A set-valued framework for birth-and-growth process

1 A set–v alued framew ork for birth–and–gro wth pro cess Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso Department o f Mathematics, Un i versit y of Milan, via Saldini 50, 10133 Milan Italy giacomo.al etti@mat.unimi. it bongio@mat .unimi.it vincenzo.c apasso@mat.unim i.it Summary . W e prop o se a set–v alued framew ork for the wel l–p osedness of birth– and–gro wth process. Our birth–and–gro wth mo del is rigorously defined as a suitable com bination, inv olving Mink owski sum and Aumann in tegral, of t w o very general set–v alued processes representing n ucleation and gro wth resp ectiv ely . The simplicit y of th e used geometrical approach leads us to a void problems arising by an analytical definition of the fron t growth suc h as b oundary regularities. In this framewo rk, gro wth is generally anisotropic and, according to a mesoscale p oint of view, it is not local, i.e. for a fixed time instan t, gro wth is th e same at eac h space p o int. In tro duction Nucleation and gro wth pro cesses arise in sev eral natural and tec hnological app l ica- tions (cf. [7, 8] and the references therein) su c h as, for example, solidification and phase–transition of materials, semiconductor crystal growth, b i omineralization, an d DNA replication (cf., e.g., [15]). A birth–and–gr o wth pr o c e ss is a RaCS family giv en by Θ t = S n : T n ≤ t Θ t T n ( X n ), for t ∈ R + , where Θ t T n ( X n ) is the RaCS obtained as the evolution up to time t > T n of the germ b orn at (random) time T n in ( ra ndom) lo ca tion X n , according t o some gro wth mo del. An analytical approach is often used to mod el birth –and – growth process, in par- ticular it is assumed that the growth is driven according to a non–negative normal velocit y , i.e. for every instan t t , a b ord er p oin t x ∈ ∂ Θ t “gro ws” along the outw ard normal unit (e.g. [3–6, 11, 13, 2 2]). Thus, gro wth is p oin tw ise isotropic; i.e. gi ven a p o int b elongi ng ∂ Θ t , the growth rate is in d ependently from ou tw ard normal direc- tion. N ote that, the existence of the outw ard normal vector imp os es a regularity condition on ∂ Θ t and also on the nucleation process (it cannot be a p oint pro cess). This p a p er is an attempt to offer an original alternative ap p ro ach b a sed on a purely geometric sto c hastic p oin t of v i ew, in order to avo id regularit y assumptions describing birth–and–grow th p rocess. In particular, Minko wski sum (already em- plo yed in [19] to describe self–similar grow th for a single con vex germ) and Aumann 2 A set – value d fr amework for birth–and–gr owth pr o c ess integ ral are used here to derive a mathematical mo del of such p rocess. This m o del, that emphasizes the geometric gro wth without regularit y assumptions on ∂ Θ t , is rig- orously defined as a suitable combination of tw o very general set–v alued pro cesse s representing nucleation { B t } t ∈ [ t 0 ,T ] and gro wth { G t } t ∈ [ t 0 ,T ] respectively Θ t = “ Θ t 0 ⊕ R t t 0 G s ds ” ∪ S s ∈ [ t 0 ,t ] dB s dΘ t = ⊕ G t dt ∪ dB t or Θ t + dt = ( Θ t ⊕ G t dt ) ∪ d B t . Roughly sp eaking, increment dΘ t , during an in fi nitesi mal time interv a l dt , is an enlargemen t due to an infinitesimal Minko wski add e nd G t dt follo wed by the union with the infinitesimal nucleati on dB t . As a consequence of Minko wski sum definition, for every instant t , eac h p oin t x ∈ Θ t (and then eac h point x ∈ ∂ Θ t ) gro ws up by G t dt and n o regularity b order assumptions are required. Then w e deal with not–lo c al growth; i.e. grow th is the same Minko wski addend for ev ery x ∈ Θ t . Nevertheless, und er mesoscale hyp othesis w e can only consider constant gro wth region as described, for example, in [6]. On the other hand, gro wth is anisotropic whenever G t is not a ball. The aim of this pap er is to ensu re t he w ell–posedness of such a mod el and, hence, to sho w that ab o ve “integra l” and “differen tial” notations are meaningful. In v i ew of wel l–p osedness, in [1], th e auth ors show how th e mo del leads to different and significan t statistical results. The article is organized as follo ws. Section 1.1 conta ins some assumptions abou t (random) closed sets and their basilar prop erties. Mo del assumptions are collected in Section 1.2 an d integrabilit y prop erti es of growth pro cess are stu died in Section 1.3. F or the sake of simplicit y , we present, in Section 1.4, main results of the pap er (that imply w ell-p osedness of th e model), whilst corresp ondent proofs are in S ecti on 1.4.1. At the last, Section 1.5 proposes a discrete time p oin t of v iew , also justifying integ ral and differential n o tations. 1.1 Preliminary results Let N , Z , R , R + b e the sets of all non–negative integer, integer, real and non–negative real n umbers respectively . Let X , X ∗ , B ∗ 1 b e a Banach space, its dual space and the unit ball of the d ual space centered in the origin resp ectiv ely . W e shall consider P 0 ( X ) = the family of all su b se ts of X , P ( X ) = P 0 ( X ) \ {∅} F 0 ( X ) = t h e family of all closed subsets of X , F ( X ) = F 0 ( X ) \ {∅} . The suffixes c and b denote conv exity and b oundedness p ro p erties resp ectiv ely (e.g. F 0 bc ( X ) denotes the family of all closed, b ounded and conv ex subsets of X ). F or all A, B ∈ P 0 ( X ) and α ∈ R + , let us define A + B = { a + b : a ∈ A, b ∈ B } = S b ∈ B b + A , (Minko wski S um) α · A = αA = { αa : a ∈ A } , (Scalar Prod uct) By definition, ∀ A ∈ P 0 ( X ), α ∈ R + , we h a v e ∅ + A = ∅ = α ∅ . It is well kn o wn that + is a comm utative and associativ e op e ration with a neut ra l element b ut ( P ( X ) , +) is not a group ( cf . [20]). The follow ing relations are u seful in the sequel (see [21]): for all ∀ A, B , C ∈ P ( X ) Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 3 ( A ∪ B ) + C = ( A + C ) ∪ ( B + C ) if B ⊆ C, A + B ⊆ A + C In the follo wing, we shall work with close d sets. In general , if A, B ∈ F 0 ( X ) then A + B does not b elong to F 0 ( X ) (e.g., in X = R let A = { n + 1 /n : n > 1 } and B = Z , t h en { 1 /n = ( n + 1 /n ) + ( − n ) } ⊂ A + B and 1 /n ↓ 0, b u t 0 6∈ A + B ). I n view of this fact, we define A ⊕ B = A + B where ( · ) denotes the closure in X . F or any A, B ∈ F ( X ) the Hausdorff distanc e (or metric ) is defined by δ H ( A, B ) = max  sup a ∈ A inf b ∈ B k a − b k X , sup b ∈ B inf a ∈ A k a − b k X ff . F or all ( x ∗ , A ) ∈ B ∗ 1 × F ( X ), the supp ort f unc tion is defin ed by s ( x ∗ , A ) = sup a ∈ A x ∗ ( a ). It can b e prov ed (cf. [2, 14]) that for eac h A, B ∈ F bc ( X ), δ H ( A, B ) = sup {| s ( x ∗ , A ) − s ( x ∗ , B ) | : x ∗ ∈ B ∗ 1 } . (1.1) Let ( Ω , F ) b e a measurable space with F complete with resp ect to some σ -finite measure, let X : Ω → P 0 ( X ) b e a set–v alued map, and D ( X ) = { ω ∈ Ω : X ( ω ) 6 = ∅} b e the domain of X X − 1 ( A ) = { ω ∈ Ω : X ( ω ) ∩ A 6 = ∅} , A ⊂ X , b e the inverse image of X Roughly speak in g, X − 1 ( A ) is the set of all ω su ch that X ( ω ) hits set A . Different d e finitions of measurabilit y for set–v alued functions are d ev eloped o ver the yea rs by sev eral authors (cf. [2, 10, 16, 17] and reference therein). H ere , X is me asur a ble if, for each O , op en subset of X , X − 1 ( O ) ∈ F . Proposition 1.1.1 (Se e [17] ) X : Ω → P 0 ( X ) is a measurable set–v alued map if and only if D ( X ) ∈ F , and ω 7→ d ( x, X ( ω )) is a measurable fun ctio n of ω ∈ D ( X ) for eac h x ∈ X . F rom now on, U [ Ω , F , µ ; F ( X )] (= U [ Ω ; F ( X )] if th e measure µ is clear) denotes the family of F ( X )– v alued measurable maps (analogous notation holds whenever F ( X ) is replaced b y another family of subsets of X ). Let ( Ω , F , P ) b e a complete probabilit y space and let X ∈ U [ Ω , F , P ; F ( X )], then X is a R a CS. It can b e pro ved (see [18]) that, if X , X 1 , X 2 are RaCS and if ξ is a measurable real– v alued fun cti on, then X 1 ⊕ X 2 , X 1 ⊖ X 2 , ξ X and (Int X ) C are RaCS. Moreov er, if { X n } n ∈ N is a sequence of R aC S then X = S n ∈ N X n is so. Let ( Ω , F , µ ) b e a fin i te measure space (although most of th e results are v alid for σ -finite measures space). The Aumann inte gr a l of X ∈ U [ Ω , F , µ ; F ( X )] is defin ed by Z Ω X dµ =  Z Ω xdµ : x ∈ S X ff , where S X = ˘ x ∈ L 1 [ Ω ; X ] : x ∈ X µ − a.e. ¯ and R Ω xdµ is th e usual Bochner in- tegral in L 1 [ Ω ; X ]. Moreo v er, R A X dµ = ˘R A xdµ : x ∈ S X ¯ for A ∈ F . I f µ is a probabilit y measure, we den ote th e Aumann integral by E X = R Ω X dµ . Let X ∈ U [ Ω , F , µ ; F ( X ) ] , it is inte gr ably b ounde d , and w e shall write X ∈ L 1 [ Ω , F , µ ; F ( X )] = L 1 [ Ω ; F ( X )], if k X k h ∈ L 1 [ Ω , F , µ ; R ]. Nov ember 13, 2018 4 A set – value d fr amework for birth–and–gr owth pr o c ess 1.2 Mo del assumptions Let us consider Θ t = “ Θ t 0 ⊕ R t t 0 G s ds ” ∪ S s ∈ [ t 0 ,t ] dB s dΘ t = ⊕ G t dt ∪ dB t or Θ t + dt = ( Θ t ⊕ G t dt ) ∪ d B t . (1.2) In fact, ab o ve equation is not a definition since, for example, problems arise handling non–countable un i on of (random) closed sets. The w ell–p os edness of (1.2) and h ence the existence of such a process are the main p urpose of this paper. F rom now on, let us consider the follo wing assumptions. (A-0) - ( X , k·k X ) is a refl exiv e Banach space with separable d ual space ( X ∗ , k·k X ∗ ), (then, X is separable to o, see [12, Lemma I I.3.16 p . 65]). - [ t 0 , T ] ⊂ R is the time observation interval (or time i n terval ), - “ Ω , F , { F t } t ∈ [ t 0 ,T ] , P ” is a filtered probabilit y space, where the filtration { F t } t ∈ [ t 0 ,T ] is assumed to ha ve the usu a l prop erties. ( Nucle ation Pr o c ess ). B = { B ( ω , t ) = B t : ω ∈ Ω , t ∈ [ t 0 , T ] } is a pro cess with non– empty closed v alues, i.e. B : Ω × [ t 0 , T ] → F ( X ) such th a t (A-1) B ( · , t ) ∈ U [ Ω , F t , P ; F ( X )], for ever y t ∈ [ t 0 , T ], i.e. B t is an adapte d (to { F t } t ∈ [ t 0 ,T ] ) process. (A-2) B t is increasing: for ev ery t, s ∈ [ t 0 , T ] with s < t , B s ⊆ B t . ( Gr owth Pr o c ess ). G = { G t = G ( ω , t ) : ω ∈ Ω , t ∈ [ t 0 , T ] } is a p rocess with non– empty closed v alues, i.e. G : Ω × [ t 0 , T ] → F ( X ) such that (A-3) for every ω ∈ Ω and t ∈ [ t 0 , T ], 0 ∈ G ( ω , t ). (A-4) for every ω ∈ Ω and t ∈ [ t 0 , T ], G ( ω , t ) is conv ex, i.e. G : Ω × [ t 0 , T ] → F c ( X ). (A-5) there ex ists K ∈ F b ( X ) such that G ( ω , t ) ⊆ K for every t ∈ [ t 0 , T ] and ω ∈ Ω . As a consequence, G ( ω , t ) ∈ F b ( X ) and k G ( ω , t ) k h ≤ k K k h , ∀ ( ω , t ) ∈ Ω × [ t 0 , T ]. In order to establish the w ell–posedness of integral R t t 0 G s ds in (1.2), let us con- sider a suitable h yp othesis of measurability for G ( a nalogously to what is). A F ( X )–v alued pro ces s G = { G t } t ∈ [ t 0 ,T ] has left c ontinuous tr aj e ctories on [ t 0 , T ] if, for every t ∈ [ t 0 , T ] with t < t , lim t → t δ H ` G ( ω , t ) , G ( ω , t ) ´ = 0 , a.s. The σ -algebra on Ω × [ t 0 , T ] generated by the pro cess es { G t } t ∈ [ t 0 ,T ] with left con- tinuous tra jectories on [ t 0 , T ], is called the pr ev isibl e (or pr e dictable ) σ -algebra and it is denoted b y P . Proposition 1.2.1 The previsible σ -algebra is also generated by the collection of random sets A × t 0 where A ∈ F t 0 and A × ( s, t ] where A ∈ F s and ( s, t ] ⊂ [ t 0 , T ]. Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 5 Proof. Let the σ -algebra generated by the ab o ve collection of sets b e denoted b y P ′ . W e shall sho w P = P ′ . Let G b e a left con tinuous pro ce ss and let α = ( T − t 0 ), consider for n ∈ N G n ( ω , t ) = 8 > > > < > > > : G ( ω , t 0 ) , t = t 0 G ` ω , t 0 + kα 2 n ´ , ` t 0 + kα 2 n ´ < t ≤ “ t 0 + ( k +1) α 2 n ” k ∈ { 0 , . . . , (2 n − 1) } It is clear th a t G n is P ′ -measurable, since G is adapted. As G is left continuous, the ab o ve sequence of left-contin uous pro cesses converg es p oint wise (with respect to δ H ) to G when n tends to infinity , so G is P ′ -measurable, thus P ⊆ P ′ . Con versely consider A × ( s, t ] ∈ P ′ with ( s, t ] ⊂ [ t 0 , T ] and A ∈ F s . Let b ∈ X \ { 0 } and G be th e pro cess G ( ω , v ) =  b, v ∈ ( s, t ] , ω ∈ A 0 , otherwise this function is adapted and left con tinuous, hence P ′ ⊆ P .  Then let u s consider the follo wing assumption. (A-6) G is P -measurable. 1.3 Gro wth pro cess prop e rties Theorem 1.3.2 is the main result in th is section. I t show s th a t ω 7→ R b a G ( ω , τ ) dτ is a RaCS with non–empty b ou n ded convex v alues. This is the first step in order to obtain w ell–p o sedness of (1.2). Proposition 1.3.1 Supp ose (A-3), . . . , (A- 6 ), and let µ λ b e the Leb esgue measure on [ t 0 , T ], then • G ( ω , · ) ∈ U ˆ [ t 0 , T ] , B [ t 0 ,T ] , µ λ ; F bc ( X ) ˜ for ev ery ω ∈ Ω . • G ( · , t ) ∈ U [ Ω , e F t − , P ; F bc ( X )] for each t ∈ [ t 0 , T ], where e F t − is th e so called history σ -algebr a i.e. e F t − = σ ( F s : 0 ≤ s < t ) ⊆ F . • G ∈ L 1 [[ t 0 , T ] , B [ t 0 ,T ] , µ λ ; F bc ( X )] ∩ L 1 [ Ω , F , P ; F bc ( X )] Proof. Assumptions (A - 3) and (A-4) imply that G is non–empty and convex. Mea- surabilit y and integrabili ty prop erti es are consequence of (A-6) and (A-5) resp ec- tively .  Theorem 1. 3. 2 Supp ose (A-3), . . . , (A-6). F or ev ery a, b ∈ [ t 0 , T ], the integral R b a G ( ω , τ ) dτ is non–empty and the set–v alued map G a,b : Ω → P ( X ) ω 7→ R b a G ( ω , τ ) dτ is measurable. Moreo ver, G a,b is a non–empt y , b ound ed conv ex RaCS. Nov ember 13, 2018 6 A set –value d fr amework for birth– and–gr owth pr o c ess In order to prov e Theorem 1.3.2, consider follo wing p rop erties for real pro cesses. A real–v alued pro cess X = { X t } t ∈ [ t 0 ,T ] is pr e dictable with resp ect to filtration { F t } t ∈ R + , if it is measurable with resp ect to th e pr e dictable σ -algebr a P R , i.e. the σ -algebra generated by the collectio n of random sets A × { 0 } where A ∈ F 0 and A × ( s, t ] where A ∈ F s . Proposition 1.3.3 (Se e [ 9, Pr op ositions 2.30, 2.32 and 2.41]) Let X = { X t } t ∈ [ t 0 ,T ] b e a predictable real–v al ued pro cess, then X is ( F ⊗ B [ t 0 ,T ] , B R )-measurable. F urth er, for every ω ∈ Ω , the tra jectory X ( ω , · ) : [ t 0 , T ] → R is ( B [ t 0 ,T ] , B R )-measurable . Lemma 1. 3. 4 Let x ∗ b e an elemen t of the unit ball in the dual space B ∗ 1 , then G 7→ s ( x ∗ , G ) is a measurable map. Proof. By definition s ( x ∗ , G ) = sup { x ∗ ( g ) : g ∈ G } . Since X is separable (A-0), there exists { g n } n ∈ N ⊂ G such that G = { g n } . Then, for eve ry x ∗ ∈ B ∗ 1 w e hav e s ( x ∗ , G ) = sup g ∈ G x ∗ ( g ) = sup n ∈ N x ∗ ( g n ) . Since x ∗ is a con tinuous map then , s ( x ∗ , · ) is measurable.  