An introduction to evolution PDEs November 16, 2018 CHAPTER 5 - MARKOV SEMIGROUP

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1 An inroucion o evoluion PDEs November 6, 8 CHAPTER 5 - MARKOV SEMIGROUP Conens. Markov semigroup. Asympoic of Markov semigroups 3.. Srong posiiviy coniion an Doeblin Theorem 3.. Geomeric sabiliy uner Harris an Lyapunov coniions 3 3. The renewal equaion 5. Markov semigroup In his chaper, we are inerese in Markov semigroups which is a class of semigroups which enjoy boh a posiiviy an a conservaiviy propery. The imporance of Markov semigroups comes from is eep relaion wih Markov processes in sochasic heory as well as from he fac ha a quie saisfacory escripion of he longime behaviour of such a semigroups can be performe. We sar wih he noion of posiiviy. I can be formulae in he absrac framework of Banach laices (X,, ) which are Banach spaces enowe wih compaible orer relaion or equivalenly wih an appropriae posiive cone X +. To be more concree, we jus observe ha he following hree examples are Banach laices when enowe wih heir usual orer relaion: X := C (E), he space of coninuous funcions which en o a infiniy (when E is no a compac se) enowe wih he uniform norm ; X := L p (E) = L p (E, E, µ), he Lebesgue space of funcions associae o he Borel σ-algebra E, a posiive σ-finie measure µ an an exponen p [, ]; X := M (E) = (C (E)), he space of Raon measures efine as he ual space of C (E). Here E enoes a σ-locally compac meric space (ypically E R ) an in he las example he posiiviy can be efine by ualiy: µ if µ, ϕ for any ϕ C (E). Lemma.. Consier X a Banach laice (one of he above examples), a boune linear operaor A on X an is ual operaor A on X. The following equivalence hols: () A is posiive, namely Af for any f X, f ; () A is posiive, namely A ϕ for any ϕ X, ϕ. The (elemenary) proof is lef as an exercise. We emphasize ha f, ϕ for any ϕ X + (resp. for any f X + ) implies f X + (resp. ϕ X +). There are wo equivalen (or ual ) ways o formulae he noion of Markov semigroup. Definiion.. On a Banach laice Y C (E) we say ha (P ) ia a (consans conservaive) Markov semigroup if () (P ) is a coninuous semigroup in Y ; () (P ) is posiive, namely P for any ; (3) (P ) is conservaive, namely Y an P = for any.

2 CHAPTER 5 - MARKOV SEMIGROUP Definiion.3. On a Banach laice X M (E) we say ha (S ) ia a (mass conservaive) Markov semigroup if () (S ) is a (srongly or weakly coninuous) coninuous semigroup in X; () (S ) is posiive, namely S for any ; (3) (S ) is conservaive, namely S f = f,, f X, where g := g,. The wo noions are ual. In paricular, if (P ) is a (consans conservaive) Markov semigroup on Y C (E), he ual semigroup (S ) efine by S := P on X := Y is a (mass conservaive) Markov semigroup. In he sequel we will only consier (mass conservaive) Markov semigroups efine on X L (E). Markov semigroup an semigroup of conracions for he L are closely linke. Proposiion.4. A Markov semigroup is a semigroup of conracions for he L norm. In he oher way roun, a mass conservaive semigroup of conracions for he L norm is posive, an hus i is a Markov semigroup. Proof of Proposiion.4. We fix f X an. We wrie S f = S f + S f S f + + S f = S f + + S T f = S f, where we have use he posiiviy propery in he hir line. We euce S f S f = f, because of he mass conservaion. The reciprocal par is lef o he reaer. We may also characerize a Markov semigroup in erms of is generaor. Theorem.5. Le S = S L be a srongly coninuous semigroup on a Banach space X L. There is equivalence beween (a) S L is a Markov semigroup; (b) L = an L saisfies Kao s inequaliy (sign f)lf L f, f D(L). Parial proof of Theorem.5. Sep. We prove (a) (b). On he one han, for any f D(L) an any ψ D(L ), we have ψ, (signf)lf = lim ψ, (signf)(s()f f) lim ψ, S()f f lim ψ, S() f f = lim S ()ψ ψ, f = L ψ, f, where we have use he inequaliy (signf)g g in he secon line an he posiiviy assumpion in he hir line. Tha inequaliy is he weak formulaion of Kao s inequaliy. On he oher han an similarly, for any f D(L), we have L, f =, Lf = lim, S()f f =, by jus using he mass conservaion propery. The reciprocal par is lef o he reaer.

