Brownian motion on the space of univalent functions via Brownian motion on (S )

Transcription

1 Brownian motion on the space of univalent functions via Brownian motion on (S ) Jürgen Angst 1 d après Airault, Fang, Malliavin, Thalmaier etc. Winter Workshop Les Diablerets, February 2010

2 1 Some probabilistic and algebraic motivations Brownian motion on some quotient spaces Representations of the Virasoro algebra 2 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 3 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process

3 Notations and preliminaries : a drop of complex analysis

4 The space of Jordan curves Let consider J J := {Γ C, Γ is a Jordan curve}, := {Γ C, Γ is a C Jordan curve}. Facts : Γ J φ : S 1 C continuous and injective such that φ(s 1 ) = Γ ; Let h ÀÓÑ Ó(S 1 ), then φ and φ h are parametrizations of the same Jordan curve Γ. Unspoken goal of the sequel : analysis on J

5 Riemann mapping theorem Γ splits the complex plane into two domains D + Γ and D Γ : F + Γ Γ D + Γ D D Γ F Let D := {z C, z < 1}, the Riemann mapping theorem ensures that : F + : D D Γ + biholomorphic, unique modëí(1,1), F : D D Γ biholomorphic. unique modëí(1,1).

6 Riemann mapping theorem Γ splits the complex plane into two domains D + Γ and D Γ : F + Γ Γ D + Γ D D Γ F Let D := {z C, z < 1}, the Riemann mapping theorem ensures that : F + : D D Γ + biholomorphic, unique modëí(1,1), F : D D Γ biholomorphic, unique modëí(1,1).

7 Restrictions to S 1 of homographic transformations ËÍ(1, 1) := Poincaré group of automorphisms of the disk restrictions to S 1 of homographic transformations z az + b bz + ā, a 2 b 2 = 1.

8 Holomorphic parametrizations By a theorem of Caratheodory, the maps F ± extend to homeomorphisms : F + : D D + Γ, F : D D Γ ; in particular, F ± S 1 define (canonical) parametrizations of Γ ; we have g Γ := (F ) 1 F + S 1 ÀÓÑ Ó(S 1 ) (orientation preserving).

9 Notations and preliminaries : a touch of algebra

10 The Lie group (S 1 ) and its Lie algebra (S 1 ) := the group of C, orientation preserving diffeomorphisms of the circle diff(s 1 ) := Lie algebra of right invariant vector fields on (S diff 0 (S 1 ) := C functions on S 1 via the identification u C (S 1, R) vector field u d dθ Lie bracket given by [u,v] diff(s 1 ) := u v uv { u diff(s 1 ), } 1 u(θ)dθ = 0 2π S 1 1 )

11 Central extensions of (S 1 ) and Virasoro algebra The central extensions of (S 1 ), that is 1 A E? (S ) 1, A Z(E) or equivalently 1 0 a e? diff(s 1 ) 0, have been classified by Gelfand-Fuchs.

12 Central extensions of (S They are of the form 1 V c,h = R diff(s 1 ), ) and Virasoro algebra and are associated to a fundamental cocyle on diff(s 1 ) : [( ω c,h (f,g) := h c ) f c ] f g dθ, where c,h > 0, via S 1 [ακ + f, βκ + g] Vc,h := ω c,h (f,g)κ + [f,g] diff(s 1 ).

13 Brownian motion on some quotient spaces Representations of the Virasoro algebra 1 Some probabilistic and algebraic motivations 2 3

14 Brownian motion on some quotient spaces Representations of the Virasoro algebra Define Brownian motions on some natural quotient spaces of (S 1 )

15 Brownian motion on some quotient spaces Representations of the Virasoro algebra Brownian motion on the space of Jordan curves Theorem (Beurling-Ahlfors-Letho, 1970, conformal welding) The application J (S 1 ), Γ g Γ = (F ) 1 F + S 1 is surjective and induces a canonical isomorphism : J ËÍ(1,1) \ (S 1 )/ËÍ(1,1). 1 Idea : to construct a Brownian motion on J, a first step consists in defining a Brownian motion on (S ) and pray that the construction passes to the quotient!

16 The space of univalent functions Brownian motion on some quotient spaces Representations of the Virasoro algebra In the same spirit, consider U := {f C (D, C), f univalent s.t. f(0) = 0, f (0) = 1}. To f U, one can associate Γ = f(s 1 ) J : D S 1 D f Γ = f(s 1 ) D Γ

17 Riemann mapping theorem again Brownian motion on some quotient spaces Representations of the Virasoro algebra The Riemann mapping theorem provides a biholomorphic mapping h f such that h f : C\D D Γ, h f( ) =. It is unique up to a rotation of C\D, i.e. up to an element of S 1 and extends to the boundary : h f : C\D D Γ, h f( ) =. Using this construction, we thus have an application U (S 1 ), f g f := f 1 h f S 1.