Proof of Theorem 1.3.2. At first, w e prov e that G a,b is a measurable map. F rom Prop osition 1.3.1, integral G a,b = R b a G ( ω , τ ) dτ is we ll defin ed for all ω ∈ Ω . Assumption (A-3) implies 0 ∈ G a,b ( ω ) 6 = ∅ for every ω ∈ Ω . H en ce, th e domain of G a,b is the whole Ω for all a, b ∈ [ t 0 , T ] D ( G a,b ) = { ω ∈ Ω : G a,b 6 = ∅} = Ω ∈ F . Thus, by Prop osition 1.1.1 and for a fi x ed coup le a , b ∈ [ t 0 , T ], G a,b is (weakly) measurable if and only if, for every x ∈ X , th e map ω 7→ d „ x, Z b a G ( ω , τ ) dτ « = δ H „ x, Z b a G ( ω , τ ) dτ « (1.3) is measurable. Equ ation (1.1) guarantees t hat (1.3) is measurable if and only if, for every x ∈ X , the map ω 7→ sup x ∗ ∈ B ∗ 1 ˛ ˛ ˛ ˛ s ( x ∗ , x ) − s „ x ∗ , Z b a G ( ω , τ ) dτ « ˛ ˛ ˛ ˛ is measurable. The ab ov e expression can b e computed on a countable family dense in B ∗ 1 (note that such family exists since X ∗ is assumed separable (A-0)): ω 7→ sup n ∈ N ˛ ˛ ˛ ˛ s ( x ∗ i , x ) − s „ x ∗ i , Z b a G ( ω , τ ) dτ « ˛ ˛ ˛ ˛ . It can be p rov ed ( [18, Theorem 2.1.12 p. 46]) that s „ x ∗ , Z b a G ( ω , τ ) dτ « = Z b a s ( x ∗ , G ( ω , τ )) dτ , ∀ x ∗ ∈ B ∗ 1 Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 7 and therefore, since s ( x ∗ i , x ) is a constant, G a,b is measurable if, for every x ∗ ∈ { x ∗ i } i ∈ N , the foll owing map ( Ω , F ) → ( R , B R ) ω 7→ R b a s ( x ∗ , G ( ω , τ )) dτ (1.4) is measurable. Note that s ( x ∗ , G ( · , · )), as a map from Ω × [ t 0 , T ] to R , is predictable since it is the comp osition of a p redictable map (A-6) with a measurable one (see Lemma 1.3.4): s ( x ∗ , G ( · , · )) : ( Ω × [ t 0 , T ] , P ) → ( F ( X ) , σ f ) → ( R , B R ) ( ω , t ) 7→ G ( ω , t ) 7→ s ( x ∗ , G ( ω , t )) thus, by Proposition 1.3.3, it is a P -measurable map and hence (1.4) is a measurable map. In view of the first p art, it remains to prov e that G a,b is a b ounded conv ex set for a.e. ω ∈ Ω . Since X is reflexive (A-0), by Prop osition 1.3.1 we have that G a,b is closed ( [18, Theorem 2.2.3]). F urther, G a,b is also con vex (see [18 , Theorem 2.1.5 and Coroll ary 2.1.6]). T o conclude the pro of, it is sufficient to sho w t hat G a,b is includ ed in a b ounded set: Z b a G ( ω , τ ) dτ =  Z b a g ( ω , τ ) d τ : g ( ω , · ) ∈ G ( ω , · ) ⊆ K ff ⊆  Z b a kd τ : k ∈ K ff = { ( b − a ) k : k ∈ K } = ( b − a ) K.  1.4 Geometric Random Pro c ess F or the sake of simplicity , let us present the main results whic h proofs will b e give n in Section 1.4.1. Let us assume conditions from (A- 0) to (A-6). F or every t ∈ [ t 0 , T ] ⊂ R , n ∈ N and Π = ( t i ) n i =0 partition of [ t 0 , t ], let us define s Π ( t ) = „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ i =1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « (1.5) S Π ( t ) = „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ i =1 ∆B t i ⊕ Z t t i − 1 G ( τ ) dτ ! (1.6) where ∆B t i = B t i \ B o t i − 1 ( B o t i − 1 denotes the interior set of B t i − 1 ) and where the integ ral is in the A umann sense with respect to th e Leb esgue measure dτ = d µ λ . W e write s Π and S Π instead of s Π ( t ) and S Π ( t ) when the dep endence on t is clear. Proposition 1.4.1 guarantees t hat b oth s Π and S Π are well defi n ed R aCS, fur- ther, Prop osition 1.4.3 sho ws s Π ⊆ S Π as a consequence of different time interv als integ ration: if th e time interv a l integra tion of G increases then the integra l of G does not decrease with respect to set-inclu sion (Lemma 1.4.2). Prop osition 1.4.4 means that { s Π } ( { S Π } ) increases (decreases) when ever a refinement of Π is considered. Nov ember 13, 2018 8 A set –value d fr amework for birth– and–gr owth pr o c ess At the same time, Proposition 1.4.5 implies that s Π and S Π b ecome closer eac h other (in the Hausdorff distance sense) when partition Π b ecomes fi ner. The “limit” is indep endent on the c hoice of the refinement as con seq u ence of Proposition 1.4.6. Corollar y 1.4. 