An algebraic approach to stochastic duality Cristian Giardin` a - PowerPoint PPT Presentation

An algebraic approach to stochastic duality Cristian Giardin` a RAQIS18, Annecy – 14 September 2018

Collaboration with Gioia Carinci (Delft) Chiara Franceschini (Modena) Claudio Giberti (Modena) Jorge Kurchan (ENS Paris) Frank Redig (Delft) Tomohiro Sasamoto (Tokyo Tech)

Outline ◮ Introduction ◮ Lie algebraic approach to duality theory ◮ su q ( 1 , 1 ) algebra ◮ Applications

Introduction

j j N N N 0 1 1 Non-equilibrium in 1d: particle transport 1 q ◮ asymmetry 0 N 1 1 Ρ� Ρ� ◮ density reservoirs 0 N

1 1 Non-equilibrium in 1d: particle transport 1 q ◮ asymmetry 0 N 1 1 Ρ� Ρ� ◮ density reservoirs 0 N j j N N ◮ current reservoirs N 0 Carinci, De Masi, G., Presutti (2016)

Non-equilibrium in 1d: energy transport Fourier law J = κ ∇ T KMP model (1982) z = ( z 1 , . . . , z N ) ∈ R N Energies at every site: + L KMP f ( z ) = � 1 N � � � dp f ( z 1 , . . . , p ( z i + z i + 1 ) , ( 1 − p )( z i + z i + 1 ) , . . . , z N ) − f ( z ) 0 i = 1 → conductivity 0 < κ < ∞ ; model solved by duality.

Stochastic Duality Definition ( η t ) t ≥ 0 Markov process on Ω with generator L , ( ξ t ) t ≥ 0 Markov process on Ω dual with generator L dual ξ t is dual to η t with duality function D : Ω × Ω dual → R if ∀ t ≥ 0 ∀ ( η, ξ ) ∈ Ω × Ω dual E η ( D ( η t , ξ )) = E ξ ( D ( η, ξ t )) η t is self-dual if L dual = L . In terms of generators: LD ( · , ξ )( η ) = L dual D ( η, · )( ξ )

Duality for Markov chain Assume state spaces Ω , Ω dual are countable sets, then the Markov generator L is a matrix L ( η, η ′ ) s.t. � L ( η, η ′ ) ≥ 0 if η � = η ′ , L ( η, η ′ ) = 0 η ′ ∈ Ω LD ( · , ξ )( η ) = L dual D ( η, · )( ξ ) amounts to LD = DL T dual Indeed � � L ( η, η ′ ) D ( η ′ , ξ ) = L dual ( ξ, ξ ′ ) D ( η, ξ ′ ) η ′ ξ ′

Duality ◮ A useful tool ◮ interacting particle systems [Spitzer, Ligget] hydrodynamic limit [Presutti, De Masi] KPZ scaling limits [Sch¨ utz, Spohn] population genetics [Kingman] ... ◮ the dual process is simpler: “from many to few”. ◮ Questions ◮ how to find a dual process and a duality function? ◮ how to construct processes with duality? * E.g.: duality for asymmetric partial exclusion process? * E.g.: what is the right asymmetric version of KMP?

Lie algebraic approach to duality theory

Algebraic approach ⋆ Write the Markov generator in abstract form, i.e. as an element of a universal enveloping algebra of a Lie algebra. 1. Duality is related to a change of representation. Duality functions are the intertwiners. 2. Dualities are associated to symmetries. Acting with a symmetry on a duality fct. yields another duality fct. Conversely, the approach can be turned into a constructive method.

1. Change of representation

Example: Wright-Fisher diffusion and Kingman coalescence Wright-Fisher diffusion ( X ( t )) t ≥ 0 with state space [ 0 , 1 ] 2 x ( 1 − x ) ∂ 2 f L WF f ( x ) = 1 ∂ x 2 ( x ) N ( t ) = number of blocks in the Kingman coalescence at time t ≥ 0 ( L King f )( n ) = n ( n − 1 ) ( f ( n − 1 ) − f ( n )) 2

