Classical and quantum Markov semigroups

Transcription

1 Classical and quantum Markov semigroups Alexander Belton Department of Mathematics and Statistics Lancaster University United Kingdom Young Functional Analysts Workshop Lancaster University 23rd April 2014

2 Classical Markov semigroups Markov processes Markov semigroups Infinitesimal generators Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 2 / 27

3 Markov processes Definition 1 Let S be a topological space. A Markov process with state space S is a collection of S-valued random variables X = (X t ) t 0 on a common probability space such that E [ f(x s+t ) X r : 0 r s ] = E [ f(x s+t ) X s ] (s,t 0) for all f B b (S) := {g : S R g is bounded and Borel measurable}. A Markov process X is time homogeneous if E [ f(x s+t ) X s = x ] = E[f(X t ) X 0 = x ] (f B b (S), s,t 0, x S). Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 3 / 27

4 Markov semigroups Definition 2 A Markov semigroup on B b (S) is a family T = (T t ) t 0 such that 1 T t : B b (S) B b (S) is a linear operator for all t 0, 2 T s T t = T s+t for all s, t 0 and T 0 = I (semigroup), 3 T t 1 for all t 0 (contraction) and 4 T t f 0 whenever f 0, for all t 0 (positive). If T t 1 = 1 for all t 0 then T is conservative. Proposition 3 Given a time-homogeneous Markov process X, setting (T t f)(x) = E [ f(x t ) X 0 = x ] (t 0, x S) defines a conservative Markov semigroup T on B b (S). Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 4 / 27

5 Proof of Proposition 3 Easy part Properties 1, 3 and 4 follow immediately from basic properties of conditional expectation, as does the fact that T is conservative. The semigroup property Note that (T s+t f)(x) = E[f(X s+t ) X 0 = x] (definition) = E [ E[f(X s+t ) X r : 0 r s] X 0 = x ] (tower property) = E [ E[f(X s+t ) X s ] X 0 = x ] (Markov property) = E [ (T t f)(x s ) X 0 = x ] (homogeneity) = T s (T t f)(x). The identity T 0 f = f is immediate. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 5 / 27

6 Feller semigroups Definition 4 Suppose the state space S is a locally compact Hausdorff space. The Markov semigroup T is Feller if 1 T t ( C0 (S) ) C 0 (S) for all t 0 and 2 T t f f 0 as t 0 for all f C 0 (S). Remark Every sufficiently well-behaved time-homogeneous Markov process is Feller: Brownian motion, Poisson processes, Lévy processes,... Theorem 5 If the state space S is metrisable then a conservative Feller semigroup gives rise to a time-homogeneous Markov process. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 6 / 27

7 Proof of Theorem 5 Probabilistic version Let p t (x,a) := (T t 1 A )(x) (t 0, x S, A Borel S) be the probability of moving from x to A in time t. Then (T t f)(x) = f(y)p t (x,dy) (t 0, f B b (S), x S) ( ) S and p s+t (x,a) = ( T s (T t 1 A ) ) (x) (semigroup property) (1) = (T t 1 A )(y)p s (x,dy) (by ( ) ) (2) = S S p t (y,a)p s (x,dy) (definition). (3) The second identity is the Chapman Kolmogorov equation. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 7 / 27

8 Proof of Theorem 5 Probabilistic version (ctd.) Let µ be a probability measure on S. If t n t 1 0 and A 1,...A n are Borel subsets of S then p t1,...,t n (A 1 A n ) = µ(dx 0 ) p t1 (x 0,dx 1 ) p tn t n 1 (x n 1,dx n ). S A 1 A n These finite-dimensional distributions are consistent, by C K. The Daniell Kolmogorov extension theorem yields a probability measure on the product space Ω := S R+ = {ω = (ω t ) t 0 : ω t S for all t 0} such the coordinate projections X t : Ω S; ω ω t form a time-homogeneous Markov process X with associated semigroup T. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 8 / 27

9 Proof of Theorem 5 Functional-analytic version Without loss of generality, suppose that S is compact. Then Ω is a compact Hausdorff space and the algebraic tensor product C(S) = lin{f 1 X t1 f n X tn : f 1,...,f n C(S), t 1,...,t n 0} t 0 is dense in C(Ω) by the Stone Weierstrass theorem. If µ is a state on C(S) then f 1 X t1 f n X tn µ(t t1 (f 1 (T tn t n 1 f n ) )) extends to a state φ on C(Ω). By the Riesz Markov theorem, there exists a probability measure on Ω corresponding to φ, and X is a time-homogeneous Markov process with respect to this measure, with associated semigroup T. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 9 / 27