7 means that, given any { Π j } j ∈ N refinement sequence of [ t 0 , t ], the random closed sets s Π j and S Π j pla y t he same role th at lo wer sums and upp er sum s hav e in classical analysis when we define th e Riemann integral. In fact, if Θ t denotes their limit v alue (see (1.7)), s Π j and S Π j are a low er and an upp er app roxima tion of Θ t respectively . Note that, as a conseq u ence of monotonicity of s Π j and S Π j , we a void problems that may arise considering uncountable un ions in integral expression in (1.2). Proposition 1.4.1 Let Π be a partition of [ t 0 , t ]. Both s Π and S Π , defined in (1.5) and (1.6), are RaCS. Lemma 1. 4. 2 Let X ∈ L 1 [ I , F , µ λ ; F ( X )], where I is a b ou n ded interv al of R , such that 0 ∈ X µ λ -almost everywhere on I and let I 1 , I 2 b e tw o other in terv als of R with I 1 ⊂ I 2 ⊂ I . Then Z I 1 X ( τ ) dτ ⊆ Z I 2 X ( τ ) dτ . Proposition 1.4.3 Let Π be a partition of [ t 0 , t ]. Then s Π ⊆ S Π almost surely . Proposition 1.4.4 Let Π and Π ′ b e tw o p artitions of [ t 0 , t ] such that Π ′ is a refinement of Π . Then, almost surely , s Π ⊆ s Π ′ and S Π ′ ⊆ S Π . Proposition 1.4.5 Let { Π j } j ∈ N b e a refinement sequence of [ t 0 , t ] (i.e. | Π j | → 0 if j → ∞ ). Then, almost surely , lim j →∞ δ H ` s Π j , S Π j ´ = 0. Proposition 1.4.6 Let { Π j } j ∈ N and { Π ′ l } l ∈ N b e tw o distinct refinement sequences of [ t 0 , t ], then, almost surely , lim j → ∞ l → ∞ δ H “ s Π j , s Π ′ l ” = 0 and lim j → ∞ l → ∞ δ H “ S Π j , S Π ′ l ” = 0 . Corollary 1.4.7 F or every { Π j } j ∈ N refinement sequence of [ t 0 , t ], the follo wing limits exist 0 @ [ j ∈ N s Π j 1 A , „ lim j →∞ s Π j « , li m j →∞ S Π j , \ j ∈ N S Π j , (1.7) and they are equals almost surely . The conv ergences is taken with resp ect t o the Hausdorff distance. W e are now ready to defin e t h e con tinuous time sto chastic p rocess. Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 9 Definition 1.4. 8 Assume (A-0), . . . , (A-6). F or every t ∈ [ t 0 , T ], let { Π j } j ∈ N b e a refin ement sequence of the time in terv al [ t 0 , t ] and let Θ t b e the RaCS defined by 0 @ [ j ∈ N s Π j ( t ) 1 A = „ lim j →∞ s Π j ( t ) « = Θ t = lim j →∞ S Π j ( t ) = \ j ∈ N S Π j ( t ) , then, the family Θ = { Θ t : t ∈ [ t 0 , T ] } is called ge ometric r andom pr o c ess G-R aP (on [ t 0 , T ]). Theorem 1. 4. 9 Let Θ b e a G-RaP on [ t 0 , T ], then Θ is a non-decreasing process with respect t o th e set inclusion, i.e. P ( Θ s ⊆ Θ t , ∀ t 0 ≤ s < t ≤ T ) = 1 . Moreo ver, Θ is adapted with resp ect to filtration { F t } t ∈ [ t 0 ,T ] . Remark 1.4.10 W e wan t to p oint out that, assumptions we considered on { B t } and { G t } are so general, that a wide family of classical random sets and ev olu- tion pro cesses can b e describ ed ( for example, Bo olean mo del is a birth–and–grow th process with “null grow th”). 1.4.1 Pro ofs of Prop osi tions in Section 1.4 Proof of Proposition 1.4.1. F or every i ∈ { 0 , . . . , n } , R t t i − 1 G ( τ ) dτ is a RaCS (Theorem 1.3.2) . Thus, measurabilit y Assump tion (A-1) on B guaran tees that, for every t i ∈ Π , B t i , ∆B t i , “ ∆B t i ⊕ R t t i G ( τ ) dτ ” , and hence s Π and S Π are RaCS.  