MEAN FIELD LIMIT with PROLIFERATION Franco Flandoli

Home , Semimartingale

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2016086 DYNAMICAL SYSTEMS SERIES B Volume 21, Number 9, November 2016 pp. 3029–3052

MEAN FIELD LIMIT WITH PROLIFERATION

Franco Flandoli∗ Universita di Pisa, Dipartimento Matematica Largo Bruno Pontecorvo 5, C.A.P. 56127 Pisa, Italy Matti Leimbach Technische Universit¨atBerlin, Institut f¨urMathematik Straße des 17. Juni 136, D-10623 Berlin, Germany

Abstract. An interacting particle system with long range interaction is con- sidered. Particles, in addition to the interaction, proliferate with a rate depending on the empirical measure. We prove convergence of the empirical measure to the solution of a parabolic equation with non-local nonlinear transport term and proliferation term of logistic type.

1. Introduction. We are concerned with a macroscopic limit result for a system of particles which interact by a mean field potential and may proliferate; we want to prove the convergence of the empirical measure to a parabolic equation with nonlinear non-local transport term, like in classical mean field models, plus a growth term corresponding to the proliferation. Our starting motivation for this research came from Mathematical Oncology, where cells interact by random dynamics and proliferate, and one would like to discover appropriate macroscopic limits (PDE), for instance because a cell-level simulation of a tumor is too demanding, being involved a number of particles of the order of 109. 1.1. The microscopic model. The microscopic system, defined on a filtered a,N d probability space (Ω, F, Ft,P ), is composed of particles at positions Xt ∈ R . We label particles by a multi-index a = (k, i1, ..., in) with i1, ..., in ∈ {1, 2} and k = 1, ..., N, where N ∈ N is the number of particles at time t = 0. The particles already alive at time t = 0 are those with label a = (k), k = 1, ..., N. Their descendants require the additional labeling; setting (a, −) := (k, i1, ..., in−1) if a = (k, i1, ..., in), we may say that a is a descendant of (a, −). Each particle “lives” only during a random time interval: particle with label a lives on a,N a,N a,N a,N a,N I = [T0 ,T1 ) ⊂ [0, ∞) where T0 ,T1 are Ft-stopping times: it was born a,N a,N a,N at time T0 (or it exists from t = 0 if T0 = 0) and “dies” at time T1 when it is replaced by two independent particles (this is a proliferation event); the number a,N a,N of alive particles can only increase. Setting X a,N := limt↑T a,N Xt , we impose T1 1 a,N (a,−),N a,N (a,−),N N T0 = T1 and X a,N = X (a,−),N . Denote by A the set of all labels a and T0 T1

2010 Mathematics Subject Classiﬁcation. Primary: 60K35, 35K57; Secondary: 60F17, 35K58. Key words and phrases. Mean ﬁeld, branching, limit dynamics, semilinear parabolic equation, interacting particle system. ∗ Corresponding author: Franco Flandoli.

3029 3030 FRANCO FLANDOLI AND MATTI LEIMBACH

N N by At the set of particle labels alive at time t, namely the set of a ∈ A such that t ∈ Ia,N ; the empirical measure at time t is deﬁned as

N 1 X St = δXa,N . N t N a∈At In the random time interval Ia,N , particle with label a interacts with all other living 2 d a,N particles through a potential V ∈ Cc R ; the dynamics of Xt is described by the gradient system 1 X dXa,N = − ∇V Xa,N − Xea,N dt + σdBa (1) t N t t t N ea∈At a d where Bt are independent Brownian motions in R and σ > 0. a,N Each particle proliferates at a random rate λt which, in most applications, depends on the density of particles it has in the neighborhood. We prescribe a general structure of the form

a,N N a,N λt = FN St ,Xt

d d where the properties of the measurable functionals FN : M+ R × R → [0, ∞) d d will be speciﬁed below (M+ R is the set of Borel ﬁnite positive measures on R ). The precise meaning of the previous sentences is that we have standard Poisson 0,a processes Nt , independent between themselves and with respect to the Brownian (k),N motions and initial conditions X0 , and we change their times randomly by a,N 0,a a,N R t a,N a,N setting Nt := N a,N where Λt = 0 1s∈Ia,N λs ds; the process Nt is a jump Λt a,N a,N a,N process with rate λt and the stopping time T1 is the time when Nt jumps from 0 to 1 (and then remains equal to 1).

1,2 d 1.2. Macroscopic limit. Denote by W+ R the subset of the Sobolev space 1,2 d β d W R made of all non negative functions and by Cb R , β ∈ (0, 1), the space of β-Höldercontinuous functions; the β-Hölderseminorm of f will be denoted by β 1,2 d [f]β := supx6=y |f (x) − f (y)| / |x − y| . We say that a map F : W+ R → β d d Cb R satisfies the mild Lipschitz conditions if it is Lipschitz in the Cb R -norm β d 1,2 d and has linear growth in the Cb R -norm, namely for every u, v ∈ W+ R

kF (u) − F (v)k∞ ≤ LF ku − vkW 1,2 (2)

[F (u)]β ≤ C (kukW 1,2 + 1) . (3) The macroscopic limit result below requires a number of natural assumptions that we list now, plus the more critical assumption (5) that we emphasize in the statement of the theorem. Let {θN } be a classical family of compact support smooth −d −1 mollifiers of the form θN (x) = N θ N x ; let us introduce the mollified empirical measure (the theoretical analog of the numerical method of kernel smoothing) N ht (x) defined as N N ht = θN ∗ St . −d−2 For technical reasons we assume sup N < ∞; moreover, at least in the case N∈N N of example (8)-(9) below, the role of θN is auxiliary, it does not appear in the model, hence the restriction does not have biological relevance. Concerning the interaction 2 d potential V , assume that V ∈ Cc R . Concerning the initial conditions, assume MEAN FIELD LIMIT WITH PROLIFERATION 3031

2 d 1 d N that u0 ∈ L R ∩ L R and S0 , φ be convergent in probability to hu0, φi, ∞ d as N → ∞, for every φ ∈ Cc R . Moreover, assume the following technical condition: N 2 lim sup P h0 2 d > R = 0. (4) R→∞ L (R ) N∈N A simple and relevant sufficient condition for this property is given by Proposition 1 below. Finally, the definition of weak solution of the PDE (7) is given below in Section 3.1. 1,2 d Theorem 1.1. Assume that, for some β ∈ (0, 1), there is a map F : W+ R → β d Cb R which satisfies the mild Lipschitz condition (2)-(3) above and there is a sequence of positive real numbers {αN } converging to zero such that

kF (θN ∗ µ) − FN (µ, ·)k∞ ≤ αN (1 + hµ, 1i) (5)

