Polarity of points for systems of linear spdes in critical - PowerPoint PPT Presentation

Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang Ecole Polytechnique F´ ed´ erale de Lausanne Based on joint work with: Carl Mueller and Yimin Xiao Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 1 / 24

Overview Introduction to the problem of polarity of points Existing results for Gaussian and non-Gaussian random fields The “standard method” for non-critical dimensions Talagrand’s idea for handling critical dimensions (fBM) Our results for a class of Gaussian processes Application to systems of linear stochastic heat and wave equations in critical dimensions Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 2 / 24

Polarity of points Polarity of points for random fields Let U = ( U ( x ) , x ∈ R k ) be an R d -valued continuous stochastic process. Fix I ⊂ R k , compact with positive Lebesgue measure. The range of U over I is the random compact set U ( I ) = { U ( x ) , x ∈ I } . Question. Fix z ∈ R d . Is z hit by U , that is, P {∃ x ∈ I : U ( x ) = z } > 0 ? Polarity. If P {∃ x ∈ I : U ( x ) = z } = 0, then z is polar for U . Typically, there is a critical value Q ( k ) such that: - if d < Q ( k ), then points are not polar. - if d > Q ( k ), then points are polar. - at the critical valued d = Q ( k ): ??? Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 3 / 24

Polarity of points Polarity of points for random fields Let U = ( U ( x ) , x ∈ R k ) be an R d -valued continuous stochastic process. Fix I ⊂ R k , compact with positive Lebesgue measure. The range of U over I is the random compact set U ( I ) = { U ( x ) , x ∈ I } . Question. Fix z ∈ R d . Is z hit by U , that is, P {∃ x ∈ I : U ( x ) = z } > 0 ? Polarity. If P {∃ x ∈ I : U ( x ) = z } = 0, then z is polar for U . Typically, there is a critical value Q ( k ) such that: - if d < Q ( k ), then points are not polar. - if d > Q ( k ), then points are polar. - at the critical valued d = Q ( k ): ??? Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 3 / 24

Polarity of points Polarity of points for random fields Let U = ( U ( x ) , x ∈ R k ) be an R d -valued continuous stochastic process. Fix I ⊂ R k , compact with positive Lebesgue measure. The range of U over I is the random compact set U ( I ) = { U ( x ) , x ∈ I } . Question. Fix z ∈ R d . Is z hit by U , that is, P {∃ x ∈ I : U ( x ) = z } > 0 ? Polarity. If P {∃ x ∈ I : U ( x ) = z } = 0, then z is polar for U . Typically, there is a critical value Q ( k ) such that: - if d < Q ( k ), then points are not polar. - if d > Q ( k ), then points are polar. - at the critical valued d = Q ( k ): ??? Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 3 / 24

The Brownian sheet First example: the Brownian sheet Let ( W ( x ) , x ∈ R k + ) denote an k -parameter R d -valued Brownian sheet, that is, a centered continuous Gaussian random field W ( x ) = ( W 1 ( x ) , . . . , W d ( x )) with covariance k � E [ W i ( x ) W j ( y )] = δ i , j min( x ℓ , y ℓ ) , i , j ∈ { 1 , . . . , d } , ℓ =1 where x = ( x 1 , . . . , x k ) and y = ( y 1 , . . . , y k ). The case k = 1: Brownian motion B = ( B ( t ) , t ∈ R + ). The case k > 1: multi-parameter extension of Brownian motion. A few references: Orey & Pruitt (1973), R. Adler (1978), W. Kendall (1980), J.B. Walsh (1986), D. & Walsh (1992), Khoshnevisan & Shi (1999) D. Khoshnevisan, Multiparameter processes, Springer (2002). Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 4 / 24

The Brownian sheet First example: the Brownian sheet Let ( W ( x ) , x ∈ R k + ) denote an k -parameter R d -valued Brownian sheet, that is, a centered continuous Gaussian random field W ( x ) = ( W 1 ( x ) , . . . , W d ( x )) with covariance k � E [ W i ( x ) W j ( y )] = δ i , j min( x ℓ , y ℓ ) , i , j ∈ { 1 , . . . , d } , ℓ =1 where x = ( x 1 , . . . , x k ) and y = ( y 1 , . . . , y k ). The case k = 1: Brownian motion B = ( B ( t ) , t ∈ R + ). The case k > 1: multi-parameter extension of Brownian motion. A few references: Orey & Pruitt (1973), R. Adler (1978), W. Kendall (1980), J.B. Walsh (1986), D. & Walsh (1992), Khoshnevisan & Shi (1999) D. Khoshnevisan, Multiparameter processes, Springer (2002). Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 4 / 24

The Brownian sheet Hitting probabilities for the Brownian sheet Let ( W ( x ) , x ∈ R k + ) denote a k -parameter R d -valued Brownian sheet. Theorem 1 (Khoshnevisan and Shi, 1999) Fix M > 0 and 0 < a ℓ < b ℓ < ∞ ( ℓ = 1 , . . . , k). Let ( ⊂ R k ) . I = [ a 1 , b 1 ] × · · · × [ a k , b k ] There exists 0 < C < ∞ such that for all compact sets A ⊂ B (0 , M ) ( ⊂ R d ), 1 C Cap d − 2 k ( A ) � P { W ( I ) ∩ A � = ∅} � C Cap d − 2 k ( A ) . (see also F. Hirsch and S. Song (1991, 1995). Example. A = { z } . � 1 if d < 2 k , Cap d − 2 k ( { z } ) = 0 if d � 2 k , so points are polar in the critical dimension d = 2 k . Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 5 / 24

