Classical and stochastic Löwner-Kufarev equations

Theorem 1. [191, 208] Given a function p(ζ,t) of positive real part normalized by p(ζ,t) = 1 + p1ζ +..., the solution to the equation (1) is unique, analytic and univalent with respect to ζ for almost all t ≥0, if and only if, the initial condition f0(ζ) is taken in the form (4), where the function w(ζ,t) is the solution to the equation (3) with the same driving function p. Concluding this section we remark that the L¨owner and L¨owner-Kufarev equa- tions are described in several monographs [4,

Theorem 2. [76] Suppose that f ∈S and 0 < α < 1. Then |a3 −αa2 2| ≤1+2−2α/(1−α). This bound is sharp for all 0 < α < 1. The choice α = 1 4 and a simple recalculation of coefficients of functions from S(2) versus S c3 = 1 2a2, c5 = 1 2(a3 −1 4a2),

Theorem 3. [166] Suppose that f ∈S and that z = φ(w) = f −1(w) = w+b2w2 +... is the inverse function. Then |bn| ≤1·3·5···(2n−1) (n+1)! 2n, with the equality for the function f(z) = z (1+z)2. Let us just mention that most of elementary estimates of functionals in the class S, such as | f(z)|, |arg f (z) z |, | f ′(z)|, can be obtained by L¨owner’s method, see e.g., [14, 72, 123, 191]. In particular, the sharp estimate arg z f ′(z) f(z)

Theorem 4. [201, 203] Let f ∈S gives a nonsingular boundary point of the set Vn and let f map the unit disk onto the plane with piecewise-analytic slits having m finite tips. Then there exist m real-valued functions u1,...,um continuous on [0,∞) and positive numbers λ1,...,λm, m ∑ k=1 λk = 1, such that a solution w = w(z,t) to the Cauchy problem for the generalized L¨owner differential equation (3,15) represents f according to the formula f(z) = limt→∞f(z,t). This representation is unique. The bo

Theorem 5. [201, 203] The boundary hypersurface ∂Vn, n ≥2, is a union of the sets Ω1,...,Ωn−1, every pair of which does not have mutual interior points. Each set Ωm, 1 ≥m ≥n−1, corresponds to the manifold Mm, M1 = R2n−4, so that the parametric representation Ωm = ( a(∞,ξ,λ): ξ ∈Mm; ξn = ±1; λ1,...,λm ≥0; m ∑ k=1 λk = 1 ) holds, where a(∞,ξ,λ) is the manifold coordinate of the system of bicharacteristics a,ψ for the Hamiltonian system (9,11) with continuous branches of the optimal controls given

Theorem 6. [164] Let the driving term λ : [0,1] →R be sufficiently regular with the above asymptotic of λ(t) lim t→1 |λ(1)−λ(t)| √ 1−t = k > 4. Then γ(1 −0) exists, is real, and γ intersects R at the same angle as the trace for λ = k√1−t. Namely, lim t→1arg(γ(t)−γ(1)) = π 1− p 1−16/k2

Theorem 7 ([36]). A semigroup of holomorphic self-maps of the unit disk (φt) is in fact real-analytic in the variable t, and is the solution of the Cauchy problem (20) ∂φt(z) ∂t = G(φt(z)), φ0(z) = z , where the map G, the infinitesimal generator of the semigroup, has the form (21) G(z) = (z−τ)(τz−1)p(z) for some τ ∈D and a holomorphic function p: D →C with Re p ≥0. Conversely, any vector field of the form (21) is semicomplete and if, for z ∈D, we take wz the solution of the initial value problem

Theorem 8 ([42]). Let τ : [0,+∞) →D be a measurable function and let p : D × [0,+∞) →C be a Herglotz function of order d ∈[1,+∞). Then the map Gτ,p : D×[0,+∞) →C given by Gτ,p(z,t) = (z−τ(t))(τ(t)z−1)p(z,t), for all z ∈D and for all t ∈[0,+∞), is a Herglotz vector field of order d on the unit disk. Conversely, if G : D×[0,+∞) →C is a Herglotz vector field of order d ∈[1,+∞) on the unit disk, then there exist a measurable function τ : [0,+∞) →D and a Herglotz function p : D ×[0,+∞) →C of order d su

