Classical and quantum Markov semigroupsbelton/www/notes/23iv14.pdf · 2014-04-23 · Classical...

Classical and quantum Markov semigroups

Alexander Belton

Department of Mathematics and StatisticsLancaster UniversityUnited Kingdom

http://www.maths.lancs.ac.uk/~belton/

[email protected]

Young Functional Analysts’ Workshop

Lancaster University

23rd April 2014

http://www.maths.lancs.ac.uk/~belton/

Classical Markov semigroups

Markov processes

Markov semigroups Infinitesimal generators

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

Markov processes

Definition 1

Let S be a topological space. A Markov process with state space S is acollection of S-valued random variables X = (Xt)t>0 on a commonprobability space such that

E[

f (Xs+t) | Xr : 0 6 r 6 s]

= E[

f (Xs+t) | Xs

]

(s, t > 0)

for all

f ∈ Bb(S) := g : S → R | g is bounded and Borel measurable.

A Markov process X is time homogeneous if

E[

f (Xs+t) | Xs = x]

= E[f (Xt) | X0 = x]

(f ∈ Bb(S), s, t > 0, x ∈ S).


Markov semigroups

Definition 2

A Markov semigroup on Bb(S) is a family T = (Tt)t>0 such that

1 Tt : Bb(S) → Bb(S) is a linear operator for all t > 0,

2 Ts Tt = Ts+t for all s, t > 0 and T0 = I (semigroup),

3 ‖Tt‖ 6 1 for all t > 0 (contraction) and

4 Tt f > 0 whenever f > 0, for all t > 0 (positive).

If Tt1 = 1 for all t > 0 then T is conservative.

Proposition 3

Given a time-homogeneous Markov process X , setting

(Tt f )(x) = E[

f (Xt) | X0 = x]

(t > 0, x ∈ S)

defines a conservative Markov semigroup T on Bb(S).


Proof of Proposition 3

Easy part

Properties 1, 3 and 4 follow immediately from basic properties ofconditional expectation, as does the fact that T is conservative.

The semigroup property

Note that

(Ts+t f )(x) = E[f (Xs+t) | X0 = x ] (definition)

= E[

E[f (Xs+t) | Xr : 0 6 r 6 s]∣

∣ X0 = x]

(tower property)

= E[

E[f (Xs+t) | Xs ]∣

∣ X0 = x]

(Markov property)

= E[

(Tt f )(Xs) | X0 = x]

(homogeneity)

= Ts(Tt f )(x).

The identity T0f = f is immediate.


Feller semigroups

Definition 4

Suppose the state space S is a locally compact Hausdorff space. TheMarkov semigroup T is Feller if

1 Tt

(

C0(S))

⊆ C0(S) for all t > 0 and

2 ‖Tt f − f ‖∞ → 0 as t → 0 for all f ∈ C0(S).

Remark

Every sufficiently well-behaved time-homogeneous Markov process isFeller: Brownian motion, Poisson processes, Levy processes, . . . .

Theorem 5

If the state space S is metrisable then a conservative Feller semigroupgives rise to a time-homogeneous Markov process.


Proof of Theorem 5

Probabilistic version

Let

pt(x ,A) := (Tt1A)(x) (t > 0, x ∈ S , ABorel⊆ S)

be the probability of moving from x to A in time t. Then

(Tt f )(x) =

∫

S

f (y)pt(x ,dy) (t > 0, f ∈ Bb(S), x ∈ S) (⋆)

and ps+t(x ,A) =(

Ts(Tt1A))

(x) (semigroup property) (1)

=

∫

S

(Tt1A)(y)ps(x ,dy) (by (⋆))

(2)

=

∫

S

pt(y ,A)ps(x ,dy) (definition). (3)

The second identity is the Chapman–Kolmogorov equation.


Proof of Theorem 5

Probabilistic version (ctd.)

Let µ be a probability measure on S . If tn > · · · > t1 > 0 and A1, . . .An

are Borel subsets of S then

pt1,...,tn(A1 × · · · × An)

=

∫

S

µ(dx0)

∫

A1

pt1(x0,dx1) · · ·

∫

An

ptn−tn−1(xn−1,dxn).

These finite-dimensional distributions are consistent, by C–K.The Daniell–Kolmogorov extension theorem yields a probability measureon the product space

Ω := SR+ = ω = (ωt)t>0 : ωt ∈ S for all t > 0

such the coordinate projections Xt : Ω → S ; ω 7→ ωt form atime-homogeneous Markov process X with associated semigroup T .


Proof of Theorem 5

Functional-analytic version

Without loss of generality, suppose that S is compact. Then Ω is acompact Hausdorff space and the algebraic tensor product

⊙

t>0

C (S) = linf1 Xt1 · · · fn Xtn : f1, . . . , fn ∈ C (S), t1, . . . , tn > 0

is dense in C (Ω) by the Stone–Weierstrass theorem.If µ is a state on C (S) then

f1 Xt1 · · · fn Xtn 7→ µ(Tt1(f1 · · · (Ttn−tn−1 fn) · · · ))

extends to a state φ on C (Ω).By the Riesz–Markov theorem, there exists a probability measure on Ωcorresponding to φ, and X is a time-homogeneous Markov process withrespect to this measure, with associated semigroup T .


