Truncated Taylor approximation of Loewner dynamics Supervised by - PowerPoint PPT Presentation

Truncated Taylor approximation of Loewner dynamics Supervised by Prof. Dmitry Belyaev and Prof. Terry Lyons Vlad Margarint Dept. of Mathematics, University of Oxford vlad.margarint@maths.ox.ac.uk Berlin WIAS August 2016 Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 1 / 23

Overview Introduction to SLE 1 The Rough Paths approach: Explicit truncated Taylor approximation 2 Perspectives 3 References 4 Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 2 / 23

Conformal maps Examples of conformal maps from upper halfplane with a slit to the upper halfplane H . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 3 / 23

Conformal maps and the Loewner equation In general, for a non-self crossing curve γ ( t ) : [0 , ∞ ) → ¯ H with γ (0) = 0 and γ ( ∞ ) = ∞ , we consider the simply connected domain H \ γ ([0 , t ]). Using the Riemann Mapping Theorem for the simply connected domain H \ γ ([0 , t ]), we have a three real parameter family of conformal maps g t : H \ γ ([0 , t ]) → H . Loewner Equation encodes the dependence between the evolution of the maps g t when the curve γ ([0 , t ]) grows. Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 4 / 23

Description of the conformal maps Setting the behaviour of the mapping at ∞ as g t ( ∞ ) = ∞ and g ′ t ( ∞ ) = 1, we write the Laurent expansion at ∞ of g t as g t ( z ) = z + b 0 + b 1 z + b 2 z 2 + . . . We fix the third paramater by choosing b 0 = 0 . The coefficient b 1 = b 1 ( γ ([0 , t ])) is called the half-plane capacity of γ ( t ) and is proved to be an additive, continous and increasing function. Hence, by reparametrizing the curve γ ( t ) such that b 1 ( γ ([0 , t ])) = 2 t , we obtain g t ( z ) = z + 2 t z + . . . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 5 / 23

Conformal maps and the Loewner equation Is there a way to use g t to find g t + dt ? In order to answer this question, we have to describe to find a way to describe the mapping m t , dt : H \ g t ( γ [ t , t + dt ]) → H . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 6 / 23

The Loewner equation and the square root map The square root map that we investigated in the beginning gives the description of the ’infinitesimal mapping’ m t , dt . ( z − U t ) 2 + 2 dt ≈ z + � 2 dt Heuristically, m t , dt ( z ) = U t + dt + z − U t . 2 dt Furthermore, g t + dt ( z ) ≈ g t ( z ) + g t ( z ) − U t . We obtain the Loewner Differential Equation 2 ∂ t g t ( z ) = , g 0 ( z ) = z . g t ( z ) − U t Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 7 / 23

Loewner equation and random curves in the upper half-plane So far, we adopted the perspective that given the curve γ t , the conformal maps g t must satisfy 2 ∂ t g t ( z ) = , g 0 ( z ) = z . g t ( z ) − U t with U t = g ( γ ( t )) . From now on, we take the dual perspective. Given the driving function U t : [0 , ∞ ) → R , we determine g t . Then, the maps g t determine the curve γ ( t ) . To output random continous curves, U t has to be a random continous driver. Moreover, the random driver U t induces a law on the curves γ ( t ) . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 8 / 23

Definition of SLE and dependence on κ Definition Let B t be a standard real Brownian motion starting from 0 . The chordal SLE( κ ) is defined as the law on curves induced by the solution to the following ordinary differential equation 2 g t ( z ) − √ κ B t ∂ t g t ( z ) = , g 0 ( z ) = z . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 9 / 23

Figure: SLE(1): Credit Prof. Vincent Beffara Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 10 / 23

Figure: SLE(3.5): Credit Prof. Vincent Beffara Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 11 / 23

Figure: SLE(4.5): Credit Prof. Vincent Beffara Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 12 / 23

Figure: SLE(6): Credit Prof. Vincent Beffara Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 13 / 23