Proof of Le m ma 1.4.2. Let y ∈ “ R I 1 X ( τ ) dτ ” , then th ere exists x ∈ S X , for whic h y = “ R I 1 x ( τ ) dτ ” . Let us d efine on I 2 ( ⊃ I 1 ) x ′ ( τ ) =  x ( τ ) , τ ∈ I 1 0 , τ ∈ I 2 \ I 1 then x ′ ∈ S X and y = “ R I 2 x ′ ( τ ) d τ ” ∈ “ R I 2 X ( τ ) dτ ” .  Proof of Proposition 1.4.3. Thesis is a consequence of Lemma 1.4.2 and Minko wski addition prop erties, in fact “ R t t i − 1 G ( τ ) dτ ” ⊆ “ R t t i G ( τ ) dτ ” implies s Π ⊆ S Π .  Proof of Prop osition 1.4.4. Let Π ′ b e a refinement of partition Π of [ t 0 , t ], i.e. Π ⊂ Π ′ . W e prov e that s Π ⊆ s Π ′ ( S Π ′ ⊆ S Π is analogous). I t is sufficient to show the thesis only for Π ′ = Π ∪ ˘ t ¯ where Π = { t 0 , . . . , t n } with t 0 < . . . < t n = t and t ∈ ( t 0 , t ). Let i ∈ { 0 , . . . , ( n − 1) } b e such that t i ≤ t ≤ t i +1 then s Π = „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ j = 1 j 6 = i + 1 ∆B t j ⊕ Z t t j G ( τ ) dτ ! ∪ " ` B t i +1 \ B o t i ´ ⊕ Z t t i +1 G ( τ ) dτ # Nov ember 13, 2018 10 A set –value d fr amework for birth– and–gr owth pr o c ess and s Π ′ = „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ j = 1 j 6 = i + 1 ∆B t j ⊕ Z t t j G ( τ ) dτ ! ∪ » ` B t \ B o t i ´ ⊕ Z t t G ( τ ) dτ – ∪ " ` B t i +1 \ B o t ´ ⊕ Z t t i +1 G ( τ ) dτ # Definitely , in order t o pro ve th at s Π ⊆ s Π ′ w e hav e to pro ve th at ( » ` B t \ B o t i ´ ⊕ Z t t G ( τ ) dτ – ∪ " ` B t i +1 \ B o t ´ ⊕ Z t t i +1 G ( τ ) dτ #) ⊇ " ` B t i +1 \ B o t i ´ ⊕ Z t t i +1 G ( τ ) dτ # . This inclusion is a consequence of “ R t t G ( τ ) dτ ” ⊇ “ R t t i +1 G ( τ ) dτ ” (Lemma 1.4.2) and of the Mink owski distribution p rop erty .  Proof of Proposi tion 1. 4.5. Let Π j = ( t i ) n i =0 b e the j -partition of the refinement sequence { Π j } j ∈ N , then δ H ` s Π j , S Π j ´ = max ( sup x ∈ s Π j d ( x, S Π j ) , sup y ∈ S Π j d ( y , s Π j ) ) where d ( x, S Π j ) = inf y ∈ S Π j k x − y k X . By Prop osition 1.4.3, s Π j ⊆ S Π j then sup x ∈ s Π j d ( x, S Π j ) = 0 and hence w e hav e to pro ve th at, whenever j → ∞ (i.e. | Π j | → 0), δ H ` s Π j , S Π j ´ = sup y ∈ S Π j d ( y , s Π j ) = sup y ∈ S Π j inf x ∈ s Π j k x − y k X − → 0 . F or every ω ∈ Ω , let y b e any element of S Π j ( ω ), then we distinguish tw o cases: (1) if y ∈ “ B t 0 ( ω ) ⊕ R t t 0 G ( ω , τ ) dτ ” , t hen it is also an element of s Π j ( ω ), and hence d ` s Π j ( ω ) , y ´ = 0. (2) if y 6∈ “ B t 0 ( ω ) ⊕ R t t 0 G ( ω , τ ) dτ ” , then there ex ist j ∈ { 1 , . . . , n } such th at y ∈ ∆B t j ( ω ) ⊕ Z t t j − 1 G ( ω , τ ) dτ ! . By definition of ⊕ , for eve ry ω ∈ Ω , there exist { y m } m ∈ N ⊆ ∆B t j ( ω ) + Z t t j − 1 G ( ω , τ ) dτ ! , such that lim m →∞ y m = y . Then, for every ω ∈ Ω , th ere ex ist h m ∈ ∆B t j ( ω ) and g m ∈ “ R t t j − 1 G ( ω , τ ) dτ ” such that y m = ( h m + g m ) and hence Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 11 y = lim m →∞ ( h m + g m ) = lim m →∞ y m where the conv ergence is in the Banac h norm, then let m ∈ N b e such that k y − y m k X < | Π j | , for eve ry m > m . Note th at, for every ω ∈ Ω and m ∈ N , b y Aumann in tegral definition, there exists a selection c g m ( · ) of G ( ω , · ) (i.e. c g m ( t ) ∈ G ( ω , t ) µ λ -a.e.) suc h that g m = Z t t j − 1 c g m ( τ ) d τ and y m = h m + Z t t j − 1 c g m ( τ ) d τ . F or every ω ∈ Ω , let us consider x m = h m + Z t t j c g m ( τ ) d τ then x m ∈ s Π j ( ω ) for all m ∈ N . Moreo ver, the follo wing chain of in eq ualities hold, for all m > m and ω ∈ Ω , inf x ′ ∈ s Π j ‚ ‚ x ′ − y ‚ ‚ X ≤ k x m − y k X ≤ k x m − y m k X + k y m − y k X ≤ ‚ ‚ ‚ ‚ ‚ Z t j t j − 1 c g m ( τ ) d τ ‚ ‚ ‚ ‚ ‚ X + | Π j | ≤ Z t j t j − 1 k c g m ( τ ) k X dτ + | Π j | ≤ Z t j t j − 1 k G ( τ ) k h dτ + | Π j | ≤ | t j − t j − 1 | k K k h + | Π j | ≤ | Π j | ` k K k h + 1 ´ j →∞ − → 0 since k K k h is a p ositive constan t. By the arbitrariness of y ∈ S Π j ( ω ) w e obtain th e thesis.  