Large deviations for Brownian intersection measures Chiranjib - PowerPoint PPT Presentation

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large deviations for Brownian intersection measures Chiranjib Mukherjee Prague, September, 2011

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Brownian intersection Brownian paths do intersect W 1 , . . . , W p independent Brownian motions in R d running until times t 1 , . . . , t p . Typically, here d ≥ 2.

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Brownian intersection Brownian paths do intersect W 1 , . . . , W p independent Brownian motions in R d running until times t 1 , . . . , t p . Typically, here d ≥ 2. Look at their path intersections : p � t = ( t 1 , . . . , t p ) ∈ (0 , ∞ ) p S t = W i [0 , t i ) i =1

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Brownian intersection Brownian paths do intersect W 1 , . . . , W p independent Brownian motions in R d running until times t 1 , . . . , t p . Typically, here d ≥ 2. Look at their path intersections : p � t = ( t 1 , . . . , t p ) ∈ (0 , ∞ ) p S t = W i [0 , t i ) i =1 Dvoretzky, Erd¨ os, Kakutani and Taylor showed S t is non-empty with positive probability iff  d = 2 , p ∈ N   d = 3 , p = 2  d ≥ 4 , p = 1 . 

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections A measure is naturally defined on S t : p � t i � � ds δ y ( W i ( s )) A ⊂ R d ℓ t ( A ) = dy 0 A i =1

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections A measure is naturally defined on S t : p � t i � � ds δ y ( W i ( s )) A ⊂ R d ℓ t ( A ) = dy 0 A i =1 For p = 1, ℓ t is the single path occupation measure: � t ℓ ( i ) t ( A ) = ds 1 A ( W s ) i = 1 , . . . , p 0

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections A measure is naturally defined on S t : p � t i � � ds δ y ( W i ( s )) A ⊂ R d ℓ t ( A ) = dy 0 A i =1 For p = 1, ℓ t is the single path occupation measure: � t ℓ ( i ) t ( A ) = ds 1 A ( W s ) i = 1 , . . . , p 0 Note: If ℓ ( i ) would have a Lebesgue density ℓ ( i ) t ( y ), so would t ℓ t and p ℓ ( i ) � ℓ t ( y ) = t ( y ) i =1

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections A measure is naturally defined on S t : p � t i � � ds δ y ( W i ( s )) A ⊂ R d ℓ t ( A ) = dy 0 A i =1 For p = 1, ℓ t is the single path occupation measure: � t ℓ ( i ) t ( A ) = ds 1 A ( W s ) i = 1 , . . . , p 0 Note: If ℓ ( i ) would have a Lebesgue density ℓ ( i ) t ( y ), so would t ℓ t and p ℓ ( i ) � ℓ t ( y ) = t ( y ) makes sense only in d = 1! i =1

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Intersection measure Intensity of the intersections A measure is naturally defined on S t : p � t i � � ds δ y ( W i ( s )) A ⊂ R d ℓ t ( A ) = dy 0 A i =1 For p = 1, ℓ t is the single path occupation measure: � t ℓ ( i ) t ( A ) = ds 1 A ( W s ) i = 1 , . . . , p 0 Note: If ℓ ( i ) would have a Lebesgue density ℓ ( i ) t ( y ), so would t ℓ t and p ℓ ( i ) � ℓ t ( y ) = t ( y ) makes sense only in d = 1! i =1 Goal: Make precise the above as t ↑ ∞ (in particular, d ≥ 2).

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Wiener Sausages Construction of intersection measure

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Wiener Sausages Construction of intersection measure Le Gall (1986) looked at Wiener sausages: ǫ, t = { x ∈ R d : | x − W i ( r i ) | < ǫ } S ( i ) i = 1 , . . . , p

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Wiener Sausages Construction of intersection measure Le Gall (1986) looked at Wiener sausages: ǫ, t = { x ∈ R d : | x − W i ( r i ) | < ǫ } S ( i ) i = 1 , . . . , p Normalise Lebesgue measure on the intersection of the sausages d ℓ ǫ, t ( y ) = s d ( ǫ ) 1 ∩ p ǫ , t ( y ) dy i =1 S ( i ) where