Assume moreover that there exists a constant CF > 0 such that

|FN (µ, x)| ≤ CF kF (u)k∞ ≤ CF (6) d d 1,2 d N for every µ, x, u ∈ M+ R × R × W+ R . Then the process ht (x) converges in probability, as N → ∞, to the unique weak solution of the PDE σ2 ∂ u = div ((∇V ∗ u ) u ) + ∆u + F (u ) u , u| = u . (7) t t t t 2 t t t t=0 0 N The topologies of convergence of ht (x) to ut (x) are 2 d • the strong topology of Lloc [0,T ] × R , 2 1,2 d • the weak topology of L [0,T ] × W R and ∞ 2 d • the weak star topology of L 0,T ; L R . Having in mind Fisher-Kolmogorov-Petroskii-Piskunov equation and in general the concept of logistic growth, the most natural example of functional F (u) is F (u)(x) = (1 − u (x)), which means that proliferation rate decreases when we approach the threshold u = 1 (the value 1 is conventional). More precisely, due to the term div ((∇V ∗ ut) ut), the solution ut may overcome any threshold, so one has to correct the classical logistic term and consider F (u)(x) = (1 − u (x))+ (proliferation is completely inhibited when the threshold is crossed). Our theory 1,2 d β d covers this case only in dimension d = 1, where W+ R ⊂ Cb R for some β ∈ (0, 1). In this case we take + FN (µ, x) = (1 − (θN ∗ µ)(x)) and assumption (5) is obviously satisfied; the others are elementary. In dimension d > 1 this example is not covered but we can treat the case F (u)(x) = (1 − (W ∗ u)(x))+ (8) where W is a Lipschitz continuous compact support probability density. In this case we simply take + FN (µ, x) = (1 − (W ∗ µ)(x)) . (9) The validity of assumptions (2)-(3)-(5)-(6) in this case is shown in Lemma 3.10 below. A variety of macroscopic limit results has been proved in the literature; let us quote, related to the present one, [11], [12], [14], [13], [8], [9], [10] and references 3032 FRANCO FLANDOLI AND MATTI LEIMBACH therein, from which we have taken several elements of inspiration. However, a result of mean field type with proliferation, in the sense described here, is not treated in the previous references. Let us also mention a different class of macroscopic limit results, for instance [18], [17], which require very different techniques. In the more applied literature, two examples of works related to our problem are [1], [3]. A feature of our approach, shared by some of the previously quoted references, is that we do not use only results of tightness of measure-valued processes but of processes. A more distinguished feature, probably shared only by [9] (which however is very different), is that we use typical tools of the theory of stochastic Partial Differential Equations, for instance the tightness criterion used for stochastic Navier-Stokes equations in [6], [2], see also [5], [4].

1.3. Motivations from mathematical oncology. Although the aim of this pa- per is mostly theoretical, we have been inspired by lectures about the emerging field of Mathematical Oncology in the choice of the problem and of some details. In this area, roughly speaking, models are classified as macroscopic, when described by partial differential equations, or microscopic, when described by stochastic ordinary differential equations or, even more often, cellular automata and other discrete stochastic models - in addition there are multiscale models with mixture of the previous two cases. Macroscopic models look at the tumor at the tissue level, microscopic ones at the cellular level. The link between the two descriptions is of interest for var- ious reasons, in particular because a precise justification of the macroscopic models is difficult from general arguments based on fluid dynamics or mechanical, a biological tissue being something different. At the cellular level it is easier to be more realistic and thus results on the macroscopic limit of cellular systems is a way to justify or improve the macroscopic models; and also to have different interpretations of the constants appearing in the models - characterizing these constants is a major problem for the real applications. After these general comments, which clarify why we investigate the macroscopic limit, let us say that most models used nowadays in Mathematical Oncology are more complex than our one, since they involve different cell types - e.g. normoxic and hypoxic tumor cells, or cells with different genetic mutations, or cells of the extracellular matrix - molecular fields like oxygen and growth factors, and possibly objects related to the angiogenic cascade. See for instance [16] as an example of complex model. Our model, chosen as a starting point, captures only a few features of such complexity: i) the interaction between cells which may incorporate for instance a certain degree of repulsion resulting from the fact that cells cannot press each other too much - see the term called “crowding effect” in [16], different from our one but corresponding to a similar mechanism; ii) cancer cell proliferation, always present in each model. Concerning proliferation in detail, most often in the literature of Mathematical Oncology it is taken of logistic form u (1 − u) (which corresponds to F (u)(x) = (1 − u (x))+ above). Such choice of F simply corresponds to the fact that cells proliferate better when the neighbor is not so crowded. However, other forms of F may be interesting as well. A phenomenon observed in vitro is that certain isolated cancer cells, even if embedded in a liquid that is rich of nutrients, do not proliferate: they need to adhere to other cells to proliferate. A functional F which charge - in the sense that decreases proliferation rate - not only the excessive presence of cells in the neighbor but also the opposite case, an excessive isolation, may be more realistic: cells which separate from the main cloud by random motion will continue MEAN FIELD LIMIT WITH PROLIFERATION 3033 their travel to meet blood vessels and lead to metastasis, but along the trip they do not proliferate so often as the cells close to the main tumor body. This, although vague, could be a motivation for investigating a general proliferation mechanism of a,N N a,N the form λt = FN St ,Xt above; more modeling work is necessary and is part of our research program.

a,N a,N 2. Preparation. Starting from this section, we drop the suffix N in Xt , I , a,N a,N a,N d Ti , λ , Nt to simplify notations. Let δ denote a point outside R , the so a a called grave state, where we assume the processes Xt live when t∈ / I . Hence, whenever a particle proliferates and therefore dies, it stays forever in the grave state δ. In the sequel, the test functions φ are assumed to be defined over Rd × {δ} and be such that φ (δ) = 0. Using Itôformula over random time intervals, one can a 2 d show that φ (Xt ), with φ ∈ C R , satisfies

a a a φ (X ) = φ X a 1t≥T a − φ X a 1t≥T a t T0 0 T1 1 Z t 2 Z t a a σ a + 1s∈Ia ∇φ (Xs ) dXs + 1s∈Ia ∆φ (Xs ) ds. 0 2 0 N With a few computations, one can see that the empirical measure St satisﬁes σ2 d SN , φ = − ∇V ∗ SN SN , ∇φ dt + SN , ∆φ dt t t t 2 t (10) N N 1,φ,N 2,φ,N + FN St , · St , φ dt + dMt + dMt

2 d for every φ ∈ Cb R and where Z t 1,φ,N σ X a a M := 1 a ∇φ (X ) dB t N s∈I s s a∈AN 0 Z t 2,φ,N 1 X a 1 X a a Mt := φ XT a 1t≥T a − φ (Xs ) λs ds. N 1 1 N a∈AN a∈AN 0