SLE phase transitions It is proved that there are two phase transitions when κ varies between 0 and ∞ . The argument uses the phase transition of the Bessel process on the real line. In order to show this, consider the process dZ t = 2 dt κ Z t − dB t , where 1 Z t := √ κ g t − B t . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 14 / 23

SLE phase transitions When started with a real initial value, the process dZ t = 2 dt κ Z t − dB t is a real valued Bessel process with parameter a = 2 κ . If κ ≤ 4, then with probability one , the hitting time of zero T x = ∞ for all non-zero x ∈ R . If κ ≥ 4, then with probability one, the hitting time of zero T x < ∞ for all non-zero x ∈ R . If 4 < κ < 8 and x < y ∈ R , then P ( T x = T y ) > 0 . If κ ≥ 8 , then with probability one, T x < T y for all reals x < y . Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 15 / 23

The Rough Paths perspective We consider the backward Loewner differential equation − 2 h t ( z ) − √ κ B t ∂ t h t ( z ) = , h 0 ( z ) = z . Figure: The images of a thin rectangle under the forward Loewner evolution (left) and backward Loewner evolution(right) for κ = 0 . Finally, we obtain the following RDE in the upper half plane: dt − √ κ dB t . dz t = − 2 z t Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 16 / 23

The Lie bracket of the two vector fields and the uncorrelated diffusions We study an approximation to the solution of the RDE dt − √ κ dB t . dz t = − 2 z t Remark ∂ x ] = − 2 √ κ ∂ y , √ κ ∂ 2 y For z = x + iy , we have that [ − 2 x x 2 + y 2 ∂ x 2 + y 2 ∂ ∂ x + . z 2 Proposition Let ǫ > 0 . At space scale ǫ and time scale ǫ 2 the increment of the horizontal Brownian motion B t and the increment of the area process between t and B t are uncorrelated. Moreover, they give the same order contribution in the approximation. Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 17 / 23

The field of ellipses We consider the field of ellipses associated with this diffusion. Note that these ellipses should be shifted along the drift. At this specific scales the directions and lengths of the axes are computed explicitly in terms of the argument θ and the parameter κ . Figure: A schematic representation of the field of ellipses. The drift direction is represented in green. Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 18 / 23

Explicit dynamics and local truncation error up to the second level Proposition Fix ǫ > 0 . The truncated second level order Taylor approximation ˜ z t of the Loewner RDE started from | z 0 | = ǫ , at time ǫ 2 > 0 is an explicit function of κ , z 0 and ǫ . Moreover, the local truncation error of the truncated Taylor approximation is O ( ǫ ) . Important: the contribution of the second order approximation term − 2 √ κ � ǫ 2 dA t is O ( ǫ ) , Z 2 0 t �� ǫ 2 � t � | Z 0 | 2 = 1 1 ǫ 2 and A ǫ 2 = 1 is O ( ǫ 3 ). since 0 B s ds − 0 sdB s 2 Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 19 / 23

Elements of the proof The diffusive part of the approximation is described by the ellipses given by �� t � A � � u � � u � B T T = 1 , v C D v where √  �  ǫ 6 − Re 1 κǫ 2 3 κ z 2  . T =  � ǫ 6 − Im 1 0 3 κ z 2 We obtain the explicit squares of semi-axis of the ellipses a 1 , 2 ( κ, θ, ǫ ) as inverses of the solutions to � � + ctg 2 ( − 2 θ ) 1 3 3 λ 2 − λ κǫ 2 + + = 0 . κǫ 6 Im 2 1 κ 2 ǫ 8 Im 2 1 κǫ 2 z 2 z 2 Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 20 / 23

Future perspectives Compare the probability of crossing a sequence of centered annuli for the Forward Loewner evolution given by the Rough Paths approach with the one given by the typical Bessel process approach. Similarly, study the dynamics given by the Rough Paths approach on the boundary. Study in polar coordinates arg ( z t ) using the logarithm mapping. Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 21 / 23

Thank you for your attention! Vlad Margarint (University of Oxford) SLE and Rough Paths Berlin WIAS August 2016 22 / 23