18 The space U as a quotient of (S Theorem (Kirillov, 1982) The application U (S Brownian motion on some quotient spaces Representations of the Virasoro algebra 1 ) 1 ), f g f = f 1 h f S 1 induces a canonical isomorphism : U (S 1 )/ S 1. Idea : as before, the construction of a Brownian motion on U appears closely related to the construction of a Brownian motion on (S 1 )...

19 Brownian motion on some quotient spaces Representations of the Virasoro algebra Unitarizing measures and representations of the Virasoro algebra

20 Brownian motion on some quotient spaces Representations of the Virasoro algebra Facts : The theory of Segal and Bargmann shows that the infinite dimensional Heisenberg group H has a representation : Ò ( H L 2 hol (H,ν) ), u ρ(u), where H is an Hilbert space and ν a Gaussian measure ; a similar Gaussian realization was proved by Frenkel in the case of Loop groups. Question : Does there exists a space M, a measure µ, and a representation of the Virasoro algebra of the following form? Ò ( V c,h L 2 hol (M,µ) ), u ρ(u).

21 Brownian motion on some quotient spaces Representations of the Virasoro algebra Heuristics : M := (S 1 )/ËÍ(1,1) is a good candidate. It carries a canonical Kählerian structure, associated to a Kähler potential K such that K = ω c,h ; heuristics again : the measure µ should look like : µ = c 0 exp ( K)dÚÓÐ. Idea : realize µ as an invariant measure for a Brownian motion + drift on M, with infinitesimal generator : G := 1 2 K. Up to technical difficulties, this method works!

22 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 1 Some probabilistic and algebraic motivations 2 3

23 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Canonical horizontal diffusion

24 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Horizontal diffusion on the tangent space Given a differentiable structure M, the canonical way to define a brownian motion on M is to do a stochastic development of a diffusion living on the tangent space T M to M, that is : 1 first define a brownian motion on the tangent space T M ; 2 then roll it without slipping from T M to M.

25 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach The notion of stochastic development diffusion on the tangent space T M diffusion on the underlying manifold M

26 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Stochastic development and metric structure Remarks : 1 the notion stochastic development, that is roll without slipping, implies a pre-existing metric structure on the manifold M, i.e. on T M ; 2 the resulting process on M will inherit from the invariance properties of the metric choosen on T M.

27 The way forward Canonical horizontal diffusion Construction via regularization An alternative pointwise approach To define a Brownian motion on (S 1 ), we thus have to : 1 choose a metric structure on (S ) ; 1 2 construct a Brownian motion on diff(s 1 ) ; 3 roll it without slipping on (S 1 ) via the exponential mapping.

28 How to choose a metric on (S Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 1 )?... if you have in mind to construct a Brownian motion on the space of smooth Jordan curves...

29 How to choose a metric on (S Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 1 )? Theorem (Airault, Malliavin, Thalmaier) There exists, up to a multiplicative constant, a unique Riemannian metric on ËÍ(1,1) \ (S )/ËÍ(1,1) 1 which is invariant under the left and right action ofëí(1, 1).

30 How to choose a metric on (S Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 1 )?... if you have in mind to construct a Brownian motion on the space of univalent functions...

31 How to choose a metric on (S Theorem Canonical horizontal diffusion Construction via regularization An alternative pointwise approach 1 )? 1 )/S 1, There exists a canonical Kähler metric on U (S i.e. on diff 0 (S 1 ). It is associated to the fundamental cocycle ω c,h defining the Virasoro algebra : u 2 := ω c,h (u,ju). Here diff 0 (S 1 ) {u C (S 1 ), S 1 udθ = 0}, and u(θ) = + k=1 (a k cos(kθ) + b k sin(kθ)), Ju(θ) := + k=1 ( a k sin(kθ) + b k cos(kθ)), ω c,h (u,ju) = + ( k=1 α2 k a 2 k + b 2 ) k, α 2 k := ( hk + c 12 (k3 k) ).

32 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach The algebra diff(s 1 ) as a Sobolev space The metric. is invariant under the adjoint action of S 1 ; the sequence α k grows like k 3/2, thus ( diff0 (S 1 ),. ) H 3/2 (S 1 ) ; an orthonormal system for ( diff 0 (S 1 ),. ) is, for k 1 : e 2k 1 (θ) := cos(kθ), e 2k (θ) := sin(kθ). α k α k

33 Brownian motion on diff(s 1 ) Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Definition The Brownian motion (u t ) t 0 on diff(s 1 ) (with H 3/2 structure) is the solution of the following Stratonovitch SDE du t (θ) = ( ) e 2k 1 (θ) dxt k + e 2k (θ) dyt k, k 1 where X k,y k,k 1 are independant, real valued, standard Brownian motions. Remark : almost surely, the above series converges uniformly on [0,T] S 1.

