Feller Semigroups

158 6 General Theory of Markov Processes

6.2 Feller Semigroups 159 Proposition 6.8 Let > 0, and setR D fRf W f 2 C₀.E/g. ThenR does not depend on the choice > 0. Furthermore,Ris a dense subspace of C₀.E/.

Proof If6D, the resolvent equation gives

Rf DR.f C./Rf/:

Hence any function of the formRf withf 2C₀.E/is also of the formRgfor some g2C₀.E/. This gives the first assertion.

Clearly,Ris a linear subspace ofC₀.E/. To see that it is dense, we simply note that, for everyf 2C₀.E/,

Rf D Z 1

0 e^tQtfdtD Z 1

0 e^tQt=fdt !

!1f inC₀.E/;

by property (ii) of the definition of a Feller semigroup and dominated convergence.

u t Definition 6.9 We set

D.L/D ff 2C₀.E/W Qtf f

t converges inC₀.E/whent#0g and, for everyf 2D.L/,

Lf Dlim

t#0

Qtff t :

ThenD.L/is a linear subspace ofC₀.E/andLWD.L/!C₀.E/is a linear operator called the generator of the semigroup .Qt/t0. The subspace D.L/is called the domainofL.

Let us start with two simple properties of the generator.

Proposition 6.10 Let f 2D.L/and s> 0. Then Qsf 2D.L/and L.Qsf/DQs.Lf/.

Proof Writing

Qt.Qsf/Qsf

t DQs

Qtf f t

and using the fact thatQsis a contraction ofC₀.E/, we get thatt¹.Qt.Qsf/Qsf/ converges toQs.Lf/, which gives the desired result. ut Proposition 6.11 If f 2D.L/, we have, for every t0,

Qtf Df C Z t

0 Qs.Lf/dsDfC Z t

0 L.Qsf/ds:

160 6 General Theory of Markov Processes

Proof Letf 2D.L/. For everyt0,

"¹.QtC"f Qtf/DQt."¹.Q"f f//!

"#0 Qt.Lf/:

Moreover, the preceding convergence is uniform when t varies over RC. This implies that, for everyx 2 E, the functiont 7! Qtf.x/is differentiable onRC

and its derivative isQt.Lf/.x/, which is a continuous function oft. The formula of the proposition follows, also using the preceding proposition. ut The next proposition identifies the domain D.L/ in terms of the resolvent operatorsR.

Proposition 6.12 Let > 0.

(i) For every g2C₀.E/, Rg2D.L/and.L/RgDg.

(ii) If f 2D.L/, R.L/f Df .

Consequently, D.L/DRand the operators RWC₀.E/!RandLWD.L/! C₀.E/are the inverse of each other.

Proof

(i) Ifg2C₀.E/, we have for every" > 0,

"¹.Q_"RgRg/ D "¹ Z ¹

0 e^tQ_"Ctgdt Z ₁

0 e^tQtgdt D "¹

.1e^"/ Z ₁

0 e^tQ_"Ctgdt Z _"

0 e^tQtgdt

!"!0 Rgg

using property (ii) of the definition of a Feller semigroup (and the fact that this property implies the continuity of the mappingt7!QtgfromRC intoC₀.E/).

The preceding calculation shows thatRg2D.L/andL.Rg/DRgg.

(ii) Letf 2D.L/. By Proposition6.11,Qtf Df CRt

0Qs.Lf/ds, hence Z ₁

0 e^tQtf.x/dtD f.x/

C

Z ₁

0 e^t Z ^t

0 Qs.Lf/.x/ds

dt D f.x/

C

Z ₁

0

e^s

Qs.Lf/.x/ds:

We have thus obtained the equality

Rf DfCRLf giving the result in (ii).

6.2 Feller Semigroups 161 The last assertions of the proposition follow from (i) and (ii): (i) shows thatR D.L/and (ii) gives the reverse inclusion, then the identities in (i) and (ii) show that

RandLare inverse of each other. ut

Corollary 6.13 The semigroup.Qt/t0is determined by the generator L (including also the domain D.L/).

Proof Letf be a nonnegative function inC₀.E/. ThenRf is the unique element of D.L/ such that . L/Rf D f. On the other hand, knowing Rf.x/ D R1

0 e^tQtf.x/dtfor every > 0determines the continuous functiont 7!Qtf.x/. To complete the argument, note thatQt is characterized by the values of Qtf for

every nonnegative functionf inC₀.E/. ut

Example It is easy to verify that the semigroup.Qt/t0of real Brownian motion is Feller. Let us compute its generatorL. We saw that, for every > 0andf 2C₀.R/,

Rf.x/D Z p1

2exp.p

2jyxj/f.y/dy:

Ifh 2 D.L/, we know that there exists anf 2 C₀.R/such thath D Rf. Taking D ¹₂, we have

h.x/D Z

exp.jyxj/f.y/dy:

