Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013
eal-valued Ornstein-Uhlenbeck process, solution to SDE dx t = 2dB t X t dt. Instantaneous distribution is known: semigroup (P t ) t 0 given by P t f (x) = f (e t x + 1 e 2t y) µ(dy) f dµ =: µ(f ), t where invariant (and reversible) measure µ is N (0, 1). Generator: Lf (x) = f (x) x f (x).
Commutation relation between semigroup and gradient: related to long-time behaviour: (P t f ) = e t P t (f ), Convergence in Wasserstein distance. Convergence in L 2 (µ): P t f µ(f ) L 2 (µ) e λ 1t f µ(f ) L 2 (µ), via Poincaré (or spectral gap) inequality: λ 1 = 1, where λ 1 best constant in inequality: ( λ 1 Var µ (f ) := λ 1 µ(f 2 ) µ(f ) 2) f Lf dµ = f 2 dµ, i.e. given by variational formula λ 1 = inf f f Lf dµ. Var µ (f )
Exponential decay in entropy: Ent µ (P t f ) e 2t Ent µ (f ), via log-sobolev inequality: Ent µ (f ) := µ(f log f ) µ(f ) log µ(f ) 2 Other consequences: Hypercontractivity. Measure concentration. Etc... f L f dµ.
Generalization to process solution to SDE dx t = 2dB t U (X t )dt, where U smooth potential. Invariant (and reversible) measure: µ(dx) e U(x) dx. Generator: Lf = f U f. Bakry-Émery s criterion through Γ 2-calculus: if U ρ, then (sub-) commutation relation between gradient and semigroup (P t f ) e ρt ( P t f ).
In particular if ρ > 0, then good ergodicity properties: λ 1 ρ, i.e. Poincaré inequality ρ Var µ (f ) f Lfdµ = f 2 dµ, meaning long-time convergence in L 2 (µ): P t f µ(f ) L 2 (µ) e ρt f µ(f ) L 2 (µ). Also log-sobolev, hypercontractivity, measure concentration, etc...
How to obtain Poincaré from Bakry-Émery? + Var µ (f ) = 2 P t f LP t f dµ dt 0 + IBP = 2 (P t f ) 2 dµ dt 0 Bakry Emery + 2 e 2ρt P t ( f 2 ) dµ dt 0 + Invariance = 2 e 2ρt dt f 2 dµ 0 ρ>0 1 = f 2 dµ. ρ Hence Sign(ρ) is important to obtain long-time convergence.
Intertwining between gradient and generators: (Lf ) = L V (f ), with L V Schrödinger operator L V with potential V := U. At level of semigroups: (P t f ) (x) = P V t (f )(x) := E x [f (X t ) exp where (P V t ) t 0 associated Feynman-Kac semigroup. ( t )] U (X s ) ds, 0 Hence Jensen s inequality entails Bakry-Émery s criterion.
General diffusion satisfying the SDE dx t = 2σ(X t )db t + b(x t )dt. Invariant (and reversible) measure: µ(dx) 1 ( x ) σ(x) 2 exp b(u) σ(u) 2 du dx. Generator of Sturm-Liouville type: x 0 Lf = σ 2 f + bf. Bakry-Émery s criterion through Γ 2 -calculus: if V σ := Lσ σ b ρ σ, then (sub-) commutation relation holds: σ(p t f ) e ρσt P t ( σf ).
Probabilistic interpretation of V σ? Using method of tangent process, i.e. differentiate w.r.t. initial point x: x Xt x = 1 + t t 2 σ (Xs x ) x Xs x db s + b (Xs x ) x Xs x ds, 0 0 hence by Itô s formula, process x X x given by x Xt x = σ(x t x ( ) t ( ) Lσ exp σ(x) 0 σ b ( t exp V σ (Xs x ) ds = σ(x x t ) σ(x) 0 ). ) (Xs x ) ds
Hence (P t f ) (x) = E [ f (Xt x ) x Xt x ] = 1 σ(x) E [ σ(x x t ) f (X x t ) exp ( t )] V σ (Xs x ) ds. 0 Denoting the weighted gradient σ f := σf, it rewrites as the intertwining [ ( t )] σ P t f (x) = E σ f (Xt x ) exp V σ (Xs x ) ds 0 =: Pt Vσ ( σ f )(x).
Can be recovered through straightforward infinitesimal version: If then Poincaré inequality: ρ σ Var µ (f ) σ L = L Vσ σ. ρ σ := inf V σ > 0, f Lf dµ = σ f 2 dµ. But if ρ σ 0, how to use intertwining to obtain Poincaré, thus L 2 -convergence? Idea: change the gradient.
