Markov Semigroups - Dipartimento di Matematica daipra/didattica/Bologna12/MarkovSem...Markov Semigroups

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  • Markov Semigroups

    Doctoral School, Academic Year 2011/12

    Paolo Guiotto

    Contents

    1 Introduction 1

    2 Preliminaries: functional setting 2

    3 Markov Processes 3

    4 Feller processes 4

    5 Strongly continuous semigroups on Banach spaces 8

    6 HilleYosida theorem 13

    7 Generators of Feller semigroups 17

    8 Examples 208.2 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    8.2.1 Case d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.3.1 Case d > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    8.11 Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    9 Exponential Formula 28

    1 Introduction

    The main aim of these notes is to connect Markov processes to semigroups of linear operators on functionsspaces, an important connection that allows to a very useful and natural way to define Markov processesthrough their associated semigroup.

    Therere lots of different definitions of Markov process in the literature. If this create a little bit ofconfusion at first sight, all of them are based of course on the same idea: a Markov process is someevolution phenomena whose future depends upon the past only by the present. Actually in most of theapplications we are interested in families of Markov processes living in some state space E characterized bya parameter x E which represents the starting point for the various processes of the family. Moreover,we could define the processes as usual stochastic processes (that is functions of time and of some randomparameter) or, and it is what we prefer here, through their laws, that is measures on the path space that,

    1

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    for a general Markov process, is the space DE [0,+[ of Evalued functions continuous from the rightand with limit from the left (so they may have jumps).

    Like for ordinary dynamical systems an eventually non linear dynamics induces naturally a lineardynamics on observables, that is numerical functions defined on the state space E. We gain somethingin description (linearity) but we have to move to an infinite dimensional context (functions space). Apriori this is not better or worse, but for some questions may be better to treat with an eventuallyinfinite dimensional but linear setting. In many applications (e.g. Markov processes arising as diffusionsor interacting particle systems) this approach gives a very quick way to define the process itself definingjust a linear (generally unbounded) operator on observables: the so called infinitesimal generator.

    2 Preliminaries: functional setting

    Along this section we will provide the preliminaries about the main settings where we will work. In allwhat follows, (E, d) will play the role of state space and will be a locally compact metric space. We willcall

    B(E) the algebra of Borel sets of E;

    B(E) the set of bounded and measurable real valued functions on E: in particular we recall thata function : E R is called measurable if (A) B(E) for any Borel set A.

    C0(E) the set of continuous real valued functions on E vanishing at infinity. By this we mean, inparticular that

    x0 E, : > 0, R() > 0, : |f(x)| 6 , x E, : d(x, x0) > R(). (2.1)

    Of course C0(E) B(E). On these spaces it is defined a natural norm

    := supxE|(x)|, B(E).

    It is a standard work to check that B(E) and C0(E) are Banach spaces with this norm. In general, afunction : E R will be called observable. Moreover, if f C0(E) the supnorm is actually a truemaximum as it is easily proved applying the Weierstrass theorem being E locally compact. Sometimesit will be useful to recall the

    Theorem 2.1 (Riesz). The topological dual of C0(E) is the space of all bounded real valued measure onB(E). In particular

    ?, =E

    (x) (dx).

    Moreover the C0(E)? = closurex : x E, where x, = (x).

    The natural space for trajectories of Evalued Markov processes is the space

    DE [0,+[:= { : [0,+[ E, right continuous and with left limit} .

    Frenchmen call this type of trajectories cadlag: continue a droite et avec limite a gauche. The space Eis called states space. We define also the classical coordinate mappings

    t : DE [0,+[ E, t() := (t), t > 0.

    Moreover, we will define

  • 3

    F the smallest algebra of DE [0,+[ such that all t are measurable;

    Ft the smallest algebra of DE [0,+[ such that all s for s 6 t are measurable.

    Clearly (Ft) is an increasing family of algebras.

    3 Markov Processes

    Definition 3.1. Let (E, d) be a metric space. A family (Px)xE of probability measures on the path space(DE [0,+[,F ) is called Markov process if

    i) Px ((0) = x) = 1, for any x E.

    ii) (Markov property) Px ((t+ ]) F | Ft) = P(t)(F ), for any F F and t > 0.

    iii) the mapping x 7 Px(F ) is measurable for any F F .

    Let (Px)xE be a Markov process. We denote by Ex the expectation w.r.t. Px, that is

    Ex[] =

    DE [0,+[

    dPx, L1(DE [0,+[,F ,Px).