Duality Wright-Fisher / Kingman The process { X ( t ) } t ≥ 0 with generator L WF and the process { N ( t ) } t ≥ 0 with generator L King are dual on D ( x , n ) = x n , i.e. ( X ( t ) n ) = E King E WF ( x N ( t ) ) x n Indeed: 2 x ( 1 − x ) ∂ 2 L WF D ( · , n )( x ) = 1 ∂ x 2 x n = n ( n − 1 ) ( x n − 1 − x n ) 2 = n ( n − 1 ) ( D ( x , n − 1 ) − D ( x , n )) 2 = L King D ( x , · )( n )

Duality Wright-Fisher / Kingman : algebraic approach Two representations of the Heisenberg algebra: [ a , a † ] = 1  a † = x  a † e ( n ) = e ( n + 1 )   a e ( n ) = ne ( n − 1 )  a = d  dx L = 1 2 a † ( 1 − a † )( a ) 2 The abstract element L = L WF in the first representation L T = L King in the second representation Duality fct. D ( x , n ) = x n is the intertwiner: d xD ( x , n ) = D ( x , n + 1 ) dx D ( x , n ) = nD ( x , n − 1 )

2. Symmetries S : symmetry of the original Markov generator, i.e. [ L , S ] = 0 d : duality function between L and L dual − → D = Sd is also duality function Indeed LD = LSd = SLd = SdL T dual = DL T dual

“Cheap” self-duality Let µ a reversible measure: µ ( η ) L ( η, ξ ) = µ ( ξ ) L ( ξ, η ) A cheap (i.e. diagonal) self-duality is 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed L ( η, ξ ) L ( ξ, ξ ′ ) d ( η, ξ ′ ) = L ( ξ, η ) � � L ( η, η ′ ) d ( η ′ , ξ ) = = µ ( ξ ) µ ( η ) η ′ ξ ′

“Cheap” duality Let µ a invariant measure: � η µ ( η ) L ( η, ξ ) = 0 Let the dual process ( ξ t ) t ≥ 0 be the time-reversed process of ( η t ) t ≥ 0 L dual ( ξ, ξ ′ ) = µ ( ξ ) − 1 L ( ξ ′ , ξ ) µ ( ξ ′ ) A cheap (i.e. diagonal) duality is 1 d ( η, ξ ) = µ ( η ) δ η,ξ Indeed L ( η, ξ ) L dual ( ξ, ξ ′ ) d ( η, ξ ′ ) = L ( η, ξ ) � � L ( η, η ′ ) d ( η ′ , ξ ) = = µ ( ξ ) µ ( ξ ) η ′ ξ ′

Construction of Markov generators with algebraic structure i) ( Lie Algebra ): Start from a Lie algebra g . ii) ( Casimir ): Pick an element in the center of g , e.g. the Casimir C. iii) ( Co-product ): Consider a co-product ∆ : g → g ⊗ g making the algebra a bialgebra and conserving the commutation relations. iv) ( Quantum Hamiltonian ): Compute H = ∆( C ) . v) ( Symmetries ): S = ∆( X ) with X ∈ g is a symmetry of H : [ H , S ] = [∆( C ) , ∆( X )] = ∆([ C , X ]) = ∆( 0 ) = 0 . vi) ( Markov generator ): Apply a “ground state transformation” to turn H into a Markov generator L .

Quantum su q ( 1 , 1 ) algebra

q -numbers For q ∈ ( 0 , 1 ) and n ∈ N 0 introduce the q -number [ n ] q = q n − q − n q − q − 1 Remark: lim q → 1 [ n ] q = n . The first q -number’s are: [ 2 ] q = q + q − 1 , [ 3 ] q = q 2 + 1 + q − 2 , [ 0 ] q = 0 , [ 1 ] q = 1 , . . .

Quantum Lie algebra su q ( 1 , 1 ) For q ∈ ( 0 , 1 ) consider the algebra with generators K + , K − , K 0 [ K 0 , K ± ] = ± K ± , [ K + , K − ] = − [ 2 K 0 ] q where [ 2 K 0 ] q := q 2 K 0 − q − 2 K 0 q − q − 1 Irreducible representations are infinite dimensional. E.g., for n ∈ N  � K + e ( n ) [ n + 2 k ] q [ n + 1 ] q e ( n + 1 ) =  � K − e ( n ) [ n ] q [ n + 2 k − 1 ] q e ( n − 1 ) =  K o e ( n ) ( n + k ) e ( n ) = Casimir element C = [ K o ] q [ K o − 1 ] q − K + K − In this representation C e ( n ) = [ k ] q [ k − 1 ] q e ( n ) k ∈ R +

