Physica D. The semigroup approach to transport processes in networks. B. Dorn a, M. Kramar Fijavž b,c, R. Nagel a,, A. Radl a.

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1 Physica D 239 (2010) Contnts lists availabl at ScincDirct Physica D journal hompag: Th smigroup approach to transport procsss in ntworks B. Dorn a, M. Kramar Fijavž b,c, R. Nagl a,, A. Radl a a Mathmatischs Institut, Univrsität Tübingn, Auf dr Morgnstll 10, D Tübingn, Grmany b Univrsity of Ljubljana, Faculty of Civil and Godtic Enginring, Jamova 2, SI-1000 Ljubljana, Slovnia c Institut of Mathmatics, Physics, and Mchanics, Jadranska 19, SI-1000 Ljubljana, Slovnia a r t i c l i n f o a b s t r a c t Articl history: Availabl onlin 21 Jun 2009 Ddicatd to th lat Aldo Bllni-Morant in rcognition of his contributions to smigroup and transport thory. W xplain how oprator smigroups can b usd to study transport procsss in ntworks. This mthod is applid to a linar Boltzmann quation on a finit as wll as on an infinit ntwork and yilds wllposdnss and information on th long trm bhavior of th solutions to th prsntd problms Elsvir B.V. All rights rsrvd. Kywords: Smigroups Transport quation Ntworks Asymptotic priodicity 1. A transport problm Transport of particls,.g., in a nuclar ractor, can b dscribd by th classical linar Boltzmann quation (s [1]). For this purpos w dnot by u(s, c, t) th particl dnsity dpnding on position s = (s 1, s 2, s 3 ) in som boundd rgion S R 3, on vlocity c blonging to C = {c = (c 1, c 2, c 3 ) R 3 c min c 2 c max }, and on tim t 0. Assuming that th particls mov with vlocity c, ar absorbd according to an absorption function σ (, ) on S C, and ar scattrd according to a scattring krnl k(,, ) on S C C, th volution in tim of th transport procss can b dscribd by th quation 3 t u(s, c, t) = i=1 C c i u(s, c, t) σ (s, c)u(s, c, t) si + k(s, c, c )u(s, c, t) dc Corrsponding author. Tl.: ; fax: addrsss: brdo@fa.uni-tubingn.d (B. Dorn), marjta.kramar@fgg.uni-lj.si (M. Kramar Fijavž), rana@fa.uni-tubingn.d (R. Nagl), agra@fa.uni-tubingn.d (A. Radl). (BE) with som initial condition u(s, c, 0) = u 0 (s, c), s S, c C, and, dpnding on th gomtry of S, som conditions at th boundary of S. Abstract functional analytic mthods hav bn applid to this quation with considrabl succss, and w rfr to th arly paprs by Birkhoff [2] or Vidav [3,4], and th monographs by Bllni-Morant [5,6], Mokhtar-Kharroubi [7] or Banasiak and Arlotti [8]. Th spcial cas of a slab gomtry, i.., if S is an intrval, is of particular intrst to us and w mntion th paprs by Boulanouar [9,10] Transport problm in ntworks In this articl w prsnt som vry rcnt rsults on transport procsss taking plac in a ntwork. In mathmatical trms this mans that w considr th abov Boltzmann quation (BE) on a collction of intrvals connctd in crtain nods. Hnc, th undrlying rgion S is givn by a dirctd graph G = (V, E) with vrtics V = {v 1,..., v n } and dirctd dgs E = { 1,..., m }. W dscrib th procss by th mass distribution on th dgs but assum that important physical phnomna (.g., rdistribution and scattring) happn in th vrtics. Our basic assumptions on th procss ar th following /$ s front mattr 2009 Elsvir B.V. All rights rsrvd. doi: /j.physd