Infinitesimal generators

Definition 6

Let T be a C0 semigroup on a Banach space E . Its infinitesimal generatoris the linear operator L in E with domain

domL =

x ∈ E : limt→0+

t−1(Ttx − x) exists

and actionLx = lim

t→0+t−1(Ttx − x).

The operator L is closed and densely defined.

Remark

If T comes from a time-homogeneous Markov process X then

E[

f (Xt+h)− f (Xt) | Xt

]

= (Thf − f )(Xt) = h(Lf )(Xt) + o(h),

so L describes the change in X over an infinitesimal time interval.


Examples

Uniform motion

If S = R and Xt = X0 + t for all t > 0 then

E[f (Xs+t)|Xs = x ] = f (x + t) = E[f (Xt)|X0 = x ](

f ∈ C0(R))

.

It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that Lf = f ′.

Brownian motion

If S = R and X is a standard Brownian motion then Ito’s formula givesthat

f (Xt) = f (X0) +

∫ t

0f ′(Xs)dXs +

1

2

∫ t

0f ′′(Xs)ds

(

f ∈ C 2(R))

.

It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that Lf = 1

2 f′′.


Examples (ctd.)

A Poisson process

If S = R and X is a Poisson process with unit intensity and unit jumpsthen

E[f (Xt)|Xs = x ] = e−(t−s)∞∑

n=0

(t − s)n

n!f (x + n) (t > s > 0).

It follows that X is a time-homogeneous Feller process with semigroupgenerator L such that (Lf )(x) = f (x + 1)− f (x) for all x ∈ R.(Note that

(Tt f − f )(x)

t=

e−t − 1

tf (x) + e−t f (x + 1) + O(t) (t → 0+).)


The Lumer–Phillips theorem

Theorem 7 (Lumer–Phillips)

A closed, densely defined operator L in the Banach space E generates astrongly 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 ifand only if ran(λI −L) = E for all λ > 0.


The Hille–Yosida–Ray theorem

Definition

Let S be a locally compact Hausdorff space. A linear operator L in C0(S)satisfies the positive maximum principle if whenever f ∈ domL and x0 ∈ Sare such that supx∈S f (x) = f (x0) > 0 then (Lf )(x0) 6 0.

Theorem 8 (Hille–Yosida–Ray)

A closed, densely defined operator L in C0(S) is the generator of a Fellersemigroup on C0(S) if and only if

ran(λI − L) = C0(S) for some λ > 0 and

L satisfies the positive maximum principle.


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 x0 ∈ S such that |f (x0)| = ‖f ‖∞; without lossof generality, suppose f (x0) > 0. Then

‖(λI − L)f ‖∞ > |λf (x0)− (Lf )(x0)|

and (Lf )(x0) 6 0, by the positive maximum principle. Consequently,

‖(λI − L)f ‖∞ > λf (x0)− Lf (x0) > λf (x0) = λ‖f ‖∞.

Hence T is a strongly continuous contraction semigroup, by L–F. Positivityis left as an exercise: show that (λI − L)−1 is positive for all λ > 0.


Proof of Theorem 8 (ctd.)

Necessity

Suppose that L generates a Feller semigroup on C0(S). By L–P, it sufficesto show that L satisfies the positive maximum principle.Given f ∈ domL, let x0 ∈ S be such that f (x0) = supx∈S f (x).Let f + := x 7→ maxf (x), 0 and note that

(Tt f )(x0) 6 (Tt f+)(x0) 6 ‖Tt f

+‖∞ 6 ‖f +‖∞ = f (x0).

Then

(Lf )(x0) = limt→0+

(Tt f − f )(x0)

t6 0,

as required.


Quantum Markov semigroups

Quantum Markov processes

Quantum Markov semigroups Infinitesimal generators


Quantum Feller semigroups

Theorem 9

Every commutative C ∗ algebra is isometrically isomorphic to C0(S), whereS is a locally compact Hausdorff space.

Definition

A quantum Feller semigroup on the C ∗ algebra A is a family (Tt)t>0 suchthat

1 Tt : A → A is a linear operator for all t > 0,

2 Ts Tt = Ts+t for all s, t > 0 and T0 = I ,

3 ‖Tt‖ 6 1 for all t > 0,

4 (Ttaij) ∈ Mn(A)+ whenever (aij) ∈ Mn(A)+, for all n > 1 and t > 0(complete positivity) and

5 ‖Ttx − x‖ → 0 as t → 0 for all x ∈ A.

If A is unital and Tt1 = 1 for all t > 0 then T is conservative.


Complete positivity

Exercise

A linear map Φ : A → B between C ∗ algebras is completely positive if andonly if

n∑

i ,j=1

b∗i Φ(a∗

i aj)bj > 0

for all n > 1, a1, . . . , an ∈ A and b1, . . . , bn ∈ B.

