Stochastic areas and windings Fabrice Baudoin (Joint with J. Wang) - PowerPoint PPT Presentation

Stochastic areas and windings Fabrice Baudoin (Joint with J. Wang) ICTP, Trieste September 17, 2019

Part I. Stochastic areas

The Lévy area formula Let Z t = X t + iY t , t ≥ 0, be a Brownian motion in the complex plane such that Z 0 = 0. Up to a factor 1 / 2, the algebraic area swept out by the path of Z up to time t is given by � t � S t = xdy − yx = X s dY s − Y s dX s , Z [ 0 , t ] 0

The Lévy area formula The Lévy’s area formula λ t sinh λ t e − | z | 2 � � e i λ S t | Z t = z 2 t ( λ t coth λ t − 1 ) E = was originally proved by Paul Lévy (1940) by using a series expansion of Z .

The Lévy area formula The Lévy’s area formula λ t sinh λ t e − | z | 2 � � e i λ S t | Z t = z 2 t ( λ t coth λ t − 1 ) E = was originally proved by Paul Lévy (1940) by using a series expansion of Z . The formula has numerous applications: Rough paths theory, Connections with the Riemann zeta function, Heat kernel on the Heisenberg group,...

The Lévy area formula The formula nowadays admits many different proofs. A particularly elegant probabilistic approach is due to Marc Yor.

The Lévy area formula The formula nowadays admits many different proofs. A particularly elegant probabilistic approach is due to Marc Yor. The first observation is that, due to the invariance by rotations of Z , one has for every λ ∈ R , � � � � t e − λ 2 � � 0 | Z s | 2 ds e i λ S t | Z t = z � E = E � | Z t | = | z | . 2 �

The Lévy area formula One considers then the new probability � t 2 ( | Z t | 2 − 2 t ) − λ 2 � λ � P λ | Z s | 2 ds / F t = exp P / F t 2 0 under which, thanks to Girsanov theorem, ( Z t ) t ≥ 0 is a Gaussian process (an Ornstein-Uhlenbeck process). The Lévy area formula then easily follows from standard computations on Gaussian measures.

The complex projective space CP n The complex projective space CP n can be defined as the set of complex lines in C n + 1 . To parametrize points in CP n , it is convenient to use the local inhomogeneous coordinates given by w j = z j / z n + 1 , 1 ≤ j ≤ n , z ∈ C n + 1 , z n + 1 � = 0.

The complex projective space CP n The complex projective space CP n can be defined as the set of complex lines in C n + 1 . To parametrize points in CP n , it is convenient to use the local inhomogeneous coordinates given by w j = z j / z n + 1 , 1 ≤ j ≤ n , z ∈ C n + 1 , z n + 1 � = 0. The map CP n S 2 n + 1 π : → ( z 1 , · · · , z n + 1 ) → ( w 1 , · · · , w n ) is a Riemannian submersion with totally geodesic fibers isometric to U ( 1 ) .

Brownian motion in CP n By using the submersion π , one can construct the Browian motion on CP n as � Z 1 ( t ) Z n + 1 ( t ) , · · · , Z n ( t ) � w ( t ) = ( w 1 ( t ) , · · · , w n ( t )) = Z n + 1 ( t ) where ( Z 1 ( t ) , · · · , Z n + 1 ( t )) is a Brownian motion on S 2 n + 1 .

Stochastic area in CP n Let ( w ( t )) t ≥ 0 be a Brownian motion on CP n started at 0 1 . The generalized stochastic area process of ( w ( t )) t ≥ 0 is defined by � t n w j ( s ) dw j ( s ) − w j ( s ) dw j ( s ) � α = i � θ ( t ) = , 2 1 + | w ( s ) | 2 w [ 0 , t ] 0 j = 1 where the above stochastic integrals are understood in the Stratonovitch, or equivalently in the Itô sense. 1 We call 0 the point with inhomogeneous coordinates w 1 = 0 , · · · , w n = 0

Stochastic area in CP n Let ( w ( t )) t ≥ 0 be a Brownian motion on CP n started at 0 1 . The generalized stochastic area process of ( w ( t )) t ≥ 0 is defined by � t n w j ( s ) dw j ( s ) − w j ( s ) dw j ( s ) � α = i � θ ( t ) = , 2 1 + | w ( s ) | 2 w [ 0 , t ] 0 j = 1 where the above stochastic integrals are understood in the Stratonovitch, or equivalently in the Itô sense. The form d α is the Kähler form on CP n . 1 We call 0 the point with inhomogeneous coordinates w 1 = 0 , · · · , w n = 0

Skew-product decomposition Theorem Let ( w ( t )) t ≥ 0 be a Brownian motion on CP n started at 0 and ( θ ( t )) t ≥ 0 be its stochastic area process. The S 2 n + 1 -valued diffusion process e − i θ ( t ) X t = 1 + | w ( t ) | 2 ( w ( t ) , 1 ) , t ≥ 0 � is the horizontal lift at the north pole of ( w ( t )) t ≥ 0 by the submersion π .

Skew-product decomposition Corollary Let r ( t ) = arctan | w ( t ) | . The process ( r ( t ) , θ ( t )) t ≥ 0 is a diffusion with generator � ∂ 2 ∂ r + tan 2 r ∂ 2 L = 1 ∂ r 2 + (( 2 n − 1 ) cot r − tan r ) ∂ � . ∂θ 2 2

Skew-product decomposition Corollary Let r ( t ) = arctan | w ( t ) | . The process ( r ( t ) , θ ( t )) t ≥ 0 is a diffusion with generator � ∂ 2 ∂ r + tan 2 r ∂ 2 L = 1 ∂ r 2 + (( 2 n − 1 ) cot r − tan r ) ∂ � . ∂θ 2 2 As a consequence the following equality in distribution holds � � ( r ( t ) , θ ( t )) t ≥ 0 = r ( t ) , B � t t ≥ 0 , 0 tan 2 r ( s ) ds where ( B t ) t ≥ 0 is a standard Brownian motion independent from r .

