Generating functionals on quantum groups Adam Skalski 1 Ami Viselter ⋆ ,2 1 IMPAN, Warsaw 2 University of Haifa Quantum Groups and their Analysis University of Oslo August 8, 2019 Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 1 / 25
Convolution semigroups of measures G – locally compact group Convolution of measures For positive Borel measures µ, ν on G , define their convolution µ ⋆ ν by �� � � ( µ ⋆ ν )( A ) := I A ( gh ) d µ ( g ) d ν ( h ) ( ∀ measurable A ) . G G � For convenience, write µ ( f ) := G f d µ. Definition A convolution semigroup of probability measures on G is a family ( µ t ) t ≥ 0 of probability measures on G satisfying µ 0 = δ e µ s ⋆ µ t = µ s + t ( ∀ s , t ≥ 0 ) . and It is w ∗ -continuous if µ t ( f ) − − t → 0 + µ 0 ( f ) = f ( e ) for all f ∈ C 0 ( G ) . − − → It is symmetric if every µ t is invariant under inversion. Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25
Convolution semigroups of measures G – locally compact group Convolution of measures For positive Borel measures µ, ν on G , define their convolution µ ⋆ ν by �� � � ( µ ⋆ ν )( A ) := I A ( gh ) d µ ( g ) d ν ( h ) ( ∀ measurable A ) . G G � For convenience, write µ ( f ) := G f d µ. Definition A convolution semigroup of probability measures on G is a family ( µ t ) t ≥ 0 of probability measures on G satisfying µ 0 = δ e µ s ⋆ µ t = µ s + t ( ∀ s , t ≥ 0 ) . and It is w ∗ -continuous if µ t ( f ) − − t → 0 + µ 0 ( f ) = f ( e ) for all f ∈ C 0 ( G ) . − − → It is symmetric if every µ t is invariant under inversion. Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25
Convolution semigroups of measures G – locally compact group Convolution of measures For positive Borel measures µ, ν on G , define their convolution µ ⋆ ν by �� � � ( µ ⋆ ν )( A ) := I A ( gh ) d µ ( g ) d ν ( h ) ( ∀ measurable A ) . G G � For convenience, write µ ( f ) := G f d µ. Definition A convolution semigroup of probability measures on G is a family ( µ t ) t ≥ 0 of probability measures on G satisfying µ 0 = δ e µ s ⋆ µ t = µ s + t ( ∀ s , t ≥ 0 ) . and It is w ∗ -continuous if µ t ( f ) − − t → 0 + µ 0 ( f ) = f ( e ) for all f ∈ C 0 ( G ) . − − → It is symmetric if every µ t is invariant under inversion. Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25
Convolution semigroups of measures G – locally compact group Convolution of measures For positive Borel measures µ, ν on G , define their convolution µ ⋆ ν by �� � � ( µ ⋆ ν )( A ) := I A ( gh ) d µ ( g ) d ν ( h ) ( ∀ measurable A ) . G G � For convenience, write µ ( f ) := G f d µ. Definition A convolution semigroup of probability measures on G is a family ( µ t ) t ≥ 0 of probability measures on G satisfying µ 0 = δ e µ s ⋆ µ t = µ s + t ( ∀ s , t ≥ 0 ) . and It is w ∗ -continuous if µ t ( f ) − − t → 0 + µ 0 ( f ) = f ( e ) for all f ∈ C 0 ( G ) . − − → It is symmetric if every µ t is invariant under inversion. Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25
Convolution semigroups of measures Probabilistic importance w ∗ -cont. conv. semigroups of ←→ G -valued Lévy processes prob. measures on G Definition A G -valued Lévy process is a family X = ( X t ) t ≥ 0 of random variables from a probability space to G such that: X 0 = 0; 1 X has independent and stationary increments; 2 X is continuous. 3 Given a Lévy process X , define ( µ t ) t ≥ 0 to be its family of distributions: for t ≥ 0, µ t is the probability measure on G defined by µ t := P ◦ X − 1 . t Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25
Convolution semigroups of measures Probabilistic importance w ∗ -cont. conv. semigroups of ←→ G -valued Lévy processes prob. measures on G Definition A G -valued Lévy process is a family X = ( X t ) t ≥ 0 of random variables from a probability space to G such that: X 0 = 0; 1 X has independent and stationary increments; 2 X is continuous. 3 Given a Lévy process X , define ( µ t ) t ≥ 0 to be its family of distributions: for t ≥ 0, µ t is the probability measure on G defined by µ t := P ◦ X − 1 . t Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25