Proof of Prop osition 1.4.6. Let Π j and Π ′ l b e tw o partitions of the tw o distinct refinement sequ ences { Π j } j ∈ N and { Π ′ l } l ∈ N of [ t 0 , t ]. Let Π ′′ = Π j ∪ Π ′ l b e th e refinement of b oth Π j and Π ′ l . Then Proposition 1.4.4 and Proposition 1.4.3 imp ly that s Π j ⊆ s Π ′′ ⊆ S Π ′′ ⊆ S Π ′ l . Therefore s Π j ⊆ S Π ′ l for ev ery j, l ∈ N . Then 0 @ [ j ∈ N s Π j 1 A ⊆ \ l ∈ N S Π ′ l . Analogously [ l ∈ N s Π ′ l ! ⊆ \ j ∈ N S Π j . Proposition 1.4.5 concludes the proof.  In order to prove Theorem 1.4.9, let us consider the follo wing Lemma. Lemma 1. 4. 11 Let s, t ∈ [ t 0 , T ] with t 0 < s < t and let Π s and Π t b e tw o partition of [ t 0 , s ] and [ t 0 , t ] resp ectively , such that Π s ⊂ Π t . Then s Π s ( s ) ⊆ s Π t ( t ) and S Π s ( s ) ⊆ S Π t ( t ) . Nov ember 13, 2018 12 A set –value d fr amework for birth– and–gr owth pr o c ess Proof. The proofs of the tw o inclusions are similar. Let us pro ve that s Π s ( s ) ⊆ s Π t ( t ). Since Π s ⊂ Π t , then Π s = ( t i ) n i =0 and Π t = Π s ∪ ( t i ) n + m i = n +1 with m ∈ N . By Lemma 1.4.2, w e hav e that s Π s ( s ) = „ B t 0 ⊕ Z s t 0 G ( τ ) dτ « ∪ n [ i =1 „ ∆B t i ⊕ Z s t i G ( τ ) dτ « ⊆ „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ i =1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « ⊆ „ B t 0 ⊕ Z t t 0 G ( τ ) dτ « ∪ n [ i =1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « ∪ n + m [ i = n +1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « i.e. s Π s ( s ) ⊆ s Π t ( t ).  Proof of Theorem 1.4.9. F or every s, t ∈ [ t 0 , T ] with s < t , let { Π s i } i ∈ N and ˘ Π t i ¯ i ∈ N b e t wo refinement sequences of [ t 0 , s ] and [ t 0 , t ] respectively , such that Π s i ⊂ Π t i for every i ∈ N . Then, by Lemma 1.4.11, S Π s i ⊆ S Π t i . N o w, as i tends to infinity , we obtain Θ s = \ i →∞ S Π s i ⊆ \ i →∞ S Π t i = Θ t . F or the second part, note that Theorem 1.3.2 still holds replacing F t instead of F , so that for every s ∈ [ t 0 , T ], the family n R t s G ( ω , τ ) dτ o t ∈ [ s,T ] is an adapted pro cess to the filtration { F t } t ∈ [ t 0 ,T ] . This fact together with Assump tion (A-1) guarantees that { S Π } t ∈ [ s,T ] is adapted for every partition Π of [ s, T ] and hence Θ is adap t ed too.  1.5 Discrete time case and infinitesimal notations Let us consider Θ s and Θ t with s < t . Let ˘ Π s j ¯ j ∈ N and ˘ Π t j ¯ j ∈ N b e tw o refin ement sequences of [ t 0 , s ] and [ t 0 , t ] resp ectively , such that Π s j ⊂ Π t j for every j ∈ N (i.e. Π s j = ( t i ) n i =0 and Π t j = Π s j ∪ ( t i ) n + m i = n +1 with n, m ∈ N ). I t is easy to compute s Π t j = „ s Π s j ⊕ Z t s G ( τ ) dτ « ∪ n + m [ i = n +1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « . Then, by Definition 1.4.8, whenever ˛ ˛ Π t j ˛ ˛ → 0, w e obtain Θ t = „ Θ s ⊕ Z t s G ( τ ) dτ « ∪ lim | Π t j | → 0 n + m [ i = n +1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « . (1.8) The follo wing notations G k = Z t s G ( τ ) dτ and B k = l im | Π t j | → 0 n + m [ i = n +1 „ ∆B t i ⊕ Z t t i G ( τ ) dτ « Nov ember 13, 2018 Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso 13 lead us to the set-v alued discrete time sto chastic p rocess Θ k =  ( Θ k − 1 ⊕ G k ) ∪ B k , k ≥ 1 , B 0 , k = 0 . In view of this, w e are able to justify infinitesimal notations introduced in (1.2). 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