Fourier transforms of measures on the Brownian graph Jonathan M. - PowerPoint PPT Presentation

Fourier transforms of measures on the Brownian graph Jonathan M. Fraser The University of Manchester Joint work with Tuomas Sahlsten (Bristol, UK) and Tuomas Orponen (Helsinki, Finland) ICERM 10th March 2016 Jonathan M. Fraser Fourier transforms

My co-authors Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform of a measure µ on R d is a function ˆ µ : R d → C defined by � ˆ µ ( x ) = exp ( − 2 π i x · y ) d µ ( y ) . Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. dim H K = sup { s � 0 : ∃ µ on K such that I s ( µ ) < ∞} where �� d µ ( x ) d µ ( y ) I s ( µ ) = | x − y | s Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. dim H K = sup { s � 0 : ∃ µ on K such that I s ( µ ) < ∞} where �� d µ ( x ) d µ ( y ) I s ( µ ) = | x − y | s or (using Parseval and convolution formulae) � µ ( x ) | 2 | x | s − d dx I s ( µ ) = C ( s , d ) R d | ˆ (0 < s < d ) Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. dim H K = sup { s � 0 : ∃ µ on K such that I s ( µ ) < ∞} where �� d µ ( x ) d µ ( y ) I s ( µ ) = | x − y | s or (using Parseval and convolution formulae) � µ ( x ) | 2 | x | s − d dx I s ( µ ) = C ( s , d ) R d | ˆ (0 < s < d ) . . . and so if µ is supported on K , then µ ( x ) | � | x | − s / 2 | ˆ Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. dim H K = sup { s � 0 : ∃ µ on K such that I s ( µ ) < ∞} where �� d µ ( x ) d µ ( y ) I s ( µ ) = | x − y | s or (using Parseval and convolution formulae) � µ ( x ) | 2 | x | s − d dx I s ( µ ) = C ( s , d ) R d | ˆ (0 < s < d ) . . . and so if µ is supported on K , then µ ( x ) | � | x | − s / 2 ⇒ I s − ( µ ) < ∞ | ˆ Jonathan M. Fraser Fourier transforms

Fourier transforms and dimension The Fourier transform gives much geometric information about the measure. dim H K = sup { s � 0 : ∃ µ on K such that I s ( µ ) < ∞} where �� d µ ( x ) d µ ( y ) I s ( µ ) = | x − y | s or (using Parseval and convolution formulae) � µ ( x ) | 2 | x | s − d dx I s ( µ ) = C ( s , d ) R d | ˆ (0 < s < d ) . . . and so if µ is supported on K , then µ ( x ) | � | x | − s / 2 ⇒ I s − ( µ ) < ∞ ⇒ dim H K � s | ˆ Jonathan M. Fraser Fourier transforms

Simple example For example, if µ is Lebesgue measure on the unit interval, then a quick calculation reveals that for x ∈ R � � 1 � � � 1 � � π | x | − 1 . | ˆ µ ( x ) | = exp ( − 2 π ixy ) dy � � � 0 Jonathan M. Fraser Fourier transforms

Examples with no decay Jonathan M. Fraser Fourier transforms

Examples with no decay The middle 3rd Cantor set also supports no measures with Fourier decay! Jonathan M. Fraser Fourier transforms

Fourier dimension How much dimension can be realised by Fourier decay? Jonathan M. Fraser Fourier transforms

Fourier dimension How much dimension can be realised by Fourier decay? � µ ( x ) | � | x | − s / 2 � dim F K = sup s � 0 : ∃ µ on K such that | ˆ Jonathan M. Fraser Fourier transforms

Fourier dimension How much dimension can be realised by Fourier decay? � µ ( x ) | � | x | − s / 2 � dim F K = sup s � 0 : ∃ µ on K such that | ˆ dim F K � dim H K Sets with equality are called Salem sets . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The image of a Cantor set K ⊆ R under Brownian motion is a random fractal: B ( K ) = { B ( x ) : x ∈ K } . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The image of a Cantor set K ⊆ R under Brownian motion is a random fractal: B ( K ) = { B ( x ) : x ∈ K } . • McKean proved in 1955 that ‘Brownian motion doubles dimension’, i.e. a . s . dim H B ( K ) = min { 2 dim H K , 1 } . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The image of a Cantor set K ⊆ R under Brownian motion is a random fractal: B ( K ) = { B ( x ) : x ∈ K } . • McKean proved in 1955 that ‘Brownian motion doubles dimension’, i.e. a . s . dim H B ( K ) = min { 2 dim H K , 1 } . • Kahane proved in 1966 that such image sets are almost surely Salem. Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The level sets of Brownian motion are random fractals: L y ( B ) = B − 1 ( y ) = { x ∈ R : B ( x ) = y } . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The level sets of Brownian motion are random fractals: L y ( B ) = B − 1 ( y ) = { x ∈ R : B ( x ) = y } . • Taylor/Perkins proved in 1955/1981 that a . s . ∗ dim H L y ( B ) = 1 / 2 . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The level sets of Brownian motion are random fractals: L y ( B ) = B − 1 ( y ) = { x ∈ R : B ( x ) = y } . • Taylor/Perkins proved in 1955/1981 that a . s . ∗ dim H L y ( B ) = 1 / 2 . • Kahane proved in 1983 that such level sets are almost surely ∗ Salem. Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The graph of Brownian motion is a random fractal: G ( B ) = { ( x , B ( x )) : x ∈ R } . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The graph of Brownian motion is a random fractal: G ( B ) = { ( x , B ( x )) : x ∈ R } . • Taylor proved in 1953 that a . s . dim H G ( B ) = 3 / 2 . Jonathan M. Fraser Fourier transforms