Theorem 9. ([42, Theorem 1.1]) For any evolution family (ϕs,t) of order d ∈ [1,+∞] there exists an (essentially) unique Herglotz vector field G(z,t) of order d such that for every z ∈D and every s ≥0 the function [s,+∞) ∋t 7→wz,s(t) := ϕs,t(z) solves the initial value problem (22). Conversely, given any Herglotz vector field G(z,t) of order d ∈[1,+∞], for ev- ery z ∈D and every s ≥0 there exists a unique solution [s,+∞) ∋t 7→wz,s(t) to the initial value problem (22). The formula ϕs,t(z) := wz,s(t)

Theorem 10. Let G(z,t) be a Herglotz vector field of order d in D and let (ϕs,t) be its associated evolution family. The following are equivalent: (1) there exists a function g ∈Ld loc([0,+∞),C) and an infinitesimal generator H such that G(z,t) = g(t)H(z) for all z ∈D and almost all t ≥0, (2) ϕs,t ◦ϕu,v = ϕu,v ◦ϕs,t for all 0 ≤s ≤t and 0 ≤u ≤v. In order to end up the picture started with the classical L¨owner theory, we should put in the frame also the L¨owner chains. The general notion of a L¨own

Theorem 11. ([53, Theorem 1.3]) For any L¨owner chain ( ft) of order d ∈[1,+∞], if we define ϕs,t := f −1 t ◦fs whenever 0 ≤s ≤t, then (ϕs,t) is an evolution family of the same order d. Conversely, for any evolution family (ϕs,t) of order d ∈[1,+∞], there exists a L¨owner chain ( ft) of the same order d such that ft ◦ϕs,t = fs whenever 0 ≤s ≤t. In the situation of this theorem we say that the L¨owner chain ( ft) and the evolu- tion family (ϕs,t) are associated with each other. It was proved in [5

Theorem 12. [55, Theorem 5.1] The following two assertions hold: (A): For any Ld-evolution family (ϕs,t) over the canonical domain system (Dt) there exists an essentially unique semicomplete weak holomorphic vector field G : D →C of order d and a null-set N ⊂[0,+∞) such that for all s ≥0 the following statements hold: (i): the mapping [s,+∞) ∋t 7→ϕs,t ∈Hol (Ds,C) is differentiable for all t ∈ [s,+∞)\N; (ii): dϕs,t/dt = G(·,t)◦ϕs,t for all t ∈[s,+∞)\N. (B): For any semicomplete weak holomorphic ve

Theorem 13. [56, Theorem 1.9] Let ( ft) be a L¨owner chain of order d over a canonical domain system (Dt) of order d. If we define (28) ϕs,t := f −1 t ◦fs, 0 ≤s ≤t < ∞, then (ϕs,t) is an evolution family of order d over (Dt). An interesting consequence of this result is thatany L¨owner chain over a canon- ical system of annuli satisfies a PDE driven by a semicomplete weak holomorphic vector field. Moreover, the concrete formulation of this PDE clearly resembles the celebrated L¨owner-Kufarev PDE ap

Theorem 14. [56, Theorem 1.10] Let (ϕs,t) be an evolution family of order d ∈ [1,+∞] over the canonical domain system Dt := Ar(t) with r(t) > 0 (a non-degenerate system). Let r∞:= limt→+∞r(t). Then there exists a L¨owner chain ( ft) of order d over (Dt) such that (1) fs = ft ◦ϕs,t for all 0 ≤s ≤t < +∞, i.e. ( ft) is associated with (ϕs,t); (2) I( ft ◦γ) = I(γ) for any closed curve γ ⊂Dt and any t ≥0; (3) If 0 < r∞< 1, then ∪t∈[0,+∞) ft(Dt) = Ar∞; (4) If r∞= 0, then ∪t∈[0,+∞) ft(Dt) is either D∗,

Theorem 14. It follows from this theorem that the standard L¨owner chain ( ft) associated with a given evolution family, is defined uniquely up to a rotation (and scaling if ∪t∈[0,+∞) ft(Dt) = C∗).

Theorem 15. [56, Theorem 1.13] Let (Dt),(ϕs,t)
be a non-degenerate evolution family and denote as before r∞:= limt→+∞r(t). In the above notation, the following statements hold: (i) the conformal type of the evolution family (ϕs,t) is Aρ for some ρ > 0 if and only if r∞> 0; (ii) the conformal type of the the evolution family (ϕs,t) is D∗if and only if r∞= 0 and ϕ0,t does not converge to 0 as t →+∞; (iii) the conformal type of the the evolution family (ϕs,t) is C \ D if and only if r∞= 0 and ˜ϕ

Lemma 1. Let the function w(z,t) be a solution to the Cauchy problem (3). If the driving function p(·,t), being from the Carath´eodory class for almost all t ≥0, is C∞smooth in the closure ˆD of the unit disk D and summable with respect to t, then the boundaries of the domains Ω(t) = w(D,t) ⊂D are smooth for all t and w(·,t) extended to S1 is injective on S1.