Consider the Jacobi generator � ∂ ∂ 2 �� � � � L α,β = 1 α + 1 β + 1 ∂ r 2 + cot r − tan r ∂ r , α, β > − 1 2 2 2 We denote by q α,β ( r 0 , r ) the transition density with respect to the t Lebesgue measure of the diffusion with generator L α,β . Theorem For λ ≥ 0 , r ∈ [ 0 , π/ 2 ) , and t > 0 we have � � � t e − λ 2 � e i λθ ( t ) | r ( t ) = r � 0 tan 2 r ( s ) ds | r ( t ) = r E = E 2 q n − 1 ,λ e − n λ t ( 0 , r ) t = . q n − 1 , 0 (cos r ) λ ( 0 , r ) t

Limit distribution Theorem When t → + ∞ , the following convergence in distribution takes place θ ( t ) → C n , t where C n is a Cauchy distribution with parameter n .

The complex hyperbolic space The complex hyperbolic space CH n is the open unit ball in C n .

The complex hyperbolic space The complex hyperbolic space CH n is the open unit ball in C n . Let H 2 n + 1 = { z ∈ C n + 1 , | z 1 | 2 + · · · + | z n | 2 − | z n + 1 | 2 = − 1 } be the 2 n + 1 dimensional anti-de Sitter space.

The complex hyperbolic space The complex hyperbolic space CH n is the open unit ball in C n . Let H 2 n + 1 = { z ∈ C n + 1 , | z 1 | 2 + · · · + | z n | 2 − | z n + 1 | 2 = − 1 } be the 2 n + 1 dimensional anti-de Sitter space. The map H 2 n + 1 CH n π : → � � z 1 z n ( z 1 , · · · , z n + 1 ) → z n + 1 , · · · , z n + 1 is an indefinite Riemannian submersion whose one-dimensional fibers are definite negative.

Stochastic area in CH n To parametrize CH n , we will use the global inhomogeneous coordinates given by w j = z j / z n + 1 where ( z 1 , . . . , z n ) ∈ M with k = 1 | z k | 2 − | z n + 1 | 2 < 0 } . M = { z ∈ C n , 1 , � n Definition Let ( w ( t )) t ≥ 0 be a Brownian motion on CH n started at 0 2 . The generalized stochastic area process of ( w ( t )) t ≥ 0 is defined by � t n w j ( s ) dw j ( s ) − w j ( s ) dw j ( s ) � α = i � θ ( t ) = , 2 1 − | w ( s ) | 2 w [ 0 , t ] 0 j = 1 where the above stochastic integrals are understood in the Stratonovitch sense or equivalently Itô sense. 2 We call 0 the point with inhomogeneous coordinates w 1 = 0 , · · · , w n = 0

Skew product decomposition Theorem Let ( w ( t )) t ≥ 0 be a Brownian motion on CH n started at 0 and ( θ ( t )) t ≥ 0 be its stochastic area process. The H 2 n + 1 -valued diffusion process e i θ t t ≥ 0 Y t = 1 − | w ( t ) | 2 ( w ( t ) , 1 ) , � is the horizontal lift at ( 0 , 1 ) of ( w ( t )) t ≥ 0 by the submersion π .

Skew-product decomposition Theorem Let r ( t ) = tanh − 1 | w ( t ) | . The process ( r ( t ) , θ ( t )) t ≥ 0 is a diffusion with generator � ∂ 2 ∂ r + tanh 2 r ∂ 2 � L = 1 ∂ r 2 + (( 2 n − 1 ) coth r + tanh r ) ∂ . ∂θ 2 2 As a consequence the following equality in distribution holds � � ( r ( t ) , θ ( t )) t ≥ 0 = r ( t ) , B � t t ≥ 0 , (1) 0 tanh 2 r ( s ) ds where ( B t ) t ≥ 0 is a standard Brownian motion independent from r .

Limit law Theorem When t → + ∞ , the following convergence in distribution takes place θ ( t ) √ t → N ( 0 , 1 ) where N ( 0 , 1 ) is a normal distribution with mean 0 and variance 1.

Part II. Stochastic windings

Winding form In the punctured complex plane C \ { 0 } , consider the one-form α = xdy − ydx x 2 + y 2 .

Winding form In the punctured complex plane C \ { 0 } , consider the one-form α = xdy − ydx x 2 + y 2 . For every smooth path γ : [ 0 , + ∞ ) → C \ { 0 } one has the representation � � � γ ( t ) = | γ ( t ) | exp t ≥ 0 . i α , γ [ 0 , t ]

Winding form In the punctured complex plane C \ { 0 } , consider the one-form α = xdy − ydx x 2 + y 2 . For every smooth path γ : [ 0 , + ∞ ) → C \ { 0 } one has the representation � � � γ ( t ) = | γ ( t ) | exp t ≥ 0 . i α , γ [ 0 , t ] It is therefore natural to call α the winding form around 0 since the integral of a smooth path γ along this form quantifies the angular motion of this path.

Asymptotic Brownian Winding The integral of the winding form along the paths of a two-dimensional Brownian motion Z ( t ) = X ( t ) + iY ( t ) which is not started from 0 can be defined using Itô’s calculus and yields the Brownian winding functional: � t � X ( s ) dY ( s ) − Y ( s ) dX ( s ) ζ ( t ) = α = . X ( s ) 2 + Y ( s ) 2 Z [ 0 , t ] 0