3 CHAPTER 5 - MARKOV SEMIGROUP 3. Asympoic of Markov semigroups.. Srong posiiviy coniion an Doeblin Theorem. We consier he case of a srong posiiviy coniion. Theorem. (Doeblin). Consier a Markov semigroup S such ha S T f α ν f, f X +, for some consans T > an α (, ) an some probabiliy measure ν. There hols for some consans C an a <. Proof of Theorem.. wrie S f L C e a f L,, f X, f =, We fix f X such ha f = an we efine η := αν f + = αν f. We S T f = S T f + η S T f + η S T f + η + S T f η = S T f + η + S T f η, where in he las equaliy we have use he Doeblin coniion. Inegraing, we euce S T f S T f + α ν f + + S T f α ν f f + α f + + f α f ( α) f. By inucion, we obain a := [log( α)]/t an C := exp[ a T ]... Geomeric sabiliy uner Harris an Lyapunov coniions. We consier now a semigroup S wih generaor L an we assume ha (H) here exiss some weigh funcion m : R [, ) saisfying m(x) as x an here exis some consans α >, b > such ha L m α m + b; (H) for any R > here exis a consan T > an a posiive an no zero measure ν such ha S T f ν f, f X +. B R Proposiion. (Doeblin). Consier a Markov semigroup S on X := L (m) which saisfies (H) an (H). There hols for some consans C an a <. S f L (m) C e a f L (m),, f X, f =, We sar wih a varian of he key argumen in he above Doeblin s Proposiion. Lemma.3 (Doeblin s varian). Uner assumpion (H), if f L (m), wih m(x) as x, saisfies (.) f L 4 m(r) f L (m) an f =, we hen have S T f L ( ν ) f L.

4 4 CHAPTER 5 - MARKOV SEMIGROUP Proof of Lemma.3. Togeher wih (H), we ge From he hypohesis (.), we have f ± = f ± f ± B R BR c f f m m(r) 4 S T f ± ν 4 f =: η. f. We euce an nex S T f S T f + η + S T f η = S T f + η + S T f η = S T f η, S T f S T f η = f ν f, which is nohing bu he announce esimae. Proof of Theorem.. Sep. We spli he proof in several seps. We fix f L (m), f = an we enoe f := S f. From (H), we have from wha we euce f L (m) α f L (m) + b f L, f L (m) e α f L (m) + ( e α) b α f L. In oher wors, we have prove (.) S T f L (m) γ f L (m) + K f L, wih γ (, ) an K >. We fix R > large enough such ha K/A ( γ)/ wih A := m(r)/4. We also recall ha (.3) S T f L f L. We efine f β := f L + β f L (m), β >, an we observe ha he following alenaive hols (.4) f L (m) A f L or (.5) f L (m) > A f L. Sep. We observe ha uner coniion (.4), here hols (.6) S T f L γ f L, γ (, ), an more precisely γ := ν /, which is nohing bu he conclusion of Lemma.3. Sep 3. We claim ha uner coniion (.4), here hols ( γ + ) (.7) S T f β γ f β, γ := max, γ for β > small enough. Inee, using (.) an (.7), we compue an we ake β > such ha γ + Kβ γ. Sep 4. S T f β = S T f L + β S T f L (m) (γ + Kβ) f L + γβ f L (m), We claim ha uner coniion (.5), here hols (.8) S T f L (m) γ 3 f L (m), γ 3 := γ +.