2 B. Dorn t al. / Physica D 239 (2010) On ach dg th particls ar flowing only in on dirction (with vlocity c satisfying 0 < c min c c max ). No mass is gaind or lost during th procss. In particular, no absorption taks plac along th dgs, and in ach nod w hav a Kirchhoff law incoming matrial = outgoing matrial. In ach vrtx v i th incoming matrial is distributd into th outgoing dgs j according to som wights w ij 0 satisfying j:v i j w ij = 1 for ach i = 1,..., n. (1) Th scattring taks plac only in th vrtics. Th graph structur is dscribd by th outgoing incidnc matrix Φ = ( φ ) ij with n m { φ ij := 1, if v i is starting point of j, v i j, 0, othrwis, and th incoming incidnc matrix Φ + = ( φ + ) ij with n m { φ + ij := 1, if v i is ndpoint of j, j vi, 0, othrwis. Th dgs ar normalizd as j = [0, 1] and paramtrizd contrary to th dirction of th flow on thm, i.., th matrial flows from 1 to 0. Undr ths assumptions w first considr th simpl procss whr all particls on th dg j mov with th sam spd c j > 0 and thr is no scattring. In this cas th particl dnsity on th jth dg at position s [0, 1] and at tim t is givn by u j (s, t) and th Boltzmann quation on th ntwork (dscribd by th graph G) taks th following form. t u j (s, t) = c j s u j (s, t), s (0, 1), t 0, (BEN) u j (s, 0) = f j (s), s (0, 1), (IC) m φ ij u j (1, t) = w ij φ + ik u k (0, t), t 0 (BC) k=1 for i = 1,..., n and j = 1,..., m. A mor sophisticatd modl is obtaind if w allow diffrnt individual vlocitis of th particls on th sam dg and assum scattring in th vrtics. This mans that th particls may chang thir vlocitis in th vrtics and ar thn rdistributd among th outgoing dgs according to th wights. In ach of th vrtics th scattring obys th sam ruls and is dscribd by a scattring oprator J L(L 1 [c min, c max ]). In mathmatical trms this problm is formulatd as t u j(s, c, t) = c s u j(s, c, t), x (0, 1), c [c min, c max ], t 0, (BEN S ) u j (s, c, 0) = f j (s, c), s (0, 1), c [c min, c max ], (IC S ) m φ ij u j(1,, t) = w ij J φ + ik u k(0,, t), t 0, (BC S ) k=1 whr j = 1,..., m, i = 1,..., n. In th following (s Sction 2) w xplain how to trat ths problms using abstract mthods from th thory of oprator smigroups, combind with tools from graph thory. Our first aim is to prov th xistnc and uniqunss of solutions to th initial valu problms (BEN) and (BEN S ). W ar thn intrstd in th qualitativ bhavior, spcially in th long trm mass distribution in th ntwork Th rsults In a sris of paprs (s [11 17]) th problms (BEN) and (BEN S ) as wll as som gnralizations ar tratd using smigroup mthods. Th following main rsults will b discussd in Sction 3. Rsult 1 (S Thorms 5 and 9). Th problms (BEN) and (BEN S ) ar wll-posd, i.., for ach initial mass distribution (IC) or (IC S ), thr xists a uniqu solution dpnding continuously on th initial distribution. Rsult 2 (S Thorms 7 and 10). Undr appropriat assumptions on th flow vlocitis, th solutions to (BEN) and (BEN S ) bhav asymptotically priodic, i.., for ach initial distribution th solution approachs in th long run a distribution that is priodic in tim. This rsult says that thr xists a nonngativ numbr τ (th priod) such that th mass distribution on th ntwork at larg tim t is up to a small ε th sam as th mass distribution at tim t + τ. Not that τ = 0 mans convrgnc to a tim invariant distribution. As a consqunc on obtains that th tim mans always convrg to an quilibrium. Corollary 3. Th solution u(t) = (u 1 (t),..., u m (t)) to th initial valu problm (BEN) or (BEN S ), rspctivly, satisfis 1 lim t t t 0 u(s) ds = u for som stationary mass distribution u on th ntwork. In Sction 4 w gnraliz th problm (BEN) to infinit ntworks, s (BEN I ) in Sction 4.1, and prsnt th appropriat analogus to th abov rsults. 