Lemma 2. With the above notations let f(z) ∈F0. Then there exists a function p(·,t) from the Carath´eodory class for almost all t ≥0, and C∞smooth in ˆD, such that f(z) = limt→∞f(z,t) is the final point of the L¨owner-Kufarev trajectory with the driving term p(z,t). 12.1. Witt and Virasoro algebras. The complex Witt algebra is the Lie algebra of holomorphic vector fields defined on C∗= C\{0} acting by derivation over the ring of Laurent polynomials C[z,z−1]. It is spanned by the basis Ln = zn+1 ∂ ∂

Proposition 1. Let the driving term p(z,t) in the L¨owner-Kufarev ODE be from the Carath´eodory class for almost all t ≥0, C∞-smooth in ˆD, and summable with respect to t. The functions G (z), (G (z))<0, (G (z))≥0, and all coefficients Gn are time-independent for all z ∈S1.

Proposition 2. The conjugates ¯ Gk, k = 1,2,..., to the coefficients of the generating function satisfy the Witt commutation relation { ¯ Gm, ¯Gn} = (n−m) ¯ Gn+m for n,m ≥1, with respect to our Poisson structure. The isomorphism ι : ¯ψk →∂k = ∂ ∂ck , k > 0, is a Lie algebra isomorphism (T ∗(0,1) f F0,{ , }) →(T (1,0) f F0,[ , ]).

Proposition 3. The operator C1,1 : H+ →H+ is invertible. The generating function also defines a map G : T ∗F0 ⊗C →H by T ∗F0 ⊗C ∋( f(z),ψ(z)) 7→G = ¯f ′(z)ψ(z) ∈H. Observe that any solution f(z,t), ¯ψ(z,t)
of the Hamiltonian system is mapped into a single point of the space H, since all Gk, k ∈Z are time-independent by

Proposition 1. Consider a bundle π : B →T ∗F0⊗C with a typical fiber isomorphic to Gr∞(H). We are aimed at construction of a curve Γ: [0,T] →B that is traced by the solutions

Proposition 4. The operator Tn defines a graph WTn = span{e0,e1,e2,...} in the Grassmannian Gr∞of virtual dimension 0. Given any ψ = ∞ ∑ k=0 ψk+1zk ∈H+ ⊂H, the function G(z) = ∞ ∑ k=−n Gk+1zk = ∞ ∑

Proposition 5. In the autonomous case of the Cauchy problem (3), when the func- tion p(z,t) does not depend on t, the pseudo-Hamiltonian H plays the role of time-dependent energy and H (t) = ¯G0(t)+const, where ¯G0

Proposition 6. Let n = 1, and let the Baker-Akhiezer function be of the form ΨWTn[g](z) = eξ(t,z)  1+ ω z  , where ω = ω1 is given by the formula (38). Then ∂ω = ∂2A 1−A +  ∂A 1−A 2 is a solution to the KP equation with the Lax operator L = ∂2 −2(∂ω).

Proposition 7. Let |V⟩be a Virasoro primary field in F of conformal dimension λ with central charge c. Then the field [L−m +η1L−1[L−m+1 +η2L−1[L−m+2 +···+ηm−2L−1[L−2 +ηm−1L2 −1]...]|V⟩ is a primary singular field of dimension (λ +m), if and only if η1,...,ηm−1, c, and λ satisfy a system of m linear equations. For example, if m = 2, then the singular field is (44) [L−2 +η1L2 −1]|V⟩, and ( 3+2η1 +4η1λ = 0, c+8λ +12η1λ = 0, if m = 3, then the singular field is [L−3 +η2L−1L−2 +η1η2L3

Proposition 8. For SLE evolution we have • V α ⋆is a Virasoro primary field; • V = V α ⋆,t(ξt)Xt(z) possesses the Markov property, ξt = √ kBt; • M(z) = E[V α ⋆(0)X(z)] is a one-point martingale-observable; • The process Mt(z) = E  V α ⋆,t(ξt)Xt(z) Dt 