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Wiener Sausages Construction of intersection measure Le Gall (1986) looked at Wiener sausages: ǫ, t = { x ∈ R d : | x − W i ( r i ) | < ǫ } S ( i ) i = 1 , . . . , p Normalise Lebesgue measure on the intersection of the sausages d ℓ ǫ, t ( y ) = s d ( ǫ ) 1 ∩ p ǫ , t ( y ) dy i =1 S ( i ) where π − p log p ( 1  ǫ ) if d = 2   (2 πǫ ) − 2 s d ( ǫ ) = if d = 3 and p = 2 2  ω d ( d − 2) ǫ 2 − d if d ≥ 3 and p = 1 . 

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Wiener Sausages Intersection measure: scaling limit of Lebesgue measure on sausages Le Gall shows limit ǫ ↓ 0 gives the Brownian intersection measure: ǫ → 0 ℓ ǫ, t ( A ) = ℓ t ( A ) in L q for q ∈ [1 , ∞ ) lim

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics Single path measure Look at one path. Fix i ∈ { 1 , . . . , p } .

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics Single path measure Look at one path. Fix i ∈ { 1 , . . . , p } . W i running in a compact set B in R d until its first exit time τ i from B .

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics Single path measure Look at one path. Fix i ∈ { 1 , . . . , p } . W i running in a compact set B in R d until its first exit time τ i from B . Make sure the path does not leave B by time t : P t ( · ) = P ( · ∩ { t < τ i } ).

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics Single path measure Look at one path. Fix i ∈ { 1 , . . . , p } . W i running in a compact set B in R d until its first exit time τ i from B . Make sure the path does not leave B by time t : P t ( · ) = P ( · ∩ { t < τ i } ). t ℓ ( i ) Normalise the occupation measure: 1 ∈ M 1 ( B ) t

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics Single path measure Look at one path. Fix i ∈ { 1 , . . . , p } . W i running in a compact set B in R d until its first exit time τ i from B . Make sure the path does not leave B by time t : P t ( · ) = P ( · ∩ { t < τ i } ). t ℓ ( i ) Normalise the occupation measure: 1 ∈ M 1 ( B ) t t ℓ ( i ) Want to study: Behavior of 1 t , as t ↑ ∞ .

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]:

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]: t ℓ ( i ) 1 large deviation principle (LDP) in M 1 ( B ) under P t as t ↑ ∞ : t

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]: t ℓ ( i ) 1 large deviation principle (LDP) in M 1 ( B ) under P t as t ↑ ∞ : t µ ∈ M 1 ( B ). � 1 � t ℓ ( i ) P t ≈ µ = t

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]: t ℓ ( i ) 1 large deviation principle (LDP) in M 1 ( B ) under P t as t ↑ ∞ : t µ ∈ M 1 ( B ). � 1 � t ℓ ( i ) P t ≈ µ = exp [ − t ( I ( µ ) + o (1))] t ↑ ∞ t where

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]: t ℓ ( i ) 1 large deviation principle (LDP) in M 1 ( B ) under P t as t ↑ ∞ : t µ ∈ M 1 ( B ). � 1 � t ℓ ( i ) P t ≈ µ = exp [ − t ( I ( µ ) + o (1))] t ↑ ∞ t where � � � 2 1 d µ if d µ dx ∈ H 1 � � � ∇ 0 ( B ) 2 dx I ( µ ) = 2

Brownian intersection Brownian Intersection measure Main results: Large deviations Outlook Large t -asymptotics for single path measure Path densities show up [Donsker-Varadhan (1975-83)], [G¨ artner (1977)]: t ℓ ( i ) 1 large deviation principle (LDP) in M 1 ( B ) under P t as t ↑ ∞ : t µ ∈ M 1 ( B ). � 1 � t ℓ ( i ) P t ≈ µ = exp [ − t ( I ( µ ) + o (1))] t ↑ ∞ t where � � � 2 1 d µ if d µ dx ∈ H 1 � � � ∇ 0 ( B ) 2 dx I ( µ ) = 2 ∞ else