5 CHAPTER 5 - MARKOV SEMIGROUP 5 Inee we compue Sep 5. S T f L (m) γ f L (m) + K A f L (m) = γ 3 f L (m) We claim ha uner coniion (.5), here hols (.9) S T f β γ 4 f β, γ 4 := γ 3 + /β + /β. Inee, using (.3) an (.8), we compue S T f β = S T f L + β S T f L (m) f L + γ 3 β f L (m), ( ε) f L + (ε + γ 3 β) f L (m), an we choose ε (, ) such ha ε = ε/β + γ 3. Sep 6. By gahering (.7) an (.9), we see ha we have S T f β γ 5 f β, γ 5 := max(γ, γ 4 ) (, ), for some well choosen β >. By ieraion, we ge S nt f β γ5 n f β, an we hen conclue in a sanar way. 3. An example: he renewal equaion We will iscuss now he renewal equaion for which we apply some of he resuls of he preceing secions in orer o ge some insigh abou is qualiaive behavior in he large ime asympoic. We are hus ineresing by he renewal equaion { f + (3.) x f + af = f(, ) = ρ f(), f(, x) = f (x), where f = f(, x),, x, an ρ g := g(y) a(y) y. Here f ypically represens a populaion of cells (paricles) which are aging (geing holer), ie (isappear) wih rae a, born again (reappear) wih age x = an has isribuion f a iniial ime. A leas a a formal level, any soluion of (3.) saisfies f x = so ha he mass is conserve. Similarly, we have f x = ( x f af) x = [ f ] af x =, ( x f a f ) x = [ f ] a f x, so ha he sign of he soluion is preserve by observing ha g = ( g + g)/ an using he above wo informaions. Tha seems o iniquae ha if (3.) efines a semigroup, his one is a L Markov semigroup. Preliminarily, we consier he (simpler) ranspor equaion wih bounary coniion { f + (3.) x f + af = f(, ) = ρ(), f(, x) = f (x), wih f an ρ are given aa. We observe ha when f is smooh (C ) an saisfies (3.), we have from wha we euce s [f( + s, x + s)ea(x+s) ] =, A(x) := x f(, x)e A(x) = f( s, x s)e A(x s), a(y) y,

6 6 CHAPTER 5 - MARKOV SEMIGROUP when boh erms are well efine. Choosing eiher s = or s = x, we ge (3.3) f(, x) = f (x ) e A(x ) A(x) x> + ρ( x) e A(x) x<. In he oher way roun, we may check ha for any smooh funcions a, f, ρ, he above formula gives a classical soluion o (3.) a leas in he region {(, x) R +, x }, an hus a weak soluion o (3.) in he sense (3.4) f ( ϕ x ϕ + aϕ) x f (x)ϕ(, x) x ρ()ϕ(, ) =, for any ϕ C c (R +). I is worh noicing ha his las equaion is also he weak formulaion of he evoluion equaion wih source erm f + x f + af = ρ()δ, f(, x) = f (x), efine on he all line (ha is for any x R). A leas a a formal level, for any soluion f o (3.), we may compue so ha (3.5) sup [,T ] f x = [ f ] a f x ρ(), T f() L f L + ρ(). Lemma 3.. Assume a L. For any f L (R + ) an α L (, T ) here exiss a unique weak soluion f C([, T ]; L (R + )) associae o equaion (3.). Proof Lemma 3.. Sep. Exisence. When a C b (R + ) an f, ρ Cc (R + ) he soluion is explicily given hanks o he characerisics formula (3.3). In he general case, we consier hree sequences (a ε ), (f,ε ) an (ρ ε ) of C b (R + ) an Cc (R + ) which converge appropriaely, namely a ε a a.e. an (a ε ) boune in L, f,ε f in L (R + ) an ρ ε ρ in L (, T ), an we see immeiaely from (3.5) ha he funcions (f ε ) an f efine hanks o he characerisics formula (3.3) saisfy f ε f in C([, T ]; L ). As a consequence, we may pass o he limi in (3.) an we euce ha f is a weak soluion o equaion (3.). Sep. Uniqueness. Consier wo weak soluions f an f o equaion (3.). The ifference f := f f saisfies (3.6) f ( ϕ x ϕ + aϕ) x =, for any ϕ C c (R +) an hus also for any ϕ C c (R +) W, (R +). Inroucing he semigroup (S g)(x) := g(x ) e A(x ) A(x) x>, associae o equaion (3.) wih no bounary erm, is ual is (S ψ)(x) := ψ(x + ) e A(x) A(x+), ψ L (R + ), an (S ) is well-efine as a semigroup in C c W, (R + ). Now, for ψ C c (R +), we efine an we compue x ϕ(, x) = ϕ(, x) := T from wha we euce = ϕ(, x) = ψ(, x) + T T (S s ψ(s, ))(x) s ψ(s, x + s ) e A(x) A(x+s ) s C c (R +) W, (R +), [ x ψ(s, x + s ) + ψ(s, x + s )(a(x) a(x + s ))] e A(x) A(x+s ) s, T [ x ψ(s, x + s ) + ψ(s, x + s )a(x + s )] e A(x) A(x+s ) s = ψ(, x) x ϕ(, x) + a(x)ϕ(, x).