34 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Stochastic development via exponential map 1 The stochastic development of the diffusion (u t ) t 0 on diff(s 1 ) to a Brownian motion g t on (S ) writes formally : ( ) dg t = ( du t )g t, g 0 =Á, in other words dg t = ( k 1 e 2k 1 (g t ) dx k t + e 2k(g t ) dy k t ), g 0 =Á. Problem : the classical Kunita s theory of stochastic flow works with a regularity H 3/2+ε for any ε > 0, but not in the critical case H 3/2.

35 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Malliavin s approach : construction via regularization

36 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Regularized Brownian motion on diff(s 1 ) Malliavin s approach of the problem is to regularize the horizontal diffusion, i.e. consider the following SDE for 0 < r < 1 : du r t (θ) = ( k 1 rk e 2k 1 (θ) dxt k + e 2k(θ) dy k ) t, ( ) r dgt r = ( dur t ) gr t, gr 0 =Á. Theorem (Airault, Malliavin, Thalmaier) 1 The limit g t ÀÓÑ Ó(S 1 For any 0 < r < 1, the equation ( ) r admits a unique solution t gt r (S ). The limit g t (θ) := lim r 1 gt r (θ) exists uniformly in θ and defines a solution of ( ). ) only!

37 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach Alternative approach : finite dimensional approximation

38 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach The pointwise approach by S. Fang Fang s approach of the problem is to consider the following approximating SDE s, for n 1 : du n t (θ) = n ( k=1 e2k 1 (θ) dxt k + e 2k (θ) dy k ) t, ( ) n dgt n = ( du n t ) gt n, g0 n =Á. Theorem (Fang) 1 For any n 1, the equation ( ) n admits a unique solution t gt n (S ). For θ given, the limit g n (θ) := lim n + gt n (θ) exists uniformly in [0,T] and defines a solution of equation ( ).

39 Canonical horizontal diffusion Construction via regularization An alternative pointwise approach The pointwise approach by S. Fang Theorem (Fang) There exists a version of g t such that, almost surely, g t ÀÓÑ Ó(S 1 ) for all t. Moreover, there exists c 0 > 0 such that g t (θ) g t (θ ) C t θ θ e c 0 t. In other words, the mappings g t are δ t -Hölderian homeomorphisms with δ t going to zero when t +. ) with its H 3/2 metric struc- ). Morality : Brownian motion on (S ture must be realized in the bigger spaceàóñ Ó(S 1 1

40 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process 1 Some probabilistic and algebraic motivations 2 3

41 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Brownian motion on the space of univalent functions

42 What a wonderful world... Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process We have defined a Brownian motion g t on the group of diffeormorphisms of the circle. To construct a Brownian motion φ t on U, the space of univalent functions, we would like to factorize g t thanks to the notion of conformal welding : g t = (φ t ) 1 h t. The classical theory of conformal welding is well developed for diffeomorphisms of the circle that have a quasi-conformal extension to the unit disk.

43 What a wonderful world... or not Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process The class of diffeomorphisms preserving the point at infinity and admitting a quasi-conformal extension to the half-plane is caracterized by the quasi-symmetry property : h(θ + θ ) h(θ ) sup θ,θ S 1 h(θ) h(θ θ ) <. Theorem (Airault, Malliavin, Thalmaier) 1 ) does not sa- ) Almost surely, the Brownian motion g t on (S tisfy the above quasi-symmetry property : lim sup h 0 1 log h ( sup t [0,1], θ S 1 log g t (θ + h) g t (θ) g t (θ) g t (θ h) = log 2.

44 From BM( (S 1 )) to BM(U ) Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process The problem : classical theory of conformal welding cannot be applied directly here... The solution : 1 extend the stochastic flow g t to a stochastic flow of diffeomorphisms in the unit disk D ; 2 use conformal welding inside the disk, i.e. on a disk of radius 0 < ρ < 1 ; 3 pray and let ρ goes to 1...

45 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Beurling-Ahlfors extension

46 Beurling-Ahlfors extension Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Smooth vector fields on the circle are of the form u(θ)d/dθ where u C (S 1 ) C 2π (R). Given u C (S 1 ), Beurling-Ahlfors extension provides a vector field U on H := {ζ = x + iy C, y > 0} via : U(ζ) = U(x+iy) := u(x sy)k(s)ds 6i u(x sy)sk(s)ds, where K(s) := (1 s ) 1 [ 1,1] (s).