Convolution semigroups of measures Probabilistic importance w ∗ -cont. conv. semigroups of ←→ G -valued Lévy processes prob. measures on G Definition A G -valued Lévy process is a family X = ( X t ) t ≥ 0 of random variables from a probability space to G such that: X 0 = 0; 1 X has independent and stationary increments; 2 X is continuous. 3 Given a Lévy process X , define ( µ t ) t ≥ 0 to be its family of distributions: for t ≥ 0, µ t is the probability measure on G defined by µ t := P ◦ X − 1 . t Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25
Convolution semigroups of measures ( µ t ) t ≥ 0 – w ∗ -cont. conv. semigroup of prob. measures on G . Definition The generating functional of ( µ t ) t ≥ 0 is the functional γ on C 0 ( G ) given by µ t ( f ) − µ 0 ( f ) γ ( f ) := lim t t → 0 + with domain consisting of all f ∈ C 0 ( G ) s.t. the limit exists. What can we say about D ( γ ) ? Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 4 / 25
Convolution semigroups of measures ( µ t ) t ≥ 0 – w ∗ -cont. conv. semigroup of prob. measures on G . Definition The generating functional of ( µ t ) t ≥ 0 is the functional γ on C 0 ( G ) given by µ t ( f ) − µ 0 ( f ) γ ( f ) := lim t t → 0 + with domain consisting of all f ∈ C 0 ( G ) s.t. the limit exists. What can we say about D ( γ ) ? Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 4 / 25
Convolution semigroups of measures Elementary C 0 -semigroup tools � D ( γ ) is dense in C 0 ( G ) . Theorem (Hunt, Hazod–Siebert) D ( γ ) contains a dense subalgebra of C 0 ( G ) . G – Lie group: C 2 0 ( G ) ⊆ D ( γ ) . Generally: C ∞ c ( G ) ⊆ D ( γ ) . The form of γ is completely understood – the Lévy–Khintchine formula: When G = R n , there exist a positive matrix a ∈ M n ( R ) , b ∈ R n and a R n \{ 0 } min( � y � 2 , 1 ) d ν ( y ) < ∞ s.t. for � positive measure on R n \ { 0 } with all f ∈ C 2 0 ( G ) , n n ∂ 2 f ∂ f � � γ ( f ) = a ij ( 0 ) + b k ( 0 ) ∂ x i ∂ x j ∂ x k i , j = 1 k = 1 n � ∂ f � + f − f ( 0 ) − χ B 1 ( 0 ) ( y ) ( 0 ) y ℓ d ν ( y ) . ∂ x ℓ R n \{ 0 } ℓ = 1 Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25
Convolution semigroups of measures Elementary C 0 -semigroup tools � D ( γ ) is dense in C 0 ( G ) . Theorem (Hunt, Hazod–Siebert) D ( γ ) contains a dense subalgebra of C 0 ( G ) . G – Lie group: C 2 0 ( G ) ⊆ D ( γ ) . Generally: C ∞ c ( G ) ⊆ D ( γ ) . The form of γ is completely understood – the Lévy–Khintchine formula: When G = R n , there exist a positive matrix a ∈ M n ( R ) , b ∈ R n and a R n \{ 0 } min( � y � 2 , 1 ) d ν ( y ) < ∞ s.t. for � positive measure on R n \ { 0 } with all f ∈ C 2 0 ( G ) , n n ∂ 2 f ∂ f � � γ ( f ) = a ij ( 0 ) + b k ( 0 ) ∂ x i ∂ x j ∂ x k i , j = 1 k = 1 n � ∂ f � + f − f ( 0 ) − χ B 1 ( 0 ) ( y ) ( 0 ) y ℓ d ν ( y ) . ∂ x ℓ R n \{ 0 } ℓ = 1 Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25
Convolution semigroups of measures Elementary C 0 -semigroup tools � D ( γ ) is dense in C 0 ( G ) . Theorem (Hunt, Hazod–Siebert) D ( γ ) contains a dense subalgebra of C 0 ( G ) . G – Lie group: C 2 0 ( G ) ⊆ D ( γ ) . Generally: C ∞ c ( G ) ⊆ D ( γ ) . The form of γ is completely understood – the Lévy–Khintchine formula: When G = R n , there exist a positive matrix a ∈ M n ( R ) , b ∈ R n and a R n \{ 0 } min( � y � 2 , 1 ) d ν ( y ) < ∞ s.t. for � positive measure on R n \ { 0 } with all f ∈ C 2 0 ( G ) , n n ∂ 2 f ∂ f � � γ ( f ) = a ij ( 0 ) + b k ( 0 ) ∂ x i ∂ x j ∂ x k i , j = 1 k = 1 n � ∂ f � + f − f ( 0 ) − χ B 1 ( 0 ) ( y ) ( 0 ) y ℓ d ν ( y ) . ∂ x ℓ R n \{ 0 } ℓ = 1 Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25
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