Theorem 16. Let M be a complete hyperbolic complex manifold and let H : M → TM be an holomorphic vector field on M. Then the following are equivalent. (1) H is an infinitesimal generator, (2) For all z,w ∈M with z ̸= w it holds (dkM)(z,w) ·(H(z),H(w)) ≤0. This apparently harmless characterization contains instead all the needed infor- mation to get good growth estimates. In particular, it is equivalent to the Berkson- Porta representation formula in the unit disc. 14.4. Ld-Herglotz vector fields an

Theorem 17. Let M be a complete hyperbolic complex manifold. Then for any Her- glotz vector field G of order d ∈[1,+∞] there exists a unique Ld-evolution family (ϕs,t) over M such that for all z ∈M (57) ∂ϕs,t ∂t (z) = G(ϕs,t(z),t) a.e. t ∈[s,+∞). Conversely for any Ld-evolution family (ϕs,t) over M there exists a Herglotz vector field G of order d such that (57) is satisfied. Moreover, if H is another weak holo- morphic vector field which satisfies (57) then G(z,t) = H(z,t) for all z ∈M and almost ev

Theorem 18. [20] Let M be a complex manifold. Then any algebraic evolution family (ϕs,t) on M admits an associated algebraic L¨owner chain ( ft : M →N). Moreover if (gt : M →Q) is a subordination chain associated with (ϕs,t) then there exist a holomorphic mapping Λ: rg( ft) →Q such that gt = Λ◦ft, ∀t ≥0. The mapping Λ is univalent if and only if (gt) is an algebraic L¨owner chain, and in that case rg(gt) = Λ(rg( ft)). The previous theorem shows that the range rg( ft) of an algebraic L¨owner chai

Theorem 19. [20] Let M be a complete hyperbolic manifold with a given Her- mitian metric and d ∈[1,+∞]. Let (ϕs,t) be an algebraic evolution family on M and let ( ft : M →N) be an associated algebraic L¨owner chain. Then (ϕs,t) is a Ld-evolution family on M if and only if ( ft) is a Ld-L¨owner chain. Once the general L¨owner equation is established and L¨owner chains have been well defined, even the L¨owner-Kufarev PDE can be generalized:

Theorem 20. [20] Let M be a complete hyperbolic complex manifold, and let N be a complex manifold of the same dimension. Let G : M ×R+ →TM be a Herglotz vector field of order d ∈[1,+∞] associated with the Ld-evolution family (ϕs,t). Then a family of univalent mappings ( ft : M →N) is an Ld-L¨owner chain associated with (ϕs,t) if and only if it is locally absolutely continuous on R+ locally uniformly with respect to z ∈M and solves the L¨owner-Kufarev PDE ∂fs ∂s (z) = −(d fs)zG(z,s), a.e. s ≥0,z ∈

Theorem 21. Let D ⊂Cn be a complete hyperbolic starlike domain (for instance the unit ball). Let (ϕs,t) be an Ld-evolution family, d ∈[1,+∞]. Then the L¨owner range Lr(ϕs,t) is biholomorphic to a Runge and Stein open domain in Cn.

Theorem 22. [19] Let D ⊂CN be a complete hyperbolic starlike domain. Let G : D×R+ →CN be a Herglotz vector field of order d ∈[1,+∞]. Then there exists a family of univalent mappings ( ft : D →CN) of order d which solves the L¨owner PDE (59) ∂ft ∂t (z) = −d ft(z)G(z,t), a.a. t ≥0,∀z ∈D. Moreover, R := ∪t≥0 ft(D) is a Runge and Stein domain in CN and any other solu- tion to (59) is of the form (Φ◦ft) for a suitable holomorphic map Φ : R →CN. In general, one can infer some property of the L¨owner ra

Theorem 23. Let M be a complete hyperbolic complex manifold and assume that M/aut(M) is compact. Let (ϕs,t) be an algebraic evolution family on M. Then (1) If there exists z ∈M, s ≥0 such that β s z (v) ̸= 0 for all v ∈TzM with v ̸= 0 then Lr(ϕs,t) is biholomorphic to M. (2) If there exists z ∈M, s ≥0 such that dimC{v ∈TzM : β s z (v) = 0} = 1 then Lr(ϕs,t) is a fiber bundle with fiber C over a closed complex submanifold of M. REFERENCES [1] M. Abate, Iteration theory of holomorphic maps on taut m