Co-product Co-product ∆ : U q ( su ( 1 , 1 )) → U q ( su ( 1 , 1 )) ⊗ 2 K ± ⊗ q − K o + q K o ⊗ K ± ∆( K ± ) = K o ⊗ 1 + 1 ⊗ K o ∆( K o ) = The co-product is an isomorphism s.t. [∆( K o ) , ∆( K ± )] = ± ∆( K ± ) [∆( K + ) , ∆( K − )] = − [ 2 ∆( K o )] q From co-associativity (∆ ⊗ 1 )∆ = ( 1 ⊗ ∆)∆ ∆ n : U q ( su ( 1 , 1 )) → U q ( su ( 1 , 1 )) ⊗ ( n + 1 ) , i.e. for n ≥ 2 i ) ⊗ K ± ∆ n − 1 ( K ± ) ⊗ q − K o n + 1 + q ∆ n − 1 ( K 0 ∆ n ( K ± ) = n + 1 ∆ n − 1 ( K o ) ⊗ 1 + 1 ⊗ n ⊗ K 0 ∆ n ( K o ) = n + 1

Quantum Hamiltonian � � ∆( C i ) = q K 0 q − K 0 K + ⊗ K − i + 1 + K − ⊗ K + i + 1 − B i ⊗ B i + 1 i i + 1 i i

Quantum Hamiltonian � � ∆( C i ) = q K 0 q − K 0 K + ⊗ K − i + 1 + K − ⊗ K + i + 1 − B i ⊗ B i + 1 i i + 1 i i ( q k + q − k )( q k − 1 + q − ( k − 1 ) ) � q K 0 i − q − K 0 � � q K 0 i + 1 − q − K 0 � B i ⊗ B i + 1 = i ⊗ i + 1 2 ( q − q − 1 ) 2 ( q k − q − k )( q k − 1 − q − ( k − 1 ) ) q K 0 i + q − K 0 q K 0 i + 1 + q − K 0 � � � � + i i + 1 ⊗ 2 ( q − q − 1 ) 2

Quantum Hamiltonian � � ∆( C i ) = q K 0 q − K 0 K + ⊗ K − i + 1 + K − ⊗ K + i + 1 − B i ⊗ B i + 1 i i + 1 i i L − 1 � 1 ⊗ ( i − 1 ) ⊗ ∆( C i ) ⊗ 1 ⊗ ( L − i − 1 ) + c q , k 1 ⊗ L � � H := i = 1 c q , k = ( q 2 k − q − 2 k )( q 2 k − 1 − q − ( 2 k − 1 ) ) � � ⊗ L i = 1 e ( 0 ) s . t . H · = 0 i ( q − q − 1 ) 2

Markov processes with su q ( 1 , 1 ) symmetry

Symmetries of H Lemma L � q K 0 1 ⊗ · · · ⊗ q K 0 ⊗ q − K 0 i + 1 ⊗ . . . ⊗ q − K 0 i − 1 ⊗ K ± K ± := L i i = 1 L � K 0 ⊗ K 0 := 1 ⊗ · · · ⊗ 1 i ⊗ 1 ⊗ · · · ⊗ 1 . � �� � � �� � i = 1 ( i − 1 ) times ( L − i ) times are symmetries of H . Proof. Let a ∈ { + , − , 0 } , then K a = ∆ L − 1 ( K a 1 ) [ H , K a ] = [∆( C 1 ) , ∆( K a 1 )] = ∆([ C 1 , K a For L = 2 : 1 ]) = ∆( 0 ) = 0 For L > 2 : induction .

Ground state transformation Lemma Let H be a matrix with H ( η, η ′ ) ≥ 0 if η � = η ′ . Suppose g is a positive ground state, i.e. H g = 0 and g ( η ) > 0. Let G be the matrix G ( η, η ′ ) = g ( η ) δ η,η ′ . Then L = G − 1 H G is a Markov generator. Indeed L ( η, η ′ ) = H ( η, η ′ ) g ( η ′ ) g ( η ) Therefore � L ( η, η ′ ) ≥ 0 η � = η ′ L ( η, η ′ ) = 0 if η ′