10 Infinitesimal generators Definition 6 Let T be a C 0 semigroup on a Banach space E. Its infinitesimal generator is the linear operator L in E with domain { } doml = x E : lim t 0+ t 1 (T t x x) exists and action Lx = lim t 0+ t 1 (T t x x). The operator L is closed and densely defined. Remark If T comes from a time-homogeneous Markov process X then E [ f(x t+h ) f(x t ) X t ] = (Th f f)(x t ) = h(lf)(x t )+o(h), so L describes the change in X over an infinitesimal time interval. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 10 / 27

11 Examples Uniform motion If S = R and X t = X 0 +t for all t 0 then E[f(X s+t ) X s = x] = f(x +t) = E[f(X t ) X 0 = x] ( f C0 (R) ). It follows that X is a time-homogeneous Feller process with semigroup generator L such that Lf = f. Brownian motion If S = R and X is a standard Brownian motion then Itô s formula gives that f(x t ) = f(x 0 )+ t 0 f (X s )dx s t 0 f (X s )ds ( f C 2 (R) ). It follows that X is a time-homogeneous Feller process with semigroup generator L such that Lf = 1 2 f. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 11 / 27

12 Examples (ctd.) A Poisson process If S = R and X is a Poisson process with unit intensity and unit jumps then E[f(X t ) X s = x] = e (t s) (t s) n f(x +n) (t s 0). n! n=0 It follows that X is a time-homogeneous Feller process with semigroup generator L such that (Lf)(x) = f(x +1) f(x) for all x R. (Note that (T t f f)(x) t = e t 1 f(x)+e t f(x +1)+O(t) (t 0+).) t Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 12 / 27

13 The Lumer Phillips theorem Theorem 7 (Lumer Phillips) A closed, densely defined operator L in the Banach space E generates a strongly continuous contraction semigroup on E if and only if ran(λi L) = E for some λ > 0 and the operator L is dissipative: (λi L)x λ x for all λ > 0 and x doml. Remark If the operator L is dissipative then ran(λi L) = E for some λ > 0 if and only if ran(λi L) = E for all λ > 0. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 13 / 27

14 The Hille Yosida Ray theorem Definition Let S be a locally compact Hausdorff space. A linear operator L in C 0 (S) satisfies the positive maximum principle if whenever f doml and x 0 S are such that sup x S f(x) = f(x 0 ) 0 then (Lf)(x 0 ) 0. Theorem 8 (Hille Yosida Ray) A closed, densely defined operator L in C 0 (S) is the generator of a Feller semigroup on C 0 (S) if and only if ran(λi L) = C 0 (S) for some λ > 0 and L satisfies the positive maximum principle. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 14 / 27

15 Proof of Theorem 8 Sufficiency Suppose L satisfies the positive maximum principle. Let f doml and λ > 0. To see that (λi L)f λ f, note first that there exists x 0 S such that f(x 0 ) = f ; without loss of generality, suppose f(x 0 ) 0. Then (λi L)f λf(x 0 ) (Lf)(x 0 ) and (Lf)(x 0 ) 0, by the positive maximum principle. Consequently, (λi L)f λf(x 0 ) Lf(x 0 ) λf(x 0 ) = λ f. Hence T is a strongly continuous contraction semigroup, by L F. Positivity is left as an exercise: show that (λi L) 1 is positive for all λ > 0. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 15 / 27

16 Proof of Theorem 8 (ctd.) Necessity Suppose that L generates a Feller semigroup on C 0 (S). By L P, it suffices to show that L satisfies the positive maximum principle. Given f doml, let x 0 S be such that f(x 0 ) = sup x S f(x). Let f + := x max{f(x),0} and note that (T t f)(x 0 ) (T t f + )(x 0 ) T t f + f + = f(x 0 ). Then as required. (T t f f)(x 0 ) (Lf)(x 0 ) = lim 0, t 0+ t Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 16 / 27

17 Quantum Markov semigroups Quantum Markov processes Quantum Markov semigroups Infinitesimal generators Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 17 / 27

18 Quantum Feller semigroups Theorem 9 Every commutative C algebra is isometrically isomorphic to C 0 (S), where S is a locally compact Hausdorff space. Definition A quantum Feller semigroup on the C algebra A is a family (T t ) t 0 such that 1 T t : A A is a linear operator for all t 0, 2 T s T t = T s+t for all s, t 0 and T 0 = I, 3 T t 1 for all t 0, 4 (T t a ij ) M n (A) + whenever (a ij ) M n (A) +, for all n 1 and t 0 (complete positivity) and 5 T t x x 0 as t 0 for all x A. If A is unital and T t 1 = 1 for all t 0 then T is conservative. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 18 / 27

19 Complete positivity Exercise A linear map Φ : A B between C algebras is completely positive if and only if n bi Φ(ai a j )b j 0 i,j=1 for all n 1, a 1,..., a n A and b 1,..., b n B. Theorem 10 A positive linear map φ : A B between C algebras is completely positive if A is commutative (Stinespring) or B is commutative (Arveson). Theorem 11 (Kadison) A CP unital linear map Φ : A B between unital C algebras is such that Φ(a a) Φ(a) Φ(a) (a A) (CP-Schwarz) Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 19 / 27

