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Duality of Markov processes with respect to a function Sabine Jansen Ruhr-Universität Bochum, Germany joint work with Noemi Kurt (TU Berlin) Probab. Surveys 11, 59 –120 (2014) Mini-workshop Duality of Markov processes and applications to spatial population models, Berlin, 6-7 November 2014 Background Duality is a useful tool in a variety of areas – e.g., random walks with absorbing / reflecting barriers, queuing theory Feller, Asmussen, diffusions, interacting particle systems Liggett, hydrodynamic limits, population genetics... Basic idea: in order to understand the long-time behavior of some complicated process, follow a simpler, dual process backwards in time. Related: duality wrt a measure (potential theory Hunt ’57-58). General theory available. Question: general theory for duality wrt a function? Outline 1. Duality with respect to a measure 2. Duality of linear operators. Existence and uniqueness 3. Spectrum 4. Convex geometry: cone duality. 5. Quantum mechanics Definition I E and F Polish spaces (Borel σ-algebra). I (Xt ) and (Yt ) Markov processes with state spaces E and F I H : E × F → R bounded, measurable (duality function). (Xt ) and (Yt ) are dual with respect to the function H if for all t > 0, x ∈ E and y ∈ F , Ex H(Xt , y ) = Ey H(x, Yt ) Ex : (Xt ) started in x, Ey : (Yt ) started in y . In terms of the semi-groups (Pt ) and (Qt ): Z 0 0 Z Pt (x, dx )H(x , y ) = E Qt (y , dy 0 )H(x, y 0 ) F Finite state spaces: matrix equation (AT : transposed) Pt H = HQtT . Special case: H = diag(1/µ(x)): µ(x)Pt (x, y ) = µ(y )Qt (y , x). Duality wrt measure ≈ duality wrt diagonal function. Pathwise duality Fix a time horizon T > 0, starting points x ∈ E , y ∈ F . Dual processes can be defined on common probability space so that X0 = x, Y0 = y P-a.s. and ∀t ∈ [0, T ] : EH(x, YT ) = EH(Xt , YT −t ) = EH(XT , y ). = weak pathwise duality. Often, processes can be coupled in such a way that ∀t ∈ [0, T ] : H(x, YT ) = H(Xt , YT −t ) = H(XT , y ) P-a.s. = strong pathwise duality. Examples: I H(x, y ) = 1(x ≤ y ) on totally ordered spaces Clifford, Sudbury ’85: ∀t ∈ [0, T ] : x ≤ YT ⇔ Xt ≤ YT −t ⇔ XT ≤ y . P-a.s. I Interacting particle systems with “graphical representations”. I Stochastic recursions: common “driving sequence” (Wn ), recurrence relations Xn+1 = f (Xn , Wn ), Yn+1 = g (Yn , WT −n ). Ex. random walks, Wn = “up” or “down” Lindley ’52, books by Feller, Asmussen. Duality with respect to a measue Discrete state spaces: if µ(x) > 0 for all x and (Pt ) and (Qt ) are time reversals 1 of each other wrt µ, then H(x, y ) = µ(x) δx,y is a duality function. What about continuous state spaces? Definition: µ σ-finite measure on E = F . Processes (Xt ), (Yt ) with semi-groups (Pt ), (Qt ) and resolvents (Rλ ), (R̂λ ) are in duality wrt µ if I I For all λ, x, y , Rλ (x, ·) and R̂λ (x, ·) are abs. cont. wrt. µ. R R For all λ and f , g ≥ 0, E (Pt f )g dµ = E f (Qt g )dµ. Theorem Let (Pt ), (Qt ) be in duality wrt µ. Write Rλ (x, dy ) = rλ (x, y )µ(dy ), R̂λ (x, dy ) = rλ (y , x)µ(dy ) Functions y → 7 rλ (x, y ) and x 7→ rλ (x, y ) λ-excessive wrt (Rλ ) resp. (R̂λ ). Then for all λ > 0, rλ (x, y ) is a duality function for (Xt ), (Yt ). Blumenthal, Getoor: Markov processes and potential theory. Duality of linear operators Question: Relation with notions of duality in functional analysis? Natural framework: dual pairs / duality wrt bilinear form. Def.: V , W vector spaces. Bilinear form h·, ·i : V × W → R called non-degenerate if right null space is trivial NR = {w ∈ W | ∀v ∈ V : hv , w i = 0} = {0} and left null space is trivial. (V , W , h·, ·i) is a dual pair. Linear operators S : V → V , T : W → W are dual wrt h·, ·i if ∀v ∈ V ∀w ∈ W : hSv , w i = hv , Tw i. Notion useful in study