By differentiating under the integral sign (we leave the justification to the reader), we get thathis differentiable onR, and

h⁰.x/D Z

sgn.yx/exp.jyxj/f.y/dy

with the notation sgn.z/D1fz>0g1fz0g(the value of sgn.0/is unimportant). Let us also show thath⁰is differentiable onR. Letx₀2R. Then, forx>x₀,

h⁰.x/h⁰.x0/DZ

sgn.yx/exp.jyxj/sgn.yx₀/exp.jyx₀j/ f.y/dy D

Z x x₀

exp.jyxj/exp.jyx₀j/ f.y/dy C

Z

RnŒx₀;xsgn.yx₀/

exp.jyxj/exp.jyx₀j/ f.y/dy: It follows that

h⁰.x/h⁰.x₀/ xx₀ !

x#x₀ 2f.x0/Ch.x0/:

162 6 General Theory of Markov Processes We get the same limit whenx"x₀, and we thus obtain thathis twice differentiable, andh⁰⁰D 2fCh.

On the other hand, sincehDR₁₌₂f, Proposition6.12shows that .1

2 L/hDf henceLhD fC ¹₂hD ¹₂h⁰⁰.

Summarizing, we have obtained that

D.L/ fh2C².R/Whandh⁰⁰2C₀.R/g and that, ifh2D.L/, we haveLhD ¹₂h⁰⁰.

In fact, the preceding inclusion is an equality. To see this, we may argue in the following way. Ifgis a twice differentiable function such thatgandg⁰⁰are inC₀.R/, we setf D ¹₂.gg⁰⁰/2C₀.R/, thenhDR₁₌₂f 2D.L/. By the preceding argument, his twice differentiable andh⁰⁰ D 2fCh. It follows that.hg/⁰⁰Dhg. Since the functionhgbelongs toC₀.R/, it must vanish identically and we getgDh2D.L/. Remark In general, it is very difficult to determine the exact domain of the generator. The following theorem often allows one to identify elements of this domain using martingales associated with the Markov process with semigroup .Qt/t0.

We consider again a general Feller semigroup.Qt/t0. We assume that on some probability space, we are given, for everyx2E, a process.X_t^x/t0which is Markov with semigroup.Qt/t0, with respect to a filtration.Ft/t0, and such thatP.X₀^x D x/D1. To make sense of the integrals that will appear below, we also assume that the sample paths of.X_t^x/t0 are càdlàg (we will see in the next section that this assumption is not restrictive).

Theorem 6.14 Let h;g2C₀.E/. The following two conditions are equivalent:

(i) h2D.L/and LhDg.

(ii) For every x2E, the process

h.Xt^x/ Z t

0 g.Xs^x/ds is a martingale, with respect to the filtration.Ft/.

Proof We first prove that (i))(ii). Leth2D.L/andgDLh. By Proposition6.11, we have then, for everys0,

QshDhC Z s

0 Qrgdr:

6.2 Feller Semigroups 163

It follows that, fort0ands0,

EŒh.XtCs^x /jFtDQsh.Xt^x/Dh.Xt^x/C Z s

0 Qrg.Xt^x/dr:

On the other hand, E

h Z ^tCs

t

g.X^xr/drˇˇˇFt

iD Z tCs

t

EŒg.Xr^x/jFtdrD Z tCs

t

Qrtg.X^xt/dr D

Z s

0 Qrg.X^xt/dr:

The fact that the conditional expectation and the integral can be interchanged (in the first equality of the last display) is easy to justify using the characteristic property of conditional expectations. It follows from the last two displays that

E h

h.X_tCs^x / Z _tCs

0 g.X^x_r/drˇˇˇFt

iDh.X^x_t/ Z t

0 g.X_r^x/dr giving property (ii).

Conversely, suppose that (ii) holds. Then, for everyt0, E

hh.Xt^x/ Z t

0 g.X^xr/dr

iDh.x/

and on the other hand, from the definition of a Markov process, E

h h.X_t^x/

Z t

0 g.X_r^x/dr

iDQth.x/ Z t

0 Qrg.x/dr: Consequently,

Qthh

t D 1

t Z t

0 Qrgdr!

t#0 g

inC₀.E/, by property (ii) of the definition of a Feller semigroup. We conclude that

h2D.L/andLhDg. ut

Example In the case ofd-dimensional Brownian motion, Itô’s formula shows that, ifh2C².R^d/,

h.Xt/ 1 2

Z t

0 h.Xs/ds

is a continuous local martingale. This continuous local martingale is a martingale if we furthermore assume thathandhare inC₀.R^d/(hence bounded). It then follows

164 6 General Theory of Markov Processes

from Theorem6.14thath2D.L/andLhD ¹₂h. Recall that we already obtained this result by a direct computation ofLwhendD1(in fact in a more precise form since here we can only assert thatD.L/ fh 2 C².R^d/ W handh 2 C₀.R^d/g, whereas equality holds ifdD1).