20 Stinespring s theorem Theorem 12 (Stinespring) Let Φ : A B be a linear map, where A is a unital C algebra and B B(H). Then Φ is completely positive if and only if there exists a representation π : A B(K) and a bounded operator V : H K such that Φ(a) = V π(a)v (a A). Corollary 13 If Φ : A B is as above, with Φ(1) = I, then n v i, ( Φ(ai a j ) Φ(a i ) Φ(a j ) ) v j 0 i,j=1 for all n 1, a 1,..., a n A and v 1,..., v n H. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 20 / 27

21 Proof of Corollary 13 Note first that I = Φ(1) = V π(1)v = V V, so V has norm 1. Hence n v i,φ(ai a j)v j = i,j=1 = = = n Vv i,π(ai a j)vv j i,j=1 n 2 π(a i )Vv i i=1 n V 2 π(a i )Vv i i=1 n 2 Φ(a i )v i i=1 n v i,φ(a i ) Φ(a j )v j. i,j=1 Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 21 / 27

22 Infinitesimal generators Theorem 14 Let T be a quantum Feller semigroup on B(H) which is uniformly continuous: lim t 0+ T t I = 0. The generator L is bounded, -preserving and conditionally completely positive: if n 1, a 1,..., a n and v 1,..., v n H then n v i,l(ai a j )v j 0 i,j=1 whenever n a i v i = 0. i=1 Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 22 / 27

23 Proof of Theorem 14 Easy parts The boundedness of L is standard semigroup theory. The fact that L(a ) = L(a) for all a A follows by continuity of the involution. Proof of conditional complete positivity Let a 1,..., a n A and v 1,..., v n H. By Corollary 13, n v i, ( T t (ai a j ) T t (a i ) T t (a j ) ) v j 0. i,j=1 Differentiating with respect to t gives that n v i, ( L(ai a j ) L(a i ) a j ai L(a j ) ) v j 0; i,j=1 if n i=1 a iv i = 0, the second and third terms vanish. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 23 / 27

24 Characterisation of bounded generators Theorem 15 (Lindblad, Evans) Let L be a -preserving bounded linear map on the unital C algebra A. Then T t = exp(tl) is completely positive for all t 0 if and only if L is conditionally completely positive (in the appropriate sense). Since CP unital linear maps between unital C algebras are automatically contractive, this characterises the generators of uniformly continuous conservative quantum Feller semigroups on unital C algebras. Theorem 16 (Gorini Kossakowski Sudarshan, Lindblad) A bounded map L on B(H) is the generator of a uniformly continuous conservative quantum Feller semigroup composed of normal maps if and only if L(X) = i[h,x] 1 2( L LX 2L (X I)L+XL L ) ( X B(H) ), where H = H B(H) and L B(H;H K) for some Hilbert space K. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 24 / 27

25 Quantum Markov processes Random variables Let S be a compact Hausdorff space. If X : Ω S is a random variable then j X : A B; f f X is a unital -homomorphism, where A = C(S) and B = L (Ω,F,P). Definition 17 A non-commutative random variable is a unital -homomorphism j between unital C algebras. A family (j t : A B) t 0 of non-commutative random variables is a dilation of the quantum Feller semigroup T on A if there exists a conditional expectation E : B A such that T t = E j t for all t 0. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 25 / 27

26 Construction of dilations Many authors have tackled this problem: Evans and Lewis; Davies; Accardi, Frigerio and Lewis; Vincent-Smith; Kümmerer; Sauvageot; Bhat and Parthasarathy;... Essentially, one attempts to mimic the functional-analytic proof of Theorem 5. The state f 1 X t1 f n X tn µ(t t1 (f 1 (T tn t n 1 f n ) )) becomes a sesquilinear form (f 1 f n,g 1 g n ) µ(t t1 (f 1 (T tn t n 1 (f ng n )) g 1 )). The key to proving positivity of this form is the complete positivity of the semigroup maps. There are many technical details which must be addressed. This would require another talk. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 26 / 27

27 References Classical D. Applebaum, Lévy processes and stochastic calculus, second edition, Cambridge University Press, T.M. Liggett, Continuous time Markov processes, American Mathematical Society, L.C.G. Rogers and D. Williams, Diffusions, Markov processes and martingales, volumes I and II, second edition, Cambridge University Press, Quantum D.E. Evans and J.T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A No.24 (1977), v+104 pp. F. Fagnola, Quantum Markov semigroups and quantum flows, Proyecciones 18 no.3 (1999), 144 pp. Alexander Belton (Lancaster University) Classical and quantum Markov semigroups YFAW, 23iv14 27 / 27