    The Markov property has a more flexible and general form by means of conditioned expectations:

    Ex [((t+ ])) | Ft] = E(t) [] , L. (3.1)

    We now introduce the fundamental object of our investigations: lets define

    S(t)(x) := Ex [((t))] DE [0,+[

    ((t)) dPx(), B(E). (3.2)

    We will see immediately that any S(t) is well defined for t > 0. The family (S(t))t>0 is called Markovsemigroup associated to the process (Px)xE . This is because of the following

    Proposition 3.2. Let (Px)xE be a Markov process on E and (S(t)t>0 be the associated Markov semi-group. Then:

    i) S(t) : B(E) B(E) is a bounded linear operator for any t > 0 and S(t) 6 for any B(E), t > 0 (that is S(t) 6 1 for any t > 0).

    ii) S(0) = I.

    iii) S(t+ r) = S(t)S(r), for any t, r > 0.

    iv) S(t) > 0 a.e. if > 0 a.e.: in particular, if 6 a.e., then S(t) 6 S(t) a.e..

    v) S(t)1 = 1 a.e. (here 1 is the function constantly equal to 1).

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    Proof i) It is standard (about the measurability proceed by approximation: the statement is true for = A,A B(E) by iii) of the definition of Markov process because

    S(t)A(x) = Px((t) A) = Px (t (A)) ,

    and F := t (A) F ; hence it holds for sum of A, that is for simple functions; for general take firstf > 0 and approximate it by an increasing sequence of simple functions). Linearity follows by the linearity of theintegral. Clearly

    |S(t)(x)| 6 Ex [|((t))|] 6 , x E, = S(t) 6 , t > 0.

    In other words S(t) L (B(E)) and S(t) 6 1.

    ii) Evident.

    iii) This involves the Markov property:

    S(t+ r)(x) = Ex [((t+ r))] = Ex [Ex [((t+ r)) | Ft]](3.1)= Ex

    [E(t) [((r))]

    ]= Ex [S(t)((r))]

    = S(r) [S(t)] (x).

    iv), v) Evident.

    4 Feller processes

    To treat with bounded measurable observables is in general quite difficult because of their poor properties,so its better to restrict to continuous observables:

    Definition 4.1 (Feller property). Let (S(t))t>0 be the Markov semigroup associated to a Markov process(Px)xE where (E, d) is locally compact. We say that the semigroup fulfills the Feller property if

    S(t)f C0(E), f C0(E), t > 0.

    This property turns out to give the strongly continuity of the semigroup:

    Theorem 4.2 (strong continuity). Let (S(t))t>0 be the Markov semigroup associated to a Markov process(Px)xE where (E, d) is locally compact. If (S(t))t>0 fulfills the Feller property, it is then stronglycontinuous on C0(E), that is

    S() C ([0,+[; C0(E)), C0(E).

    Proof First we prove right weak-continuity, that is

    limtt0+

    S(t)(x) = S(t0)(x), x E, Cb(E).

    This follows immediately as an application of dominated convergence and because trajectories are right continuous.Indeed

    limtt0+

    S(t)(x) = limtt0+

    DE [0,+[

    ((t)) Px(d).

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    Now: |((t))| 6 which is Pxintegrable, and (t) (t0) as t t0 because DE [0,+[. In asimilar way we have

    limtt0

    S(t)(x), x E, C0(E), t0 > 0.

    So: fixing x E we have that the function t 7 S(t)(x) has limit from the left and from the right at any pointof [0,+[. It is a standard first year Analysis exercise to deduce that S()(x) has at most a countable numberof discontinuities, hence is measurable.

    Now, define

    R(x) :=

    +0

    etS(t)(x) dt, > 0, x E.

    We will see later what is the meaning of this. The integral is well defined and convergent because

    |etS(t)(x)| = et|S(t)(x)| 6 et.

    We say that R C0(E) if C0(E). Indeed: everything follows immediately as application of dominatedconvergence because any S(t) C0(E) and because of the usual bound |S(t)(x)| 6 .

    We will show now strong continuity for of type R, that is S()R C ([0,+[; C0(E)). This willbe done in steps: the first is to prove strong right continuity, that is

    S()RC0(E) S(t0)R, as t t0 + .

    We start with the case t0 = 0. Notice that

    S(t)R(x) = S(t)

    +0

    erS(r)(x) dr =

    +0

    erS(r + t)(x) dr = et +t

    erS(r)(x) dr,

    hence

    S(t)R(x)R(x) = (et 1) +t

    erS(r)(x) dr +

    t0

    erS(r)(x) dr,

    therefore

    S(t)RR 6(et 1

    ) +t

    er dr + t0

    er dr 6et 1 + t 0,

    as t 0+. For generic t0 we have

    S(t)R S(t0)R = S(t0) (S(t t0)RR) 6 S(t t0)RR 0,

    as t t0+. Now we can prove the left continuity at t0: assuming now t < t0,

    S(t)R S(t0)R = S(t) (S(t0 t)RR) 6 S(t0 t)RR 0, t t0 .

    We will now show that the set of R ( > 0) is dense in C0(E). To this aim take ? C0(E)? such that

    0 = ?, R =E

    R(x) d(x