The Brownian sheet Hitting probabilities for the Brownian sheet Let ( W ( x ) , x ∈ R k + ) denote a k -parameter R d -valued Brownian sheet. Theorem 1 (Khoshnevisan and Shi, 1999) Fix M > 0 and 0 < a ℓ < b ℓ < ∞ ( ℓ = 1 , . . . , k). Let ( ⊂ R k ) . I = [ a 1 , b 1 ] × · · · × [ a k , b k ] There exists 0 < C < ∞ such that for all compact sets A ⊂ B (0 , M ) ( ⊂ R d ), 1 C Cap d − 2 k ( A ) � P { W ( I ) ∩ A � = ∅} � C Cap d − 2 k ( A ) . (see also F. Hirsch and S. Song (1991, 1995). Example. A = { z } . � 1 if d < 2 k , Cap d − 2 k ( { z } ) = 0 if d � 2 k , so points are polar in the critical dimension d = 2 k . Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 5 / 24

Other Gaussian processes Anisotropic Gaussian random fields (Bierm´ e, Lacaux & Xiao, 2007) Let ( V ( x ) , x ∈ R k ) be a centered continuous Gaussian random field with values in R d with i.i.d. components: V ( x ) = ( V 1 ( x ) , . . . , V d ( x )). Set ∆( x , y ) = � V 1 ( x ) − V 1 ( y ) � L 2 Let I be a “rectangle”. Assume the two conditions: (C1) There exists 0 < c < ∞ and H 1 , . . . , H k ∈ ]0 , 1[ such that for all x ∈ I , c − 1 � ∆(0 , x ) � c , and for all x , y ∈ I , k k c − 1 � | x j − y j | H j � ∆( x , y ) � c � | x j − y j | H j j =1 j =1 ( H j is the H¨ older exponent for coordinate j ). (C2) There is c > 0 such that for all x , y ∈ I , k | x j − y j | 2 H j . � Var( V 1 ( y ) | V 1 ( x )) � c j =1 Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 6 / 24

Other Gaussian processes Anisotropic Gaussian random fields (Bierm´ e, Lacaux & Xiao, 2007) Let ( V ( x ) , x ∈ R k ) be a centered continuous Gaussian random field with values in R d with i.i.d. components: V ( x ) = ( V 1 ( x ) , . . . , V d ( x )). Set ∆( x , y ) = � V 1 ( x ) − V 1 ( y ) � L 2 Let I be a “rectangle”. Assume the two conditions: (C1) There exists 0 < c < ∞ and H 1 , . . . , H k ∈ ]0 , 1[ such that for all x ∈ I , c − 1 � ∆(0 , x ) � c , and for all x , y ∈ I , k k c − 1 � | x j − y j | H j � ∆( x , y ) � c � | x j − y j | H j j =1 j =1 ( H j is the H¨ older exponent for coordinate j ). (C2) There is c > 0 such that for all x , y ∈ I , k | x j − y j | 2 H j . � Var( V 1 ( y ) | V 1 ( x )) � c j =1 Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 6 / 24

Other Gaussian processes Anisotropic Gaussian random fields (Bierm´ e, Lacaux & Xiao, 2007) Let ( V ( x ) , x ∈ R k ) be a centered continuous Gaussian random field with values in R d with i.i.d. components: V ( x ) = ( V 1 ( x ) , . . . , V d ( x )). Set ∆( x , y ) = � V 1 ( x ) − V 1 ( y ) � L 2 Let I be a “rectangle”. Assume the two conditions: (C1) There exists 0 < c < ∞ and H 1 , . . . , H k ∈ ]0 , 1[ such that for all x ∈ I , c − 1 � ∆(0 , x ) � c , and for all x , y ∈ I , k k c − 1 � | x j − y j | H j � ∆( x , y ) � c � | x j − y j | H j j =1 j =1 ( H j is the H¨ older exponent for coordinate j ). (C2) There is c > 0 such that for all x , y ∈ I , k | x j − y j | 2 H j . � Var( V 1 ( y ) | V 1 ( x )) � c j =1 Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 6 / 24

Other Gaussian processes Anisotropic Gaussian fields Theorem 2 (Bierm´ e, Lacaux & Xiao, 2007) Fix M > 0 . Set k 1 � Q = H j . j =1 Then there is 0 < C < ∞ such that for every compact set A ⊂ B (0 , M ) , C − 1 Cap d − Q ( A ) � P { V ( I ) ∩ A � = ∅} � C H d − Q ( A ) . Example. A = { z }   1 if d < Q , ∞ if d < Q ,   Cap d − Q ( { z } ) = 0 if d = Q , H d − Q ( { z } ) = 1 if d = Q , 0 if d > Q , 0 if d > Q .   If d = Q , Theorem 2 says: 0 � P {∃ x ∈ I : V ( x ) = z } � 1 (not informative)! Polarity of points for systems of linear spde’s in critical dimensions Robert C. Dalang 7 / 24