N We deduce that ht (x) satisﬁes σ2 dhN (x) = div θ ∗ ∇V ∗ SN SN (x) + ∆hN (x) t N t t 2 t N N 1,N 2,N + θN ∗ FN St , · St (x) dt + dMt (x) + dMt (x) where Z t 1,N σ X a a M (x) := − 1 a ∇θ (x − X ) · dB , t N s∈I N s s a∈AN 0 Z t 2,N 1 X a 1 X a a Mt (x) := θN (x − XT a )1t≥T a − θN (x − Xs )λs ds. N 1 1 N N 0 N a∈A a∈As The total relative mass Card AN SN := SN d = SN , 1 = t t t R t N 3034 FRANCO FLANDOLI AND MATTI LEIMBACH plays a central role. Since, in our model, the number of particles may only increase, we have the inequality N N St ≤ ST for all t ∈ [0,T ] , (11) N that we shall use very often. The quantity ST is, moreover, exponentially inte- N grable, uniformly in N, see Lemma 3.11 below. Sometimes, however, having ST in the inequalities could spoil properties associated to adaptedness, so we also introduce, for every R > 0, the stopping time

N n N N o τR := inf r ≥ 0 : Sr > R or h0 2 d > R (12) L (R ) N (τR = +∞ if the set is empty). We also repeatedly use the identity Z N N ht (x)dx = St (13) d R which follows from Fubini Theorem. Other simple rules of calculus we often use are N N N N θN ∗ fSt (x) ≤ kfk∞ ht (x) , f ∗ St (x) ≤ kfk∞ St (14) for every bounded measurable f : Rd → R. Finally, we often use the inequalities Z Z 1 2 1 2 2 |∇θN (x)| dx ≤ C, |θN (x)| dx ≤ CN (15) N d N d R R which holds with a suitable constant C > 0. The bound on the ﬁrst term comes from Z −d−2 Z −d−2 Z 1 2 N −d −1 2 N 2 |∇θN (x)| dx = N ∇θ N x dx = |∇θ (x)| dx N d N d N d R R R −d−2 the assumption that θ has compact support and the assumption supN N /N < ∞; the bound on the second term is similar.

3. Tightness. In this section, about FN , we only use (6).

Deﬁnition 3.1. We deﬁne, for a function f : R → R, f(t−) := lim f(s), the left limit of f at t, s%t Jf(t) := f(t) − f(t−), the jump size of f at t.

Further, let (Xt)t≥0 be a stochastic process. We deﬁne its quadratic variation, when the limit exists and is independent of the partitions, n−1 X 2 [X]t = P- lim Xtk ∧t − Xtk∧t , k→∞ j+1 j j=0

k k where the maximal distance of two consequent sites in the partition {0 = t0 < t1 < k ··· < tn = T } converges to 0 as k → ∞. We denote the continuous part of the quadratic variation by [X]c, i.e.

c X 2 [X]t = [X]t − JXs . s≤t In the following we need a lemma, also known as generalized Itˆoformula, see for example [15, page 245]. MEAN FIELD LIMIT WITH PROLIFERATION 3035

Lemma 3.2. Let X be a one-dimensional semimartingale such that Xt,Xt− take values in an open set U ⊂ R and f : U → R twice continuously diﬀerentiable. Then f(X) is a semimartingale and Z t Z t 0 1 00 c f(Xt) =f(X0) + f (Xs−)dXs + f (Xs)d [X]s 0 2 0 X 0 + (Jf(Xs) − f (Xs−)JXs) . s≤t We use the generalized Itˆoformula to derive the following energy estimate. Lemma 3.3. 2 Z t 1 N 2 σ N 2 ht L2 + ∇hr L2 dr 2 2 0 Z t 1 N 2 N N N = h0 L2 − θN ∗ ∇V ∗ Sr Sr , ∇hr dr 2 0 Z t N N N 1 1,N + θN ∗ FN Sr , · Sr , hr dr + [M ]t L1 0 2 Z Z t N 1,N 2,N + hr−(x)d Mr (x) + Mr (x) dx d R 0 Z 2 1 1 X a + θ (x − X a ) 1 a dx. 2 N T t≥T1 2 d N 1 R a∈AN

d 2 Proof. For every x ∈ R , we apply the generalized Itˆoformula to ((h(x)) )t≥0 and integrate over x ∈ Rd: Z Z t Z N 2 N 2 N N N c ht L2 = h0 L2 + 2 hr−(x)dhr (x)dx + h (x) t dx d 0 d Z R R X N 2 N N + J hr (x) − 2hr−(x)Jhr (x) dx d R r≤t Z Z t Z N 2 1,N 2 N N = kh0 kL2 + M (x) t dx + σ hr (x)∆hr (x)dxdr d d R 0 R Z t Z N N N + 2 hr (x) div θN ∗ ∇V ∗ Sr Sr (x)dxdr d 0 R Z t Z N N N + 2 hr (x)θN ∗ FN Sr , · Sr (x)dxdr d 0 R Z Z t N 1,N 2,N + 2 hr−(x)d Mr (x) + Mr (x) dx d 0 Z R X N 2 N N + J hr (x) − 2hr−(x)Jhr (x) dx d R r≤t N c 1,N N N where we have used the fact that h (x) t = M (x) t and that hr−(x) = hr (x) N N for a.e. r (hence we may replace hr−(x) by hr (x) in ordinary Lebesgue integrals). It remains to understand the last line of the previous formula. At every jump time r, we have N 2 N N J hr (x) − 2hr−(x)Jhr (x) 3036 FRANCO FLANDOLI AND MATTI LEIMBACH

N 2 N 2 N N N = hr (x) − hr− (x) − 2hr− (x) hr (x) − hr− (x)

N 2 N 2 N N N N 2 N 2 =hr (x) + hr− (x) − 2hr (x) hr− (x) = hr (x) − hr− (x) = Jhr (x) . N 1 a Moreover, Jh a (x) = θN (x − X a ) hence T1 N T1 2 2 X N 2 X N 1 X a Jhr (x) = JhT a (x) 1t≥T a = θN (x − XT a ) 1t≥T a . 1 1 N 2 1 1 r≤t a∈AN a∈AN