2. Th mthod 2.1. Th smigroup approach in a nutshll W now brifly xplain th basic ida of smigroup thory and rfr to [18,19] for furthr rading. Th smigroup approach to th systms (BEN) and (BEN S ), as wll as to many othr linar initial valu problms (IVP) govrnd by partial, intgro or functional diffrntial quations (s [18, Chap. VI]), procds as follows. First, choos a Banach spac X, usually som function spac, containing th initial valus and all possibl stats of th systm. This choic is ssntial and should rspct physical intuition as wll as mathmatical nds. W call th spac X th stat spac of (IVP). Nxt, writ th diffrntial xprssion in (IVP) as a linar oprator A : D(A) X X. A carful choic of th domain D(A) of A must includ th boundary conditions in (IVP), if thr ar any. Th original problm thn taks th following form: Find a function v : R + X such that d v(t) = Av(t), t 0, dt (ACP) v(0) = v 0 X. In this way th possibly complicatd (IVP) is transformd into an quivalnt simpl Banach spac valud ordinary diffrntial quation calld abstract Cauchy problm (s [18, Df. II.6.1]). It is for such abstract Cauchy problms that smigroup thory provids sophisticatd tools allowing th following stps.

3 1418 B. Dorn t al. / Physica D 239 (2010) Using th Hill Yosida thorm (or on of its many variants, s [18, Sc. II.3]) on provs that A is th gnrator of a strongly continuous oprator smigroup (T(t)) t 0 on X. This implis that (ACP) is wll-posd (c.f. [18, Cor. II.6.9]) and th uniqu solution to (ACP) is obtaind by th smigroup as v(t) = T(t)v 0 for all t 0. A dtaild analysis of th spctrum σ (A) and th rsolvnt R(λ, A) of th gnrator A yilds information on rgularity, positivity or othr qualitativ proprtis of (T(t)) t 0, hnc of th solutions to (ACP). In particular, this spctral analysis prmits conclusions on th bhavior of (T(t)) t 0 as t and phnomna such as stability, priodicity or hyprbolicity can b charactrizd. In th final stp on may translat th information gaind on th smigroup (T(t)) t 0 back into th languag of th original (IVP). In th following w xplain this stratgy for flows in ntworks dscribd by (BEN), (BEN S ), and (BEN I ) Toolbox from graph thory Hr w collct th trminology and rsults from graph thory ndd in ordr to dscrib th ntwork in which th transport procsss tak plac. W will s that th bhavior of th solutions to our problms dply rlis on th ntwork structur. In our mathmatical modl this structur appars as boundary condition rflctd in th domain of th appropriat gnrator A. A good dscription of th ntwork is thus ssntial for our study. Our main rfrncs for graph thory ar [20 22] Adjacncy matrics In addition to th incidnc matrics from Sction 1, an quivalnt way to dscrib th structur of a wightd graph is via th so-calld adjacncy matrics. Th ntris of th wightd (transposd) adjacncy matrix A ar { (A) ij = w jk, if thr is an dg k = (v j, v i ), v j k ω vi, jk 0, othrwis. Anothr option is th wightd (transposd) adjacncy matrix of th lin graph B cf. [22, Sction 8.2/8.3]. Its ntris ar givn as w ki, if j has its had in v k and i (B) ij = has its tail in v k, j v k i ω, ki 0, othrwis. Not that th wightd adjacncy matrics A and B dscrib not only th graph structur but contain also full information on th distribution of th flow. By (1), thy ar both column stochastic. Whil th matrix A is mor intuitiv and usful to study th spctrum of th gnrator A, using B turns out to b most fficint to dscrib th domain D(A) and vn th smigroup (T(t)) t Paths and cycls A (dirctd) path btwn two vrtics v, w V is a st of dgs l1,..., lk E such that v is th tail of l1, th had of li is th tail of li+1 for all i = 1,..., k 1, and w is th had of lk. W call v th starting point, w th ndpoint of th path. Th lngth of a path is dfind as