Theorem 10

A positive linear map φ : A → B between C ∗ algebras is completelypositive 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)


Stinespring’s theorem

Theorem 12 (Stinespring)

Let Φ : A → B be a linear map, where A is a unital C ∗ algebra andB ⊆ B(H). Then Φ is completely positive if and only if there exists arepresentation π : A → B(K) and a bounded operator V : H → K suchthat

Φ(a) = V ∗π(a)V (a ∈ A).

Corollary 13

If Φ : A → B is as above, with Φ(1) = I , then

n∑

i ,j=1

〈vi ,(

Φ(a∗i aj)− Φ(ai)∗Φ(aj)

)

vj〉 > 0

for all n > 1, a1, . . . , an ∈ A and v1, . . . , vn ∈ H.


Proof of Corollary 13

Note first that I = Φ(1) = V ∗π(1)V = V ∗V , so V ∗ has norm 1. Hence

n∑

i ,j=1

〈vi ,Φ(a∗

i aj)vj〉 =n

∑

i ,j=1

〈Vvi , π(a∗

i aj)Vvj〉

=∥

∥

∥

n∑

i=1

π(ai )Vvi

∥

∥

∥

2

>

∥

∥

∥V ∗

n∑

i=1

π(ai )Vvi

∥

∥

∥

2

=∥

∥

∥

n∑

i=1

Φ(ai)vi

∥

∥

∥

2

=n

∑

i ,j=1

〈vi ,Φ(ai )∗Φ(aj)vj〉.


Infinitesimal generators

Theorem 14

Let T be a quantum Feller semigroup on B(H) which is uniformlycontinuous:

limt→0+

‖Tt − I‖ = 0.

The generator L is bounded, ∗-preserving and conditionally completelypositive: if n > 1, a1, . . . , an and v1, . . . , vn ∈ H then

n∑

i ,j=1

〈vi ,L(a∗

i aj)vj〉 > 0

whenevern

∑

i=1

aivi = 0.


Proof of Theorem 14

Easy parts

The boundedness of L is standard semigroup theory. The fact thatL(a∗) = L(a)∗ for all a ∈ A follows by continuity of the involution.

Proof of conditional complete positivity

Let a1, . . . , an ∈ A and v1, . . . , vn ∈ H. By Corollary 13,

n∑

i ,j=1

〈vi ,(

Tt(a∗

i aj)− Tt(ai )∗Tt(aj)

)

vj〉 > 0.

Differentiating with respect to t gives that

n∑

i ,j=1

〈vi ,(

L(a∗i aj)− L(ai)∗aj − a∗i L(aj)

)

vj〉 > 0;

if∑n

i=1 aivi = 0, the second and third terms vanish.


Characterisation of bounded generators

Theorem 15 (Lindblad, Evans)

Let L be a ∗-preserving bounded linear map on the unital C ∗ algebra A.Then Tt = exp(tL) is completely positive for all t > 0 if and only if L isconditionally completely positive (in the appropriate sense).

Since CP unital linear maps between unital C ∗ algebras are automaticallycontractive, this characterises the generators of uniformly continuousconservative 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 continuousconservative quantum Feller semigroup composed of normal maps if andonly if

L(X ) = i [H,X ] − 12

(

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.


Quantum Markov processes

Random variables

Let S be a compact Hausdorff space. If X : Ω → S is a random variablethen

jX : A → B; f 7→ 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 jbetween unital C ∗ algebras.

A family (jt : A → B)t>0 of non-commutative random variables is adilation of the quantum Feller semigroup T on A if there exists aconditional expectation E : B ։ A such that Tt = E jt for all t > 0.


Construction of dilations

Many authors have tackled this problem: Evans and Lewis; Davies;Accardi, Frigerio and Lewis; Vincent-Smith; Kummerer; Sauvageot; Bhatand Parthasarathy; . . . . Essentially, one attempts to mimic thefunctional-analytic proof of Theorem 5. The state

f1 Xt1 · · · fn Xtn 7→ µ(Tt1(f1 · · · (Ttn−tn−1 fn) · · · ))

becomes a sesquilinear form

(f1 ⊗ · · · ⊗ fn, g1 ⊗ · · · ⊗ gn) 7→ µ(Tt1(f∗

1 · · · (Ttn−tn−1(f∗

n gn)) · · · g1)).

The key to proving positivity of this form is the complete positivity of thesemigroup maps.There are many technical details which must be addressed. This wouldrequire another talk.


References

Classical

D. Applebaum, Levy processes and stochastic calculus, secondedition, Cambridge University Press, 2009.

T.M. Liggett, Continuous time Markov processes, AmericanMathematical Society, 2010.

L.C.G. Rogers and D. Williams, Diffusions, Markov processes andmartingales, volumes I and II, second edition, Cambridge UniversityPress, 2000.

Quantum

D.E. Evans and J.T. Lewis, Dilations of irreversible evolutions inalgebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. ANo.24 (1977), v+104 pp.

F. Fagnola, Quantum Markov semigroups and quantum flows,Proyecciones 18 no.3 (1999), 144 pp.


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Classical and quantum Markov semigroupsbelton/www/notes/23iv14.pdf · 2014-04-23 · Classical...

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