In the next lemmata, we generically denote by C > 0 any constant depending −d−2 N only on σ, T , k∇V kL∞ , CF , supN N /N, moments of ST (see Lemma 3.11), R 2 R 2 d |θ (x)| dx and d |∇θ (x)| dx (recall that θ has compact support). R R Lemma 3.4. Z t 2 Z t N N N σ N 2 θN ∗ ∇V ∗ Sr Sr , ∇hr dr ≤ ∇hr L2 dr 0 4 0 Z t N 2 N 2 + C ST hr L2 dr, 0 Z t Z t N N N N 2 θN ∗ FN Sr , · Sr , hr dr ≤ C hr L2 dr. 0 0 Proof. By H¨olderinequality we have Z t 2 Z t N N N σ N 2 θN ∗ ∇V ∗ Sr Sr , ∇hr dr ≤ ∇hr L2 dr 0 4 0 Z t C N N 2 + 2 θN ∗ ∇V ∗ Sr Sr L2 dr σ 0 and then we handle the second term by means of the bound (using also (11)) N N N N θN ∗ ∇V ∗ Sr Sr (x) ≤ k∇V kL∞ Sr hr (x) which follows from (14). The left-hand-side of the second inequality of the lemma is bounded above by Z t Z t N N N N 2 ≤ θN ∗ FN Sr , · Sr , hr dr ≤ CF hr L2 dr 0 0 N N N because θN ∗ FN Sr , · Sr (x) ≤ CF hr (x) by (14) and (6). Lemma 3.5. " # 1,N E sup [M ]t L1 ≤ C t∈[0,T ]

" Z 2 # 1 X a E sup θN (x − X a ) 1t≥T a dx ≤ C. 2 T1 1 t∈[0,T ] d N R a∈AN Proof. Since 2 Z t 1,N σ X a 2 M (x) = 1 a |∇θ (x − X )| ds t N 2 s∈I N s a∈AN 0 MEAN FIELD LIMIT WITH PROLIFERATION 3037

a d we have (using the change of variable x 7→ x−Xs in the Lebesgue integral over R ) Z 2 Z t Z 1,N σ X a 2 a M (x) t dx = 2 1s∈I |∇θN (x − Xs )| dx ds d N d R a∈AN 0 R 2 Z t Z 2 Z Z t σ X 2 σ 2 N a = 2 1s∈I |∇θN (x)| dx ds = |∇θN (x)| dx Ss ds. N d N d a∈AN 0 R R 0 We conclude the ﬁrst estimate of the lemma using Lemma 3.11 and (15). Similarly, since Z 2 Z 1 X a 1 N 2 θ (x − X a ) 1 a dx ≤ S θ (x) dx 2 N T t≥T1 t N d N 1 N d R a∈AN R we deduce the second estimate of the lemma. Lemma 3.6. " N # Z Z t∧τR N 1,N 2,N E sup hs−(x)d Ms (x) + Ms (x) dx d t∈[0,T ] R 0 "Z T 2 # N ≤ 2 + CRE h N ds . s∧τ 2 0 R L Proof. Obviously the expected value on the left-hand-side above is bounded by

 N 2 Z Z t∧τR N 1,N ≤ 2 + E  sup hs (x)dMs (x)dx  d t∈[0,T ] R 0

 N 2 Z Z t∧τR N 2,N + E  sup hs−(x)dMs (x)dx  d t∈[0,T ] R 0

N N (we have also replaced hs−(x) by hs (x) in the integral with respect to the continuous 1,N 1,N martingale Ms ). Concerning the Ms -term, we have  N 2 Z Z t∧τR N 1,N E  sup hs (x)dMs (x)dx  d t∈[0,T ] R 0

 N 2 2 Z t∧τR σ X N a a = E sup 1 a g (X ) · dB 2  s∈I s s s  N t∈[0,T ] a∈AN 0 N R N (we have used stochastic Fubini theorem) where g (y) := d h (x)∇θN (x − y)dx, s R s  2 2 Z T ∧τR σ X N a a ≤ C E 1 a g (X ) · dB N 2  s∈I s s s  a∈AN 0 " # 2 Z T ∧τR σ X N a 2 = C E 1s∈Ia g (X ) ds . N 2 s s a∈AN 0 1 N 2 N 2 But N gs (y) ≤ C hs L2 , by (15), hence  2 t∧τ N " T ∧τ N # Z Z R Z R N 1,N 2 N N 2 E  sup hs (x)dMs (x)dx  ≤ Cσ E Ss hs L2 ds . d t∈[0,T ] R 0 0 3038 FRANCO FLANDOLI AND MATTI LEIMBACH

2,N Concerning the Ms -term, we have

 N 2 Z Z t∧τR N 2,N E  sup hs−(x)dMs (x)dx  d t∈[0,T ] R 0  2 t∧τ N 1 X Z Z R = E sup hN (x) θ x − Xa d (N a − Λa) dx 2  s− N s− s s  N t∈[0,T ] d a∈AN R 0  2 t∧τ N 1 X Z R = E sup gN (Xa ) · d (N a − Λa) 2  es− s− s s  N t∈[0,T ] a∈AN 0

t∧τ N where gN (y) := R hN (x)θ (x − y)dx. The process P R R gN (Xa ) · es d s N a∈AN 0 es− s− a a R d (Ns − Λs ) is a martingale with respect to the ﬁltration Gt := FΛt , hence the last expression is bounded by  2 C X Z T ∧τR ≤ E gN (Xa ) · d (N a − Λa) . N 2  es− s− s s  a∈AN 0 a b Since the jumps of Ns and Ns , for a 6= b, never occur at the same time, we have " ! !# Z T ∧τR Z T ∧τR N a a a N b b b,N E ges− Xs− d (Ns − Λs ) ges− Xs− d Ns − Λs = 0. 0 0 Hence the last expression is equal to  2 T ∧τ N C X Z R = E gN (Xa ) · d (N a − Λa) . (16) N 2  es− s− s s  a∈A 0 It is known that  2 T ∧τ N " T ∧τ N # Z R Z R N a a a N a 2 a E  ges− Xs− d (Ns − Λs )  = E ges− Xs− dΛs 0 0 " N # Z T ∧τR N a 2 N a =E ges (Xs ) 1s∈Ia FN Ss ,Xs ds 0 " N # Z T ∧τR N a 2 ≤CF E ges (Xs ) 1s∈Ia ds 0 hence (16) is bounded above by " N # Z T ∧τR C 1 X N a 2 ≤ E g (X ) 1s∈Ia ds N N es s 0 a∈A " N # Z T ∧τR Z C N 2 N = E gs (x) Ss (dx) ds . N d e 0 R 2 2 As above for gN (y), using (15) we have 1 gN (y) ≤ C hN . Hence we get s N es s L2 " N # Z T ∧τR N 2 N ≤ CE hs L2 Ss ds . 0 MEAN FIELD LIMIT WITH PROLIFERATION 3039

N The result of the lemma follows by estimating Ss by R, being the integral in s N only up to T ∧ τR . Corollary 1. Z T ! N 2 N 2 lim sup P sup ht 2 + ∇hr 2 dr > R = 0. R→∞ L L N∈N t∈[0,T ] 0 Proof. Step 1. From lemmata 3.3 and 3.4 we have