Denote the weighted gradient a f := af for positive function a. We have the intertwining a Lf = L a ( a f ) V a a f =: L Va a ( a f ), where new Sturm-Liouville generator is where b a := b + 2σσ 2σ 2 a a and potential L a f = σ 2 f + b a f, V a := L a(a) a = b + 2σ2 h h, with h := σ a, b.
At level of semigroups, ( t )] a P t f (x) = Pa,t( Va a f )(x) := E x [ a f (Xt a ) exp V a (Xs a ) ds, 0 with (P a,t ) t 0 semigroup of process X a associated to L a. If a = σ then h = 0 thus L a = L and In particular if V a = L a(a) a b = Lσ σ b = V σ. ρ a := inf V a > 0, then convergence in (distorting) Wasserstein distance (exponentially fast, depending on function a) and also L 2 -convergence via Poincaré:
Var µ (f ) = 2 = 2 Intertwining = 2 Jensen 2 Invariance = 2 ρ a>0 = 1 ρ a + 0 + 0 + 0 + 0 + 0 σ P t f 2 dµ dt ( a P t f 2 σ ) 2 dµ dt a ( Pa,t( Va a f ) 2 σ ) 2 dµ dt a ( σ ) 2 P a,t ( a f 2 ) dµ dt a e 2ρat e 2ρat dt σ f 2 dµ, since (σ/a) 2 dµ is invariant for (P a,t ) t 0. ( a f 2 σ ) 2 dµ a
Hence λ 1 sup ρ a. a ecovers Chen-Wang s Theorem on spectral gap, obtained originally via coupling. Is our criterion optimal? Yes. Take a = 1/g where g > 0. Then V a = (Lg) g. Hence if g eigenvector associated to λ 1, V a λ 1.
Another consequence of intertwining: if then V σ := Lσ σ b > 0, λ 1 1 1 V σ dµ. Convenient when Bakry-Émery criterion does not apply (for instance V σ > 0 but tends to 0 at infinity).
Some classical examples: Let U(x) = x α α. ecall dx t = 2dB t U (X t ) dt, µ(dx) e U(x) dx. α = 2 (O.U.): with a = σ 1, λ 1 = 1 (Bakry-Émery). α = 1: with a(x) = e x /2, λ 1 = 1/4 (Bobkov-Ledoux).
α = 4: with a(x) = e U(x) 2 + εx2 3 2, λ1 ε max = 2. 1 < α < 2: Hence λ 1 V σ (x) = (α 1) x α 2 1 1 V σ dµ 0. x = (α 1) α 1 2/α Γ(1/α) Γ((3 α)/α).
Consider new process dx t = 2 σ(x t ) db t + ( 2σσ σ 2 U ) (X t ) dt, also reversible w.r.t. µ(dx) e U(x) dx. Generator: Lf = σ 2 f + ( 2σσ σ 2 U ) f. Choice of diffusion constant σ?
Heavy-tailed case: weighted Poincaré. Take U(x) := β log(1 + x 2 ), with β > 1/2, leading to Cauchy-type distribution µ(dx) Then with choice σ(x) := 1 + x 2, dx (1 + x 2 ) β. V σ (x) = 2β 1 1 + x 2 x 0.
Hence for β > 3/2, λ 1 1 1 V σ dµ (2β 1)(β 3/2) = β 1 =: C β, meaning that weighted Poincaré holds: C β Var µ (f ) σ f 2 dµ = (1 + x 2 ) f (x) 2 µ(dx).
Coming back to general case, why process X a and potential V a := L a(a) a appear in the intertwining relation? Answer: (Doob s) h-transform. b, (Pt V ) t 0 original Feynman-Kac semigroup with some potential V. New dynamics: multiply inside and divide outside by some function h. P V (h) t f := PV t (hf ). h
At level of generator (of Schrödinger type): L V (h) f = LV (hf ) h ( ) Γ(f, h) Lh = Lf + 2 f + h h V f ) ( ) = σ 2 f + (b + 2σ 2 h Lh f + h h V f. }{{}}{{} generator of h transform h potential
In case V = 0 then Doob s h-transform is Markov if and only if h is L-harmonic, i.e. Lh = 0. Group structure: hk-transform is the h-transform of the k-transform. h-transform and original dynamics have the same distribution if and only if h is constant.
If V = V σ := Lσ σ b and h := σ a, then X a is Doob s h-transform of X, and ( ) L V (σ/a) Lh f = L a f + h V σ f = L a f + L(σ/a) σ/a }{{} by h transform ( ) La (a) = L a f b f a = L Va a f. ) Lσ σ b }{{} by classical intertwining f
Probabilistic interpretation of intertwining: first perform the classical intertwining by using method of tangent process: σ P t f = P Vσ t ( σ f ), with potential V σ := Lσ σ b ; then use h-transform with h := σ/a to obtain the final intertwining a P t f = Pa,t( Va a f ), with potential V a := L a(a) b. a