th numbr of its dgs, whras th wight of a path is th product of th wights of its dgs. A cycl is a path in which th starting point coincids with th ndpoint, and th lngth of a cycl is just th lngth of this path. W will s that cycls in th ntwork and thir wightd lngths dtrmin th longtim bhavior of th solutions to our problms. v 1 v 5 v 4 v 2 v 3 Fig. 1. A finit, dirctd, wightd, strongly connctd graph Strong connctivity A graph is strongly connctd if thr is a dirctd path of finit lngth btwn any pair of distinct vrtics v i, v j V. Th strong connctivity of th graph can b charactrizd by proprtis of th adjacncy matrics A and B as wll as by th associatd smigroup (T(t)) t 0. Proposition 4 ([15, Prop. 4.9]). For a graph G, its adjacncy matrics A and B, and th corrsponding flow smigroups (T(t)) t 0 and (T S (t)) t 0 from Sction 3 blow, th following proprtis ar quivalnt. (i) G is strongly connctd. (ii) A is irrducibl (in th sns of [23, Df. I.6.1]). (iii) B is irrducibl (in th sns of [23, Df. I.6.1]). (iv) (T(t)) t 0 and/or (T S (t)) t 0 is irrducibl (in th sns of [24, Df. C-III 3.1]). Th prvious notions ar visualizd in th abov xampl (Fig. 1) of a finit, strongly connctd graph with wightd dgs. 3. Th flow smigroup W shall dmonstrat th valu of th smigroup approach combind with graph thory to our problms (BEN) and (BEN S ). In th following, th undrlying graph G is assumd to b strongly connctd Linar Boltzmann quation in ntworks with no scattring Th basic cas (BEN) has bn considrd first by Kramar and Sikolya in 2005 [11] and motivatd a sris of furthr invstigations, s.g. [12,13,25,17]. Following th approach outlind in Sction 2, thy procd by first choosing th Banach spac X = (L 1 [0, 1]) m as th stat spac of th systm. Thn th oprator d c 1 0 dx A =... with domain d 0 c m dx D(A) = {v (W 1,1 [0, 1]) m v(1) = Bv(0)} is th abstract vrsion of th transport quations (BEN). In particular, th adjacncy matrix B of th lin graph guarants that th functions v D(A) fulfill th conditions (BC). Thir first rsult yilds wll-posdnss of th systm (BEN). Thorm 5 ([11, Prop. 2.5]). Th oprator (A, D(A)) gnrats a positiv, boundd, strongly continuous smigroup (T(t)) t 0 on X. Th abstract Cauchy problm (ACP), hnc (BEN), is wll-posd, and its solutions ar givn by v(t) = T(t)v 0 for t 0.

4 B. Dorn t al. / Physica D 239 (2010) In th cas of constant vlocitis qual to 1, th gnratd smigroup allows an xplicit rprsntation. It acts on th mass distributions as a shift, combind with a jump provokd by th adjacncy matrix B of th lin graph. Corollary 6 ([15, Prop. 3.3]). If c j = 1 j, th smigroup (T(t)) t 0 is givn xplicitly as T(t)f (s) = B n f (t + s n) if n t + s < n + 1, n N 0, (2) for s [0, 1], f X and t 0. Following th lin of argumnts from Sction 2 w now nd som information on th spctrum σ (A). As it turns out, th spctrum of th (unboundd) oprator (A, D(A)) can b dscribd via a simpl charactristic quation for a scalar n n adjacncy matrix of th graph, s [11, Cor. 3.4]. Th form of th spctrum dpnds on th cycl structur of th graph as wll as on th rational linar dpndncy of th flow vlocitis on th appropriat dgs. Thr ar two possibilitis: (LD) Q Thr ( xists a ) ral numbr k > 0 such that 1 k c i c il N for all i1,..., il forming a cycl in G; (LI) Q Such a numbr k dos not xist. Thn smigroup thory allows us to dscrib th asymptotic bhavior of (T(t)) t 0. First, on can dcompos th stat spac X = X s X r into th dirct sum of th stabl spac X s, whr T(t) convrg to 0 as t, and th rvrsibl spac X r, on which th smigroup bcoms a priodic group, i.., T(t + σ )f = T(t)f