2 t∧τ N t∧τ N 1 2 σ Z R 2 1 2 Z R 2 2 N N N N N h N + ∇h 2 dr ≤ h 2 + C S h 2 dr t∧τ 2 r L 0 L r r L 2 R L 4 0 2 0 N Z t∧τR N 2 + C hr L2 dr + aN + bN,R 0 where Z 2 1 1,N 1 1 X a a aN := sup [M ]t 1 + sup θN (x − X a ) 1t≥T dx L 2 T1 1 2 t∈[0,T ] 2 t∈[0,T ] d N R a∈AN N Z Z t∧τR N 1,N 2,N bN,R := sup hs−(x)d Ms (x) + Ms (x) dx . d t∈[0,T ] R 0 N 2 Hence, writing χR = 1 hN ≤R, from Lemmata 3.5 and 3.6 we have k 0 kL2

" 2 # 1 N N 2 E χ sup h N ≤ R /2 + C R r∧τ 2 2 r∈[0,t] R L

Z t " 2 # 2 N N + C R + R + 1 E χ sup h N ds. R r∧τ 2 0 r∈[0,s] R L

By Gronwall’s lemma, we get, with C (R) := R2 + 2C exp 2C R2 + R + 1,

" 2 # N N E χ sup h N ≤ C (R) . R t∧τ 2 t∈[0,T ] R L Moreover, for the same reasons,

" N # 2 Z t∧τR σ N N 2 E χR ∇hr L2 dr 4 0

Z t " 2 # 2 2 N N ≤R /2 + C + C R + R + 1 E χ sup h N ds ≤ C1 (R) R r∧τ 2 0 r∈[0,s] R L

2 2 where C1 (R) = R /2 + C + C R + R + 1 TC (R) and we have used the previous bound in the last term. N 2 Step 2. For every R1 > 0, the probability P supt∈[0,T ] ht L2 ≥ R1 is bounded above by ! N 2 N N 2 ≤ P sup ht L2 ≥ R1, χR = 1 + P h0 L2 > R t∈[0,T ] 3040 FRANCO FLANDOLI AND MATTI LEIMBACH ! N N 2 N 2 ≤ P χR sup ht L2 ≥ R1 + P h0 L2 > R t∈[0,T ] ! N N 2 N 2 ≤ P χR sup ht L2 ≥ R1, τR ≥ T + P (τR < T ) + P h0 L2 > R t∈[0,T ] ! 2 N N N N 2 ≤ P χ sup h N ≥ R1 + P S ≥ R + P h 2 > R R t∧τ 2 T 0 L t∈[0,T ] R L " # 1 2 1 2 N N N N ≤ E χ sup h N + E S + P h 2 > R R t∧τ 2 T 0 L R1 t∈[0,T ] R L R

C (R) C N 2 ≤ + + P h0 L2 > R . R1 R

We may now ﬁnd functions R1 7→ R (R1) such that limR1→∞ R (R1) = +∞, limR1→∞ C (R (R1)) /R1 = 0, where the function C(R) has been deﬁned in step 1. We deduce, from assumption (4), that ! N 2 lim sup P sup ht 2 ≥ R1 = 0. R →∞ L 1 N∈N t∈[0,T ] This proves one half of the claim of the corollary.

R T N 2 Step 3. For every R1 > 0, by similar arguments P 0 ∇hr L2 dr ≥ R1 is bounded above by

N ! 2 Z t∧τR 2 N σ N 2 σ N 2 ≤ P χR ∇hr L2 dr > R1 + P (τR < T ) + P h0 L2 > R 4 0 4 " N # Z t∧τR 4 N N 2 1 N N 2 ≤ 2 E χR ∇hr L2 dr + E ST + P h0 L2 > R σ R1 0 R

16C1 (R) C N 2 ≤ 4 + + P h0 L2 > R . σ R1 R As above, we conclude that the second half of the claim of the corollary holds true.

N In order to show tightness of the family of the functions {h }N , in addition to the previous bound which shows a regularity in space, we also need a regularity in time. See the compactness criteria below. Lemma 3.7. Given any α ∈ (0, 1/2),

Z T Z T N N 2 ! h − h −1,2 lim sup P t s W dsdt > R = 0. R→∞ 1+2α N∈N 0 0 |t − s|

N N 2 Proof. Step 1. We need to estimate ht − hs W −1,2 in such a way that it cancels with the singularity in the denominator at t = s. First, we have Z t 2 N N 2 N N ht − hs W −1,2 ≤ C div θN ∗ ∇V ∗ Sr Sr dr s W −1,2 MEAN FIELD LIMIT WITH PROLIFERATION 3041

Z t 2 2 σ N + C ∆hr dr s 2 W −1,2 Z t 2 N N + C θN ∗ FN Sr , · Sr dr s W −1,2 2 2 1,N 1,N 2,N 2,N + C Mt − Ms + C Mt − Ms W −1,2 W −1,2 and thus by the H¨olderinequality

Z t N N 2 ≤ C (t − s) div θN ∗ ∇V ∗ Sr Sr W −1,2 dr s Z t N N 2 + C (t − s) θN ∗ FN Sr , · Sr W −1,2 dr s Z t 2 2 σ N + C (t − s) ∆hr dr s 2 W −1,2 2 2 1,N 1,N 2,N 2,N + C Mt − Ms + C Mt − Ms . W −1,2 W −1,2

Notice that L2 ⊂ W −1,2 with continuous embedding, namely there exists a constant 2 C > 0 such that kfkW −1,2 ≤ C kfkL2 for all f ∈ L . Moreover, the linear operator 2 −1,2 div is bounded from L to W , namely kdiv fkW −1,2 ≤ C kfkL2 and the operator 1,2 −1,2 ∆ is bounded from W to W , namely k∆fkW −1,2 ≤ C kdiv fkW 1,2 . Therefore N (we denote by C > 0 any constant independent of N, h. , t, s)

Z t Z t N N 2 N 2 ≤ C (t − s) θN ∗ ∇V ∗ Sr Sr L2 dr + C (t − s) hr W 1,2 dr s s Z t N N 2 + C (t − s) θN ∗ FN Sr , · Sr L2 dr s 2 2 1,N 1,N 2,N 2,N + C Mt − Ms + C Mt − Ms L2 L2

R t R T and now using (14), assumption (6) and s ≤ 0 in all terms,

Z T N N 2 N 2 N 2 ht − hs W −1,2 ≤ C (t − s) Sr + 1 hr L2 dr 0 Z T N 2 + C (t − s) hr W 1,2 dr 0 2 2 1,N 1,N 2,N 2,N + C Mt − Ms + C Mt − Ms L2 L2 N 2 N 2 ≤ C (t − s) ST + 1 sup hr L2 r∈[0,T ] Z T N 2 + C (t − s) hr W 1,2 dr 0 2 X i,N i,N + C Mt − Ms . L2 i=1,2 3042 FRANCO FLANDOLI AND MATTI LEIMBACH