Classical results on Brownian motion The graph of Brownian motion is a random fractal: G ( B ) = { ( x , B ( x )) : x ∈ R } . • Taylor proved in 1953 that a . s . dim H G ( B ) = 3 / 2 . • It remained open for a long time whether or not graphs are almost surely Salem. • Kahane explicitly asked the question in 1993 (also asked by Shieh-Xiao in 2006). Jonathan M. Fraser Fourier transforms

Our work on Brownian motion Theorem (F-Orponen-Sahlsten, IMRN ‘14) The graph of Brownian motion is almost surely not a Salem set. Jonathan M. Fraser Fourier transforms

Our work on Brownian motion Theorem (F-Orponen-Sahlsten, IMRN ‘14) The graph of Brownian motion is almost surely not a Salem set. In fact, there does not exist a measure µ supported on a graph which satsfies: µ ( x ) | � | x | − s / 2 | ˆ a . s . for any s > 1 . Therefore dim F G ( B ) � 1 < 3 / 2 = dim H G ( B ) . Jonathan M. Fraser Fourier transforms

Our work on Brownian motion Theorem (F-Orponen-Sahlsten, IMRN ‘14) The graph of Brownian motion is almost surely not a Salem set. In fact, there does not exist a measure µ supported on a graph which satsfies: µ ( x ) | � | x | − s / 2 | ˆ a . s . for any s > 1 . Therefore dim F G ( B ) � 1 < 3 / 2 = dim H G ( B ) . • Key idea : we proved a new slicing theorem for planar sets supporting measures with fast Fourier decay. • the answer to Kahane’s problem is geometric (not stochastic). Jonathan M. Fraser Fourier transforms

Slicing theorems Theorem (Marstrand) Suppose K ⊂ R 2 has dim H K > 1 , then for almost all directions θ ∈ S 1 , Lebesgue positively many y ∈ R satisfy: dim H K ∩ L θ, y > 0 . Jonathan M. Fraser Fourier transforms

Slicing theorems Theorem (Marstrand) Suppose K ⊂ R 2 has dim H K > 1 , then for almost all directions θ ∈ S 1 , Lebesgue positively many y ∈ R satisfy: dim H K ∩ L θ, y > 0 . Theorem (F-Orponen-Sahlsten, IMRN ‘14) Suppose K ⊂ R 2 has dim F K > 1 , then for all directions θ ∈ S 1 , Lebesgue positively many y ∈ R satisfy: dim H K ∩ L θ, y > 0 . Jonathan M. Fraser Fourier transforms

Our work on Brownian motion • Brownian graphs are not Salem, but what is their Fourier dimension? Jonathan M. Fraser Fourier transforms

Our work on Brownian motion • Brownian graphs are not Salem, but what is their Fourier dimension? Theorem (F-Sahlsten, 2015) Let µ be the push forward of Lebesgue measure on the graph of Brownian motion. Then almost surely µ ( x ) | � | x | − 1 / 2 � ( x ∈ R 2 ) | ˆ log | x | a . s . and, in particular, dim F G ( B ) = 1 . Jonathan M. Fraser Fourier transforms

Our work on Brownian motion • Brownian graphs are not Salem, but what is their Fourier dimension? Theorem (F-Sahlsten, 2015) Let µ be the push forward of Lebesgue measure on the graph of Brownian motion. Then almost surely µ ( x ) | � | x | − 1 / 2 � ( x ∈ R 2 ) | ˆ log | x | a . s . and, in particular, dim F G ( B ) = 1 . This time our proof was stochastic (not geometric) and relied on techniques from Itˆ o calculus , as well as adapting some of Kahane’s ideas. Jonathan M. Fraser Fourier transforms