for som σ > 0 and all t 0, f X r. W call τ := inf{σ > 0} th priod and rmark that, if τ = 0, thn T(t) Xr is th idntity for all t 0. To prov th priodic bhavior on X r, positivity and irrducibility of th smigroup play a crucial rol. Combining th bhavior on X s and X r, on obtains convrgnc to a priodic group with ithr priod τ > 0 (s [11, Thm. 4.5] or τ = 0 (s [13, Thm. 4.22]), according to th cass (LD) Q or (LI) Q. Thorm 7 ([11, Thm. 4.5] and [13, Thm. 4.22]). Th spac X can b dcomposd as X = X r X s such that th following proprtis ar fulfilld. In th cas of condition (LD) Q : (i) Th smigroup is uniformly stabl on th spac X s, i.., T(t) Xs t 0. (ii) Th smigroup is priodic on th spac X r and vn similar to a rotation group on L 1 (Γ ), whr Γ := {z C z = 1}. Th priod dpnds both on th structur of th graph and th vlocitis and can b computd as τ = 1 ( 1 {k k gcd ) i1,..., il c i1 c il } form a cycl in G, (3) whr k > 0 is any positiv ral numbr satisfying condition (LD) Q. In th cas of condition (LI) Q : (iii) Th smigroup is strongly stabl on th spac X s, i.., T(t)f t 0 f X s. (iv) Th spac X r = fix(t(t)) t 0 = kr A (4) is on dimnsional and spannd by a strictly positiv ignvctor of A. W xplain th significanc of th abov thorm. Indpndnt of th initial mass distribution on th ntwork, th systm approachs for larg tim t (uniformly or strongly) th flow on a circl of lngth τ 0. Thrfor, th flow is asymptotically priodic with priod τ which is zro in cas (LI) Q. Similar intrprtations hold for Thorms 10, 12 and 14 blow. If all vlocitis ar qual to 1, thn th numbr τ in (3) is th gratst common divisor of th cycl lngths in th graph. Th ntwork givn in Fig. 1 has cycls of lngths 3 and 4. Hnc, undr th assumption of unit vlocitis, this ntwork bhavs asymptotically priodic with priod Linar Boltzmann quation in ntworks with scattring Th cas dscribd by (BEN S ) has bn studid by Radl [14] and xtnds th situation in [11]. Th main rsults ar provd undr th assumption that th scattring oprator J is an intgral oprator on L 1 [c min, c max ] satisfying th following. Gnral Assumption 8. Th oprator J : L 1 [c min, c max ] L 1 [c min, c max ] has th form f J f := whr th scattring krnl cmax c min k : [c min, c max ] [c min, c max ] R is masurabl and boundd with k(, s)f (s) ds, k(r, s) > 0 for almost all r, s [c min, c max ]. Morovr, cmax c min k(r, s) dr = 1 for all s [c min, c max ]. (5) Undr ths assumptions, J is a compact oprator. Not that condition (5) assurs that no loss or gain of mass occurs by scattring in th vrtics. Th appropriat stat spac to modl problm (BEN S ) is X S = L 1 ( [0, 1], (L 1 [c min, c max ] m ) ) = ( L 1 ([0, 1] [c min, c max ]) ) m. Similarly as bfor, w dfin th oprator (A S, D(A S )) corrsponding to (BEN S ) as D(A S ) = {v W 1,1 ( [0, 1], (L 1 [c min, c max ] m ) ) v(1) = B J v(0)}, (A S v) j (x, c) = c x v j(x, c), x [0, 1], whr c [c min, c max ], j = 1,..., m, B J := B J = ( (B) ij J ) m m L ( (L 1 [c min, c max ]) n). Again, wll-posdnss of (BEN S ) can b shown using rsults on positiv smigroups and th Phillips gnration thorm. Thorm 9 ([14, Thm. 4.6]). Th oprator (A S, D(A S )) gnrats a positiv, boundd, strongly continuous smigroup (T S (t)) t 0 on X S. Th corrsponding abstract Cauchy problm (ACP), hnc (BEN S ), is wll-posd and its solutions ar givn by v(t) = T S (t)v 0 for t 0. Analogously to th prvious cas, th spctrum of A S can b obtaind by a charactristic oprator quation. Combining this information with sophisticatd mthods from th thory of positiv oprators, th smigroup bhavs asymptotically as follows.