2 hN −hN R T R T k t s kW −1,2 Accordingly, we split the estimate of P 0 0 |t−s|1+2α dsdt > R in four more elementary estimates, that now we handle separately; the final result will be a consequence of them. R T R T 1 The number Cα = 0 0 |t−s|2α dsdt is finite, hence the first term is bounded by (renaming the constant C)

Z T Z T N N 2 ! C (t − s) ST + 1 supr∈[0,T ] hr L2 P 1+2α dsdt > R 0 0 |t − s| ! N N 2 =P ST + 1 sup hr L2 > R/C r∈[0,T ] ! N p N 2 p ≤P ST + 1 > R/C + P sup hr L2 > R/C r∈[0,T ] and both these terms are, uniformly in N, small for large R, due to Lemma 3.11 and the estimates proved in the tightness part. The second addend, the one with R T N 2 C (t − s) 0 hr W 1,2 dr, is similar. Step 2. Concerning the martingale terms, we now prove that 2 i,N i,N E Mt − Ms ≤ C |t − s| L2 for some constant C > 0, i = 1, 2. By Chebishev inequality it follows that 2  i,N  Z T Z T M − M i,N t s 2 lim sup P  L dsdt > R = 0 R→∞  1+2α  N∈N 0 0 |t − s| and the proof will be complete. For notational convenience, we abbreviate, for i = 1, 2, 1 X M i,N (x) = M i,a(x). t N t a∈AN Note, that for every x ∈ Rd the processes M 1,a(x) and M 2,a(x) are martingales. It follows, with computations similar to those of Lemma 3.6, for t ≥ s 2 Z 1 2 1,N 1,N X 1,a 1,a E Mt − Ms = 2 E Mt (x) − Ms (x) dx L2 d N R a∈AN Z Z t 1 X a 2 a = 2 E 1r∈I ∇θN (x − Xr ) dr dx d N R a∈AN s Z t 1 2 1 X = k∇θ k E 1 a dr ≤ C (t − s) N N L2 N r∈I s a∈AN where in the last inequality we have used (15) and Lemma 3.11. Similarly, for the second martingale, 2 Z 1 h i 2,N 2,N X 2,a 2 2,a 2 E Mt − Ms = 2 E Mt (x) − Ms (x) dx L2 d N R a∈AN MEAN FIELD LIMIT WITH PROLIFERATION 3043

Z Z t 1 X a 2 a a = 2 E 1r∈I θN (x − Xr ) λr dr dx d N R a∈AN s Z t 1 2 1 X ≤ C kθ k E 1 a dr ≤ C (t − s) . F N N L2 N r∈I s a∈AN

A version of Aubin-Lions lemma, see [7], [6], [2], states that when E0 ⊂ E ⊂ E1 are three Banach spaces with continuous dense embedding, E0,E1 reflexive, with E0 compactly embedded into E, given p, q ∈ (1, ∞) and α ∈ (0, 1), the space q α,p q L (0,T ; E0) ∩ W (0,T ; E1) is compactly embedded into L (0,T ; E). We use 2 1,2 −1,2 d this lemma with E = L (D), E0 = W (D) and E1 = W R , where D is a regular bounded domain, and with p = q = 2, α ∈ (0, 1/2). The lemma 2 1,2 α,2 −1,2 d states that L 0,T ; W (D) ∩ W 0,T ; W R is compactly embedded into L2 0,T ; L2 (D). α,p Notice that for αp > 1, the space W (0,T ; E1) is embedded into C ([0,T ]; E1), so it is not suitable for our purposes since we have to deal with discontinuous α,p processes. However, for αp < 1 the space W (0,T ; E1) includes piecewise constant functions, as one can easily check. Therefore it is a good space for càdlàgprocesses. Now, consider the space ∞ 2 d 2 1,2 d α,2 −1,2 d Y0 := L 0,T ; L R ∩ L 0,T ; W R ∩ W 0,T ; W R . 2 d Using the Fréchet topology on Lloc [0,T ] × R defined as ∞ Z T Z ! X 2 d (f, g) = 2−n 1 ∧ |(f − g)(t, x)| dxdt n=1 0 B(0,n) 2 1,2 d α,2 −1,2 d one concludes that L 0,T ; W R ∩W 0,T ; W R is compactly em- 2 d bedded into Lloc [0,T ] × R (the proof is elementary, using that if a set is com- 2 2 2 d pact in L 0,T ; L (B (0, n)) for every n then it is compact in Lloc [0,T ] × R ∞ 2 d with this topology; see a similar result in [5]). Denoting by Lw∗ 0,T ; L R 2 1,2 d ∞ 2 d 2 1,2 d and Lw 0,T ; W R the spaces L 0,T ; L R and L 0,T ; W R endowed respectively with the weak star and weak topology, we have that Y0 is compactly embedded into ∞ 2 d 2 1,2 d 2 d Y := Lw∗ 0,T ; L R ∩ Lw 0,T ; W R ∩ Lloc [0,T ] × R . (17) N N Denote by Q the laws of h on Y0. From the “boundedness in N∈N N∈N N probability” of the family Q , in Y0, stated by Corollary1 and Lemma 3.7, it N∈N follows that the family QN is tight in Y . Hence, by Prohorov theorem, from N∈N every subsequence of QN it is possible to extract a further subsequence which N∈N converges to a probability measure Q on Y . We shall prove that every such limit measure Q is a delta Dirac Q = δu concentrated to the same element u ∈ Y , hence N N the whole sequence Q converges to δu; and also the processes h N∈N N∈N converge in probability to u. 3.1. Auxiliary results. Let V , F satisfy the assumptions of Theorem 1.1 above. 2 d 1 d Definition 3.8. Given u0 ∈ L R ∩L R , u0 ≥ 0, by weak solution of equation (7) we mean a function u ≥ 0 of class ∞ 2 d 2 1,2 d L 0,T ; L R ∩ L 0,T ; W R 3044 FRANCO FLANDOLI AND MATTI LEIMBACH such that Z t hut, φi = hu0, φi − h(∇V ∗ ur) ur, ∇φi dr 0 (18) σ2 Z t Z t − h∇ur, ∇φi dr + hF (ur) ur, φi dr 2 0 0

1,2 d for almost every t ∈ [0,T ] and for all φ ∈ W R .

∞ 2 d Notice that, due to u ∈ L 0,T ; L R , and the assumption that ∇V is bounded and compact supported, we have (∇V ∗ ur) bounded, hence the first integral in the weak equation is well defined. Moreover, since F is uniformly bounded (see (6)), the last term is also well defined.

