Lectures Markov processes

Xe to PCXTEBITst ps.eus B

for Cpsf see a family of consistenttrains prob

if I not specified 7 71Homogeneous if Pat Pe s is a Markov

semigroup

ti tz tn V prob mess on X gV Poe pen ta prob distron X Tg'tBy def ft boy set

EHHH IF Jfk os.tcxs.dex

Cps t f Cx ffcysps.tkdyEffart Ifs Cpstf Xs

Map Ps t i b L bfto ft E Sta f fa E BLh Z IR hueby

o

has facie fixed

Elk.CH Ft.I ElffXtli.iffXdlFe

EIEHEXHIFta.dff.tn i ffXttlFt.ILFlCPtn tnfnXXtn1fn iCXtn.t tt l7eo7

hoCXt

he icxt fe.CH ee tehe G fokki

Elf.dk flXdl7tI pe.t.CfiPe.ezCfz tn

theorem X G B isomorphic Pst consistent

v prob measure on XEy Define foldsMoti tn V Pot Otta c t

Thene F SP Xe having these fold2 Xt is a Markovprocesstrprob psst3 Any MP with trprobyppsithdnds.mest part B V

law as Xt D

Proof 1 Canonical construction Need to checkconsistent pst consistent fold

V Pot Pta tn AE Tk An

V Pta Ptu Aoi XA FAkF An

use Pth Perth Pth it i

2 Need to check VtzsPCXTEBIFIIpstcxs.to

to do this show V A EGGS

PCXTEB.Xeo.ge tl ELtIxeo aPstCXs.Blb

Sufficient to check on

A Xs EA Xs Are Vs.ssz.ES ss

A AKELIPCXtEBXs.CA Xs EA

ECPsfXs B IKEA tea

check equality by recursion formula3 Any MP with PCXIEB v B andtrans prob

Pst has same law as XtIndeed musthave same fold's Recursion

same law D

wts a

PCXEA IG Y for XE m F YangCr F P GEF

call Z PCXEAIGwts 2 Y a s

E I ECYI TAEL Ya'sequivalently W cmy W Y

E WII O VAE.by W O a s

A w W OY EW 1 0

Pu the law of MP Xt with trans prob PstXo V unique on

Hot gamyPx Pq xEX

Lemmy TAE geom xn RICAins measurable on L and

113 1,117 olds t.PL fxlPxlAIvcdD

Example Xt o indep increments

is Markov

PCXtEBIFIJ PCXEXSEB X.ITPsfXs B

PstGB P sCB x

If Xt stationary indep increm

homogeneous MP because

Pxt xs only depend out s

Eg Wt standard Wiener process

Homogeneous Markov

X Ziatt Wt for Zindepotchtalso Markov Xt Xs alt 5 NEWS

homog

Peady texp_Y 5fdy

Def P law MP with semigroupahomogeneous

prob measure von G is invariant forptif Pu ftp.vldx is invariantunder

shift

Pu A 11765CAD is

Lemmy v invariant pt ifOpec Xx B vCB ABEL

tzoie Pelt B Hok VB g

Proposition Markovproperty Pt oMarkov semigroup

on X Ey Px law of Xt to homogMPstarted at x he byThen

µ Exhix is in mg2 Ethos XIII Ex NX

ex

m

Proof Prove 1 and 2 forOst s Ethha fixed fnktnf fuEbey

and extend to generalhebgby monotone class linearity ofexpt bold our

h Osx fncxs.it o ofnCxs tn

EChlosXllFsT EIffXs tI ifIXs t lIs7ho Xs

hncxtfr.CH he IG fe.filCPte.ys tetshe

he IGI fe.fxl.CItete.fre Cx Pte te i

rds E h X

E h X E en fixed hok

Ex.hu h.CXs equal to IhsD

Px Ps prob measure on 2190 g40

p X90 1

P w withEA whileAn

potCx XO Pta tn CAH An

P w att cAG Path Al

Fat consistent trans prob

Fat pts Pt

Pt Xx G 7 Eo DG B 1 Pth B

Ptps pt sle.pt GB JPsly Bptk.dyl.PoPEPt

Pt tzo Pe Ps 12 PtPs Pets