5 1420 B. Dorn t al. / Physica D 239 (2010) Thorm 10 ([14, Thm. 6.2]). Th spac X S can b dcomposd as X S = X S r X S s, such that th following proprtis ar fulfilld. (i) Th smigroup is strongly stabl on th spac X S s, i.., T S (t)f t 0 f X S s. (ii) Th spac X S r = fix(t S (t)) t 0 = kr A S (6) is on dimnsional and spannd by a strictly positiv ignvctor of A S. 4. Infinit ntwork flow 4.1. Linar Boltzmann quation in infinit ntworks A gnralization of (BEN) to infinit ntworks has bn initiatd in [15] and thn continud in [16], cf. also [26 28]. Hr, th undrlying graph consists of infinitly many vrtics v i and dgs j. Th infinit vrsion of (BEN) bcoms t u j (x, t) = c j x u j (x, t), x (0, 1), t 0, (BEN I ) u j (x, 0) = f j (x), x (0, 1), (IC) φ ij u j (1, t) = w ij φ + ik u k (0, t), t 0, (BC) k N whr i, j N. As stat spac in this cas on taks X I = l 1 (L 1 [0, 1]), whil th oprator A I = diag(c j ) j N d dx, with domain D(A I ) = {v l 1 (W 1,1 [0, 1]) v(1) = Bv(0)} corrsponds to (BEN I ). Similarly as bfor, on shows that A I gnrats a strongly continuous smigroup on X I, hnc (BEN I ) is wll-posd. In ordr to obtain asymptotic priodicity, and in contrast to th finit cas, on nds a strong assumption on th graph. W stat th ncssary condition as a spctral proprty of th oprator A. Dfinition 11 ([16, Sct. 5]). A strongly connctd graph with 1 Pσ (A) (implying also 1 Pσ (B)) is calld positiv rcurrnt. This positiv rcurrnc of a graph can thn b xprssd as Fostr s condition known from stochastic procsss ([29]), as asymptotic priodicity of th flow smigroup, or as a priodicity condition on th solutions to (BEN I ). This is statd in th following thorm. Thorm 12 ([16, Thm. 16]). Assum that (LD) Q holds. Thn th following assrtions ar quivalnt. (i) Th adjacncy matrix A = (p ji ) i,j N satisfis Fostr s critrion, i.., thr xist a function h : V [0, ), ε > 0 and V V finit such that p ij h(v j ) < for all v i V, (7a) v j V p ij (h(v j ) h(v i )) ε for all v i V. (7b) v j V (ii) Th graph is positiv rcurrnt (i.., 1 Pσ (A)). (iii) Th smigroup (T I (t)) t 0 is strongly asymptotically priodic in th following sns: Th spac X I can b dcomposd as X I = X I s X I r such that th following proprtis ar fulfilld. (a) On th spac X I s, th smigroup is strongly stabl, i.., T(t)f t 0 f X I. s (b) On th spac X I r, th smigroup is similar to a rotation group on L 1 (Γ ). Again, th priod dpnds both on th structur of th graph and th vlocitis and can b computd as in (3). (iv) For vry initial condition (f j ) X I th solution to (BEN I ) convrgs in th norm of L 1 ([0, 1], l 1 ) to a priodic solution. In th sam way, uniform asymptotic priodicity of th solutions can b charactrizd by a Doblin condition (s [30, Sction 1.4.1] or [31, pag 192]) or by rquiring th graph to hav an attractor, cf. [15, Df. 4.6]. Dfinition 13. Lt G = (V, E) b a dirctd graph. A finit vrtx subst V V is calld an attractor if thr xist δ > 0 and L N such that vry vrtx v V has a path of lngth L lading into th st V (i.., th had of th last vrtx of ach path has to b a vrtx in V ), and th