Remark 1. If u is a solution in the sense of the deﬁnition, then there exists 2 d a (unique) re-presentative of u of class Cw [0,T ]; L R . Indeed, given φ ∈ 1,2 d W R , from identity (18) we deduce that the a.e. deﬁned function t 7→ hut, φi 2 d is a.s. equal to a continuous function gφ. Let u : [0,T ] → L R be a bounded 2 d 1,2 d measurable representative (bounded in L R by C). Let {φn} ⊂ W R be 2 d dense in L R . Let Υ ⊂ [0,T ] be a set of measure T such that hu (t) , φni = gφn (t) 2 d for all t ∈ Υ. Then, for φ ∈ L R and t, s ∈ Υ,

|hu (t) − u (s) , φi| ≤ |gφn (t) − gφn (s)| + 2C kφn − φkL2 . From this it follows that t 7→ hu (t) , φi is uniformly continuous on Υ hence uniquely extendible to a continuous function t 7→ Lt (φ) on [0,T ]. From this it is easy to extract a re-deﬁnition of u (t) for t∈ / Υ so that t 7→ hu (t) , φi is continuous on [0,T ]. Finally, it is not diﬃcult to show that identity (18) holds for all t ∈ [0,T ] for the 2 d representative of class Cw [0,T ]; L R . Theorem 3.9. There is at most one weak solution of equation (7).

(i) (1) (2) Proof. Step 1. If ut , i = 1, 2 are two solutions and vt = ut − ut , from the equation (in weak form)

σ2 ∂ v = ∆v + div (∇V ∗ v ) u(1) + div ∇V ∗ u(2) v t t 2 t t t t t (1) (1) (2) (2) + F ut ut − F ut ut

2 −1,2 d and the property ∂tv ∈ L 0,T ; W R (see Step 2 below), we have

2 Z t 1 2 σ 2 kvtkL2 + k∇vskL2 ds 2 2 0 1 = kv k2 2 0 L2 Z t Z t D (1) E D (2) E − (∇V ∗ vs) us , ∇vs ds − ∇V ∗ us vs, ∇vs ds 0 0 Z t D (1) (1) (2) (2) E + F us us − F us us , vs ds. 0 MEAN FIELD LIMIT WITH PROLIFERATION 3045

Since Z

k∇V ∗ vsk∞ = sup ∇V (x − y) vs (y) dy ≤ k∇V kL2 kvskL2 , x

(2) (2) ∇V ∗ us ≤ k∇V kL2 us ∞ L2 the terms on the second line can be bounded by 2 Z t Z t 2 Z t 2 σ 2 (1) (2) ≤ k∇vsk 2 ds + C (∇V ∗ vs) u ds + C ∇V ∗ u vs ds L s 2 s 2 4 0 0 L 0 L 2 Z t Z t 2 2 σ 2 2 2 (1) (2) ≤ k∇vsk 2 ds + C k∇V k 2 kvsk 2 u + u ds. L L L s 2 s 2 4 0 0 L L The terms on the last line, using assumptions (2)-(6), can be bounded by Z t D E (1) (1) (2) (2) F us us − F us us , vs ds 0 Z t D E Z t D E (1) (2) (2) (1) ≤ F us − F us us , |vs| ds + F us |vs| , |vs| ds 0 0 Z t Z t (2) 2 ≤ LF ku − wk 1,2 u kvsk 2 ds + CF kvsk 2 ds W s 2 L L 0 L 0 2 Z t Z t 2 σ 2 2 (2) ≤ k∇vsk 2 ds + C kvsk 2 1 + u ds. L L s 2 8 0 0 L It is then sufficient to apply Gronwall lemma to deduce v = 0. 1,2 −1,2 d Step 2. To complete the proof, we check v ∈ W 0,T ; W R , which is needed to apply the chain rule. To this end, we check that any weak solution u of equation (7) given by the definition has this regularity property. The 1,2 −1,2 d term u0 is in W 0,T ; W R . The function r 7→ (∇V ∗ ur) ur is of class 2 2 d L 0,T ; L R , since ∇V ∗ur is bounded, as remarked after the definition; hence 2 −1,2 d the function r 7→ div ((∇V ∗ ur) ur) is of class L 0,T ; W R and thus the R t 1,2 −1,2 d function t 7→ 0 div ((∇V ∗ ur) ur) dr is of class W 0,T ; W R . The func- 2 −1,2 d 2 1,2 d tion r 7→ ∆ur is of class L 0,T ; W R because u ∈ L 0,T ; W R R t 1,2 −1,2 d and thus the function t 7→ 0 ∆urdr is of class W 0,T ; W R . Finally 2 2 d the function r 7→ F (ur) ur is of class L 0,T ; L R , since F (ur) is bounded, as R t remarked after the definition; hence the function t 7→ 0 div ((∇V ∗ ur) ur) dr is of 1,2 −1,2 d class W 0,T ; W R . It follows that the function Z t σ2 Z t Z t t 7→ u0 + div ((∇V ∗ ur) ur) dr + ∆urdr + F (ur) urdr 0 2 0 0 1,2 −1,2 d is of class W 0,T ; W R . But it easily coincides with the function t 7→ u (t), see also Remark1.

+ + Lemma 3.10. If F (u)(x) = (1 − (W ∗ u)(x)) , FN (µ, x) = (1 − (W ∗ µ)(x)) , with W a Lipschitz continuous compact support probability density, assumptions (2)-(3)-(5)-(6) are fulﬁlled.

1,2 d 2 d Proof. First, it is obvious that F maps W+ R (in fact even L+ R ) into β d Cb R for any β ∈ (0, 1) and (6) holds true (the range are functions with values in [0, 1]). The map u 7→ W ∗ u is Lipschitz continuous, being linear bounded, 3046 FRANCO FLANDOLI AND MATTI LEIMBACH

1,2 d 2 d β d from W+ R (in fact L+ R ) to Cb R , hence (2)-(3) are true for F by com- position with the Lipschitz bounded function r 7→ (1 − r)+ (we also use the fact that W ∗ u ≥ 0). Finally,

h N i γ[ST ] Lemma 3.11. There exists a γ > 0 such that supN E e < ∞.