of weak topologies, locally convex vector spaces. s Application to duality of Markov processes I M(E ) = finite signed Borel measures on E . I Bilinear form associated with H: Z BH : M(E ) × M(F ) → R, BH (µ, ν) := H(x, y )µ(dx)ν(dy ). E ×F I (Pt∗ µ)(A) = (µPt )(A) = R E µ(dx)Pt (x, A). Observation: Markov semi-groups (Pt ) and (Qt ) dual wrt H iff ∀t > 0 ∀µ ∈ M(E ) ∀ν ∈ M(F ) : BH (Pt∗ µ, ν) = BH (µ, Qt∗ ν). H duality function for (Pt ), (Qt ) ⇔ (Pt∗ ), (Qt∗ ) dual wrt bilinear form BH . Question: Given: (Pt ) and H, existence of a dual Markov semi-group (Qt )? Uniqueness? Uniqueness Functional analysis fact: The dual of a given operator with respect to a non-degenerate bilinear form is unique. Consequence: if BH is non-degenerate then the dual, if it exists, is unique. Applies to a lot of common duality functions: I Siegmund dual: H(x, y ) = 1(x ≤ y ), x, y ∈ R. I Moment dual: H(x, n) = x n , x ∈ [0, 1], n ∈ N0 . I Laplace dual: H(x, λ) = exp(−λx), λ, x ∈ [0, ∞). Uniqueness in moment problems, injectivity of the Laplace transform. Remark: Connection with kernel identities used in integrable quantum systems (Toda, Calogero-Moser systems) related to Doob h-transforms / non-colliding processes. Non-degeneracy does not come for free. Ruijsenaars ’13 On positive Hilbet-Schmidt operators Existence Functional analysis fact: Every bounded operator in a Hilbert space has a unique adjoint. Every bounded operator in a Banach space has a unique dual operator. But: for general dual pairs, a given operator need not have a dual (need continuity in suitable weak topology...) Consequence: semi-group (Pt∗ ) need not have a dual semi-group of operators Tt : M(F ) → M(F ). Additional problem: dual operator semi-group need R not come from a Markov semi-group – need to know whether (Tt ν)(A) = F ν(dy )Qt (y , B) for Markov semi-group (Qt ). Existence and uniqueness criteria available for Siegmund duality in totally ordered state spaces (stochastic monotonicity). Generalization? Key: invariance of convex sets Möhle. Existence, continued M(F ) = signed Borel measures, M1,+ (F ) = probability measures. Images nZ o V= H(·, y )ν(dy ) | ν ∈ M(F ) ⊂ L∞ (E ) F nZ o V1,+ = H(·, y )ν(dy ) | ν ∈ M1,+ (F ) ⊂ V. F Analogous subsets W1,+ ⊂ W ⊂ L∞ (F ). Finite state spaces: linear / convex combinations of rows / columns of H. Proposition Let E and F be countable, discrete state spaces and P a E × E stochastic matrix. Then P 1. PH = HQ T has a solution Q = (Q(x, y ))x,y ∈F with z |Q(y , z)| < ∞ for all y iff V is invariant. 2. Q can be chosen as a stochastic matrix iff V1,+ is invariant. Siegmund duality H(x, y ) = 1(x ≤ y ) in totally ordered state spaces: V1,+ = monotone increasing functions. Recover known relationship Siegmund duality / stochastic monotonicity. Existence, continued In continuous time and space, more delicate – Chapman-Kolmogorov and measurability issues, theorems for non-degenerate H given by Möhle. Convex combination of the columns of H: nZ o V1,+ = H(·, y )ν(dy ) | ν ∈ M1,+ (F ) ⊂ L∞ (E ). F Theorem E , F locally compact, H : E × F → R continuous, non-degenerate. Then I (Pt ) has a H-dual Markov semi-group if and only if V1,+ is invariant. I The dual (Qt ) is unique. I The dual is a Feller semi-group if and only if (Pt ) is. Example: moment duality. Spectrum Finite state spaces: if Pt H = HQtT and H is invertible, then Pt and Qt have the same eigenvalues. In infinite state spaces, non-degenerate duals need not have the same spectrum. Unless processes are reversible! Theorem: (Pt ), (Qt ) dual with non-degenerate duality function H. Assume that (Xt ) and (Yt ) have symmetrizing measures µ and ν. Then (Pt ) and (Qt ), as operators in L2 (E , µ) and L2 (F , ν), are unitarily equivalent. In particular, same spectrum. Convex geometry & cone dual: definitions Klebaner, Rösler, Sagitov ’07, Möhle ’11: duality wrt a convex set rather than a function. Simplex in Rn : non-empty convex, compact set such that every point in C is a unique convex combination of its extremal points exC . Choquet simplex: generalization to locally convex vector spaces; bijection simplex ↔ probability measures on exC , unique integral representation. Characterization via partial order defined by cone {λf | f ∈ C , λ ≥ 0}. Proposition BH non-degenerate, H ∈ C0 (E × F ), F compact ⇒ V1,+ ⊂ C0 (E ) compact (topology of uniform convergence), Choquet simplex. Def. C ⊂ L∞ (E ) simplex, invariant under (Pt ). Cone dual is process (Rt ) with state space exC , duality fct E × exC → R, (x, e) 7→ e(x). Observation: for non-degenerate BH , bijection F → exC , y 7→ H(·, y ), cone dual and usual dual can be identified. Cone dual vs. usual dual. Lifts and intertwining More interesting: BH degenerate but V1,+ simplex. For y ∈ F , let Π(y , ·) be the unique probability measure on exV1,+ such that Z H(·, y ) = Π(y , de)e(·). exV1,+ Transition kernel from F to “smaller” space exV1,+ . Theorem H ∈ C0 (E × F ), F compact, V1,+ simplex, invariant under (Pt ). 1. The cone dual (Rt ) exists and is unique. It is defined on exV1,+ (Borel σ-algebra, uniform topology); exV1,+ is Polish. 2. (Pt ) has at least one dual Markov semi-group (Qt ). 3. (Qt ) is dual to (Pt ) iff for all t, y , B Z Z Qt (y , dy 0 )Π(y 0 , B) = F Π(y , de)Rt (e, B). exV1,+ Intertwining relation. Usual duals are lifts of the cone dual. In terms of processes: cone dual (Zt ), usual dual (Yt ) satisfy P(Zt ∈ B | Yt = y ) = Π(y , B). Quantum mechanics Definition: H self-adjoint operator (Hamiltonian) in Hilbert space H . (Pt ) Markov semi-group in Polish state space E , symmetrizing σ-finite measure µ, U : H → L2 (E , µ) unitary. Call ((Pt ), E , µ, U) a stochastic representation of H if ∀t > 0 : exp(−tH) = U ∗ Pt U. Examples: Ornstein-Uhlenbeck process vs. harmonic oscillator, simple symmetric exclusion process vs. Heisenberg spin chain. Earlier theorem, rephrased If two reversible processes with non-degenerate duality function are associated with Hamiltonians H1 and H2 , then H1 and H2 are unitarily equivalent. Strategy: I interpret infinitesimal generator L as quantum Hamiltonian H. I Interacting particle systems / quantum many-body systems: Break down transitions into birth and death of particles / creation and annihilation operators. I Look for different representations of the relevant operator algebra. I Often, this yields unitary equivalences and interesting dualities. More flexible application: relax requirement of self-adjointness, reversibility. Giardina’s talk! Further aspects & conclusion I Generalized definition allowing for Feynman-Kac corrections. I Solutions of martingale problems. I Dualities and scaling limits. Swart ’06, Alkemper, Hutzenthaler ’07, J., Kurt ’12 I Intertwining of Markov processes Diaconis, Fill ’90, Carmona, Petit, Yor ’98; Huillet, Martinez ’11. I Duality for interacting particle systems as Fourier transforms on groups Holley, Stroock ’79. Conclusion: A lot of structure to explore. Invariant subsets for Siegmund and moment duality Convex subset V1,+ ⊂ L∞ (E ): H(x, y ) 1(x ≤ y ) x ∈E x ∈R y ∈F y ∈R xn x ∈ [0, 1] n ∈ N0 R V1,+ = { F H(·, y )ν(dy ) | ν ∈ M1,+ (F )} monotone increasing, right-continuous functions f , lim−∞ f = 0, lim+∞ f = 1 absolutely monotone functions f w. f (1) = 1 Linear subspace V ⊂ L∞ (E ): H(x, y ) χ(x ≤ y ) x ∈E x ∈R y ∈F y ∈R xn x ∈ [0, 1] n ∈ N0 R V = { F H(·, y )ν(dy ) | ν ∈ M(F )} bounded, right-continuous functions f with bounded variation, lim−∞ f = 0, lim+∞ f ∈ R functions f ∈ C ([0, 1]) with analytic extension F to the complex open unit disk, F in positive Wiener algebra (⊂ H ∞ Hardy space).