sum of th wights of all ths paths is δ. Thorm 14 ([15, Thm. 4.10] or [16, Thm. 17]). Assum that (LD) Q holds. Thn th following assrtions ar quivalnt. (i) Th adjacncy matrix A = (p ji ) i,j N satisfis Doblin s condition, i.., V V finit, δ > 0 and L N such that v V th probability of raching V in L stps is δ. (8) (ii) Th graph has an attractor. (iii) Th smigroup (T I (t)) t 0 is uniformly asymptotically priodic in th sns of Thorm 12 but with uniform stability on X I s. (iv) Th solutions to (BEN I ) convrg uniformly to priodic orbits for all initial conditions (f j ) X I satisfying (f j ) Exampls W giv two xampls of infinit positiv rcurrnt graphs. 1. Fig. 2 corrsponds to a so-calld birth and dath procss. It is wll-known (s,.g., [30, Thm ]) that this graph is positiv rcurrnt if and only if ( ) p1,2... p n,n+1 l 1 (N). (9) p 2,1... p n+1,n n N If this condition is satisfid, Thorm 12 shows that th flow is asymptotically priodic with asymptotic priod τ = 2. Hnc, th mass distribution bhavs asymptotically as th mass distribution on a cycl of lngth 2 (shown on th righthand sid in th figur). 2. Th nxt xampl is a grid (s Fig. 3). A possibl choic of wights to obtain a positiv rcurrnt graph is q > 3 and any 4 0 < p < 1, and in th vicinity of th vrtical boundary r > 3. 4 Th st V is placd in th uppr lft cornr of th grid, and on can choos h as th numbr of ncssary stps to rach V. Th asymptotic priod of th flow is 4. Hr, w hav convrgnc to th motion of th mass on a squar. 5. An application to a control problm Clarly, thr ar many othr rsults and intrsting opn qustions on furthr gnralizations of th original problm (BEN), s.g. [26] or [27]. W only mntion a rsult on vrtx control of transport procsss in ntworks. Using th smigroup approach on can invstigat th following vrtx control problm for (BEN).

6 B. Dorn t al. / Physica D 239 (2010) p12 p23 p34 pij i j p21 p32 p43 Fig. 2. Th birth and dath procss is an xampl of a positiv rcurrnt graph if condition (9) on th wights is fulfilld (s Exampl 1). W obtain convrgnc to a flow of priod 2. pji Fig. 3. An infinit grid with asymptotic priod 4 from Exampl 2. Givn th indicatd choic of wights, this infinit graph is positiv rcurrnt. Dscrib th maximal st of mass distributions in th ntwork that ar (approximatly) rachabl by controlling th input/output in a singl vrtx. Which vrtics allow such maximal control? Both qustions ar answrd by th following thorm. Thorm 15 ([17, Lm. 4.1] and [17, Thm. 4.4]). Assum that c j = 1 for j = 1,..., n. 1. Th maximal rachability spac associatd to (BEN) dpnds on th ntwork and is of th form L 1 ([0, 1], C) (Φ w )T C n. 2. Th problm (BEN) is maximally controllabl in th vrtx v if and only if span{v, Av,..., A n 1 v} = C n. Rfrncs [1] H.G. Kapr, C.G. Lkkrkrkr, J. Hjtmank, Spctral mthods in linar transport thory, in: Oprator Thory: Advancs and Applications, vol. 5, Birkhäusr, [2] G. Birkhoff, Ractor criticality in transport thory, Proc. Natl. Acad. Sci. USA 45 (1959) [3] I. Vidav, Existnc and uniqunss of nonngativ ignfunctions of th Boltzmann