7 CHAPTER 5 - MARKOV SEMIGROUP 7 Using hen his es funcion ϕ in (3.6), we ge an finally f = f. f ψ x =, ψ C c (R +), We are now in posiion o come back o he renewal equaion (3.). Lemma 3.. Assume a L. For any f L (R + ), here exiss a unique global weak soluion f C(R + ; L (R + )) associae o equaion (3.). We may hen associae o he renewal evoluion a Markov semigroup. Proof Lemma 3.. We efine E T := C([, T ]; L (R + )) an for any g E T, we efine f := Φ(g) E T he unique soluion o equaion (3.) associae o f an ρ() := ρ g() C([, T ]). For wo given funcions g, g E T an he wo associae images f i := Φ(g i ), we observe ha f := f f is a weak soluion o equaion (3.) associae o f() = an ρ() := ρ g() g (). The esimae (3.5) reas here sup [,T ] (f f )() L T ρ g() g ()) T a L sup [,T ] T (g g )() L. a(y) (g g )(, y) y Taking firs T small enough such ha T a L <, we ge he exisence an uniqueness of a fixe poin f = Φ(f) E T, which is nohing bu a weak soluion o he renewal equaion (3.). Ieraing he argumen, we ge he esire global weak soluion f C(R + ; L (R + )). We may apply he resuls of he firs secion in he semigroup chaper 3 in orer o ge he exisence of a semigroup S associae o he evoluion problem (3.). This semigroup is clearly posiive. Tha can be seen by consrucion for insance. Inee, if g E T,+ := { g E T, g }, hen f = Φ(g) E T,+ from he represenaion formula (3.3), an he fixe poin argumen can be mae in ha se. Nex, from (3.4), we classical euce (see chaper ) ha f ϕ R x = f ϕ R x + ( x ϕ R + aϕ R ) xs + ρ(s) s for ϕ R (x) := ϕ(x/r), ϕ Cc (R + ), [,] ϕ [,]. We ge he mass conservaion by passing o he limi as R. Lemma 3.3. Assume furhermore lim inf a a >. There exiss a unique saionary soluion F W, (R + ) o he saionary problem x F + af =, F () = ρ F, F, F =. Proof Lemma 3.3. From he firs equaion we have F (x) = Ce A(x), so ha he bounary coniion is immeiaely fulfille an he normalize coniion is fulfille by choosing C := e A(x). I is worh noicing ha he aiional assumpion implies e A(x) < so ha C > an he same is rue for F. Lemma 3.4. We sill assume a L an lim inf a a >. There exis C an α < such ha for any f L (R + ) he associae global soluionf o he renewal equaion (3.) saisfies f() f F L C e α f f F L,. Proof Lemma 3.4. We check Harris coniion. We observe ha a a / x x for some x >. We hen se T := x > an we ake f L (R + ). From (3.3), we have (3.7) f(t, x) ρ f(t x, ) e A(x) x<t/. wih ρ f(t x, ) = a a(y)f(t x, y) y x f(t x, y) y,

8 8 CHAPTER 5 - MARKOV SEMIGROUP Using he represenaion formula (3.3) again, we have so ha f(t x, y) f (y + x T ) e (A(y) A(y (x T ))) y>t x ρ f(t x, ) a Togeher wih (3.7), we obain f(t, x) a = ν(x) f (y + x T ) e (x T ) a y>t x, a x (x T ) a f (y + x T ) y>t x y e f (z) z>x+x T z e (x T ) a. f (z) z>x+x T z e (x T ) a e A(x) x<t/ f (z) z, ν(x) := a e (x T ) a e A(x) x<t/, which is precisely a Harris ype lower boun. We conclue hanks o Theorem..