47 Beurling-Ahlfors extension Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process In Fourier series, if u(x) = + n= c ne inx, then U(ζ) = U(x + iy) = + n= ( ) c n K(ny) + 6 K (ny) e inx, where ( ) sin(ξ/2) 2 K(ξ) =. ξ/2

48 Beurling-Ahlfors extension Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Consider the holomorphic chart ζ z = exp(iζ), denote log (a) = max{0, log(a)}, and define Ũ(z) := izu(ζ) := iz ( + n= ( c n K(n log ( z )) + 6 K ) (n log ( z )) e ). inx Proposition Given u L 2 (S 1 ), the vector field Ũ vanishes at the origin z = 0 an is C 1 in the unit disk D.

49 From the circle to the disk Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process We now apply the preceding machinery to the stochastic flow (u t ) t 0 on diff(s 1 ). The complex version of u t simply writes : where u t (θ) := n Z\{0} e n (θ)x n t, e n (θ) := einθ (, with α 2 n α = h n + c ) n 12 ( n 3 n ), Xt n, n 0, are independant Brownian motions.

50 From the circle to the disk Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process We thus obtain a flow Ũt of C 1 vector fields on the unit disk D, vanishing at zero : Ũ t (z) := iz ( K(n log ( z )) + 6 K (n log ( z ))) e n (x)xt n. n Z\{0} At this stage, it is possible (rhymes with technical) to control the covariance of the resulting process, i.e. to control the expectations E[Ũt(z)Ũt(z )], E[ Ũt(z) Ũt(z )]...

51 Stochastic development again Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process It is then possible to integrate the development equation : ( ( ) d Ψ t = dũt) Ψt, Ψ0 =Á. Theorem (Airault, Malliavin, Thalmaier) The equation ( ) defines a unique stochastic flow Ψ t of C 1, orientation preserving, diffeomorphisms of the unit disk D. Moreover, lim Ψ t (ρe iθ) = g t (θ) uniformly in θ, ρ 1 where g t is the solution of ( ).

52 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Stochastic conformal welding

53 Beltrami equation in a small disk Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process For 0 < ρ < 1, let D ρ := ρd and ν ρ t (z) := Ψ t Ψ t (z) if z D ρ, 0 otherwise. Let F ρ t be a solution of the following Beltrami equation, defined on the whole complex place C : F ρ t F ρ t (z) = ν ρ t (z). We normalize the solution s.t. z F ρ t (z) 1 Lp, F ρ t (0) = 0.

54 Stochastic conformal welding Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Theorem (Airault, Malliavin, Thalmaier) Let Ψ t the solution of equation ( ), i.e. the extension of g t in the unit disk. Define Then f ρ t g ρ t ρ (z) := Ft ( Ψ t ) 1 (z), z Ψ t (D ρ ), ρ (z) := Ft (z), z / Ψ t (D ρ ). f ρ t is holomorphic and univalent on Ψ t (D ρ ), ) c is holomorphic and univalent on ( Ψt (D ρ ), g ρ t and (f ρ t ) 1 g ρ t (z) = Ψ t (z), z D ρ.

55 Towards Brownian motion on U Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process We can now let ρ go to 1... Theorem (Airault, Malliavin, Thalmaier) For each 0 < r < 1, the following limit φ t (z) := lim ρ 1 f ρ t exists uniformly in z D r and it defines an univalent function φ t in the unit disk D.

56 A factorization of g t or almost Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Consider the following extra assumption : (H) φ t is continuous and injective on D. Theorem (Airault, Malliavin, Thalmaier) Suppose that the function φ t satisfies (H), then there exists a function h t univalent outside the unit disk D such that : ( (φ t ) 1 h t e iθ) = g t (e iθ), where g t is the solution of ( ), i.e. the Brownian motion on (S 1 ).

57 Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Some properties of the resulting process

58 Area of the random domain Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Question : what is the growth of φ t (D)? D φ t(d) Theorem (Airault, Malliavin, Thalmaier) Let A ρ := Ö (F ρ t t (D ρ)). Then, there exist constants c 1, c 2, c 3, independent of ρ < 1 such that ( ) P sup log(a ρ t ) c R 1T > c 2 + R exp ( c 2 ) 3. t [0,T] T t

59 Area of the random domain Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Question : what is the growth of φ t (D)? D φ t(d) Let A t = Ö (φ t Theorem (Airault, Malliavin, Thalmaier) P ( (D)). Then, there exist constants c 1, c 2, c 3 s.t. ) R exp ( c 2 ) 3. T sup log(a t ) c 1 T > c 2 + R t [0,T] t

60 A diffusion on Jordan curves Beurling-Ahlfors extension Stochastic conformal welding Some properties of the resulting process Theorem (Airault, Malliavin, Thalmaier) Let φ t be the stochastic flow of univalent functions defined above. Then t φ t (S 1 ) defines a Markov process with values in J, the space of Jordan curves.