oprator, J. Math. Anal. Appl. 22 (1968) [4] I. Vidav, Spctra of prturbd smigroups with applications to transport thory, J. Math. Anal. Appl. 30 (1970) [5] A. Bllni-Morant, Applid Smigroups and Evolution Equations, Oxford Univrsity Prss, [6] A. Bllni-Morant, A Concis Guid to Smigroups and Evolution Equations, in: Sris on Advancs in Mathmatics for Applid Scincs, vol. 19, World Scintific Publishing Co. Inc., [7] M. Mokhtar-Kharroubi, Mathmatical topics in nutron transport thory, in: Sris on Advancs in Mathmatics for Applid Scincs, vol. 46, World Scintific Publishing Co., [8] J. Banasiak, L. Arlotti, Prturbations of positiv smigroups with applications, in: Springr Monographs in Mathmatics, Springr-Vrlag, [9] M. Boulanouar, Gnration thorm for th straming oprator in slab gomtry, J. Dyn. Control Syst. 9 (2003) [10] M. Boulanouar, Th asymptotic bhavior for th straming oprator in slab gomtry, J. Dynam. Control Systms 9 (2003) [11] M. Kramar, E. Sikolya, Spctral proprtis and asymptotic priodicity of flows in ntworks, Math. Z. 249 (2005) [12] E. Sikolya, Flows in ntworks with dynamic ramification nods, J. Evol. Equ. 5 (2005) [13] T. Mátrai, E. Sikolya, Asymptotic bhavior of flows in ntworks, Forum Math. 19 (2007) [14] A. Radl, Transport procsss in ntworks with scattring ramification nods, J. Appl. Funct. Anal. 3 (2008) [15] B. Dorn, Smigroups for flows in infinit ntworks, Smigroup Forum 76 (2008) [16] B. Dorn, V. Kichr, E. Sikolya, Asymptotic priodicity of rcurrnt flows in infinit ntworks, Math. Z. (2008) (in prss). Publishd onlin [17] K.-J. Engl, M. Kramar Fijavž, R. Nagl, E. Sikolya, Vrtx control of flows in ntworks, J. Ntw. Htrog. Mdia 3 (2008) [18] K.-J. Engl, R. Nagl, On-paramtr smigroups for linar volution quations, in: Graduat Txts in Math., vol. 194, Springr-Vrlag, [19] K.-J. Engl, R. Nagl, A Short Cours on Oprator Smigroups, in: Univrsitxt, Springr-Vrlag, [20] B. Bollobás, Modrn Graph Thory, Springr-Vrlag, [21] R. Distl, Graph thory, in: Graduat Txts in Math., vol. 173, Springr-Vrlag, [22] Ch.D. Godsil, G. Royl, Algbraic graph thory, in: Graduat Txts in Math., vol. 207, Springr-Vrlag, [23] H. Schafr, Banach Lattics and Positiv Oprators, Springr Vrlag, [24] R. Nagl (Ed.), On-paramtr smigroups of positiv oprators, in: Lct. Nots Math., vol. 1184, Springr-Vrlag, [25] D. Kunsznti-Kovács, Prturbations of finit ntworks and asymptotic priodicity of flow smigroups, Smigroup Forum (2008) (in prss). Publishd onlin [26] B. Dorn, Flows in infinit ntworks A smigroup approach, Ph.D. Thsis, [27] V. Kichr, Convrgnc of positiv C 0 -smigroups to rotation groups, Ph.D. Thsis, [28] D. Kunsznti-Kovács, Ntwork prturbations and asymptotic priodicity of rcurrnt flows in infinit ntworks, SIAM J. Discrt Math. (in prss). [29] F.G. Fostr, On th stochastic matrics associatd with crtain quuing procsss, Ann. Math. Statist. 24 (1953) [30] G. Fayoll, V.A. Malyshv, M.V. Mnshikov, Topics in th Constructiv Thory of Countabl Markov Chains, Cambridg Univrsity Prss, Cambridg, [31] J.L. Doob, Stochastic Procsss, John Wily & Sons Inc., 1953.