ρ+ε x,t ) = δρ(F ρ x,t) + o(ε).

Lemma 5.6. For ρ fixed, consider the differential equation ˙z(s) = δρ+s(F ρ+s x,t )  z(s)  , z(0) = z0. (5.13) If z(s) ∈Dρ ∀s ∈[0,1 −ρ0[, (5.14) then F ρ+s

Lemma 5.7. Given r < 1, there exists ρ0 such that hypothesis (5.14) is satisfied for any z0 ∈Dr.

Theorem 5.8 (Stochastic welding theorem). Assume that (5.18) holds true. There exists a function γx,t univalent outside the disk such that  (ϕx,t)−1 ◦γx,t  (expiθ) = gx,t(expiθ) (5.19) where gx,t is defined through (1.3).

Theorem 6.1 (Loewner’s equation along a stochastic flow). The infinitesimal increments δt,εF r :=  d dt F r xε,t  ◦(F r xε,t)−1 (6.2) satisfy ¯∂(δt,εF r) = Ar xε,t, Ar xε,t := ∂f r

Theorem 6.2. Denote ar x(t) := area(F r x,t(Dr)). There exist constants c1,c2,c3, indepen- dent of r < 1, such that for any R > 0, Prob

Corollary 6.3. Let a1 x(t) be the area of ϕx,t(D), then a1 x(t) satisfies estimate (6.13).

Lemma 6.4 (Comparison lemma for the hyperbolic metric). Let G1,G2 be two simply connected domains such that G1 ⊂G2 and denote by ds2 i the corresponding Poincaré metrics. Then ds2 1 ⩾ds2 2.

Lemma 6.5. There exists a positive constant c0, independent of ζ1, ζ2, such that dD+ P (ζ1,ζ2) > c0 log  ζ0 −ζ1 ζ0 −ζ2 . (6.17) 6.3. Uniqueness of the welding We adopt the point of view of [16, p. 304]. The circle S1 is the boundary of the two closed hemispheres of the Riemann sphere. Let S1 + be the northern hemisphere and S1 −be the southern hemisphere. Given h ∈Homeo(S1), we define on S1 + ⊕S1

Theorem 6.6. Assume that there exists a welding conformal structure C0 such that Γ C0 h is a Hölder Jordan curve, then every welding structure C coincides with C0.

Theorem 6.7. To a univalent function f defined on ¯D we associate the Jordan curve Φ(f ) = f S1. Let ϕx,t be the univalent function constructed in Theorem 5.2 as solution of the stochas- tic welding problem (5.19). Then t →Φ(ϕx,t) defines a Markov processes with values in J .