Proof. On the same probability space (Ω, F,P ) one can construct two processes, a N a N Xt ; a ∈ A , which is equal in law to our particle system, and Yt ; a ∈ A , D N E D N E such that S (X)t , 1 ≤ S (Y )t , 1 a.s. (we distinguish the objects of the different particle systems by adding “(X)” or “(Y )”), where the branching rate of a N Yt ; a ∈ A is constant and dominates the branching rate of the “X” system, a,N N a,N i.e. λ(Y )t ≡ CF for all a ∈ A(Y )t and t ∈ I(Y ) . Further, one has the representation D N E 1 X (k) S (Y ) , 1 = N , T N T k=1,...,N (k) where N· is a Yule process with birth rate CF and start in 1 that describes the number of descendant of a(Y ) = (k) which are alive at time T . Furthermore, (k) NT are identically distributed and independent random variables for k = 1, ..., N. Together with Jensen’s inequality it follows

N h γ S(Y )N ,1 i h γ N (1) i h γN (1) i sup E e h t i = sup E e N T ≤ E e T . n∈N n∈N (1) Because NT is geometrically distributed with parameter CF , the right-hand side is ﬁnite if γ < 1 . 1−e−CF T

i,N Proposition 1. Assume that X0 , i = 1, ..., N, are independent identically dis- 2 d tributed r.v. with common probability density u0 ∈ L R . Let θN be molliﬁers of −d −1 2 d 1/d the form θN (x) = N θ N x , with θ ∈ L R and N ≥ CN . Then

h N 2 i sup E θN ∗ S0 L2( d) < ∞. N R The proof is an easy exercise, similar to a classical computation about the weak law of large numbers.

1 4. Passage to the limit. Given χ : [0,T ] → R of class C , with χT = 0 and given 1,2 d ψ ∈ W R , deﬁned φt := χtψ, we have MEAN FIELD LIMIT WITH PROLIFERATION 3047

Z T Z 2 ∂φt N σ N 0 = ht − ∇φt · ∇ht dxdt d ∂t 2 0 R Z T Z N N − θN ∗ ∇V ∗ St St · ∇φtdxdt d 0 R Z T Z N N N + θN ∗ FN St , · St φtdxdt + h0 , φ0 d 0 R Z Z T Z Z T 1,N 2,N − φtdMt dx + φtdMt dx. d d R 0 R 0 N N Nk Denote by Q the law of ht and assume a subsequence Q weakly converges, in the topology of the space Y deﬁned by (17), to a probability measure Q. Given φt := χtψ as above, the functional Z T Z 2 ∂φt σ u 7→ Ψφ (u) := ut − ∇φt · ∇ut d ∂t 2 0 R − ∇φt · (∇V ∗ ut) ut + F (ut) utφt dxdt + hu0, φ0i is continuous on Y . It is known (part of the so called Portmanteau Theorem) that weak convergence of QNk to Q implies Q (A) ≤ lim infQNk (A) if A is an open set; k→∞ and the set {u ∈ Y : |Ψφ (u)| > } is open since Ψφ is continuous. Hence, for every > 0,

Nk Nk Q (u : |Ψφ (u)| > ) ≤ lim infQ (u : |Ψφ (u)| > ) = lim infP Ψφ h· > . k→∞ k→∞

We shall prove that this lim inf is zero. It will follow Q (u : |Ψφ (u)| > ) = 0. Since this holds for every > 0, we will deduce Q (u :Ψφ (u) = 0) = 1. From this fact we shall prove the following result. Lemma 4.1. Q is supported on the set of weak solutions of equation (7).

Nk Proof. It remains to prove that lim infP Ψφ h· > = 0, that the assertion of k→∞ the lemma follows from the fact that Q (u ∈ Y :Ψφ (u) = 0) = 1 for every φ of the form φt = χtψ as above, and that Q is concentrated on non negative functions. We prove these claims in three diﬀerent steps.

Nk Step 1. We show that lim infP Ψφ h· > = 0. Let us write N in place of k→∞ Nk for simplicity of notation. We have Z T Z 2 N ∂φt σ N N N Ψφ h· = + ∆φt − ∇φt · ∇V ∗ ht + F ht φt ht dxdt d ∂t 2 0 R + hu0, φ0i . N Using the identity satisﬁed by ht against the test function φt, we have Z T Z N N N N N Ψφ h· = θN ∗ ∇V ∗ St St − ∇V ∗ ht ht · ∇φtdxdt d 0 R Z T Z N N N N + F ht ht − θN ∗ FN St , · St φtdxdt d 0 R Z Z T Z Z T N 1,N 2,N + u0 − h0 , φ0 + φtdMt dx + φtdMt dx. d d R 0 R 0 3048 FRANCO FLANDOLI AND MATTI LEIMBACH

N N N u0 − h0 , φ0 = u0 − S0 , φ0 + S0 , φ0 − θN (−·) ∗ φ0 From Lemmata 4.2 and 4.3 below we have Z T ! N N 2 N β N 2 Ψφ h· ≤ C (N + αN ) ST + C ST N ht W 1,2 dt + 1 0 N − N + u0 − S0 , φ0 + θN ∗ φ0 − φ0 S0 Z Z T Z Z T 1,N 2,N + φtdMt dx + φtdMt dx d d R 0 R 0 where the constant C depends only on the quantities described below, before Lemma N 4.2. In order to prove limN→∞ P Ψφ h· > ε = 0, it is sufficient to prove the same result for each one of the previous terms. N 2 We have limN→∞ P C (N + αN ) ST > ε = 0 from Chebyshev inequality N β − N and Lemma 3.11. The same applies to the terms C ST N and θN ∗ φ0 − φ0 S0 . N The term u0 − S0 , φ0 is obvious by the assumption of convergence in probabil- N ity on S0 . The two martingale terms can be treated again by Chebyshev inequality and Lemma 4.4 below. Finally Z T ! Z T ! N β N 2 N N 2 −β P C ST N ht W 1,2 dt > ε = P ST ht W 1,2 dt > εN /C 0 0 q Z T q ! N −β N 2 −β ≤P ST > εN /C + P ht W 1,2 dt > εN /C . 0 The first term goes to zero by Chebyshev inequality and Lemma 3.11. The second one goes to zero, as N → ∞, by Corollary1, by a simple argument based on the definition of limit. Step 2. We prove that the assertion of the lemma follows from the fact that n Q (u ∈ Y :Ψφ (u) = 0) = 1 for every φ of the form φt = χtψ as above. Let {χ } n 1 n be a sequence of functions χ : [0,T ] → R of class C , with χT = 0, which is dense 2 m 1,2 d n,m n m in L (0,T ). Let {ψ } be a dense sequence in W R . Set φt := χt ψ ; we have Q (u ∈ Y :Ψφn,m (u) = 0, ∀n, m ∈ N) = 1. Let us prove that the set A := {u ∈ Y :Ψφn,m (u) = 0, ∀n, m ∈ N} is contained in the set of weak solutions. m If u ∈ A, then Ψχtψ (u) = 0 for every m ∈ N and every Lipschitz continuous χ : [0,T ] → R with χT = 0. The proof of this claim by approximation with the n sequence χt is not difficult and we omit the details. Given u ∈ Y , there exists a set Υ ⊂ [0,T ) of full measure such that

1 Z t0+ n Z Z m m lim n ψ utdxdt = ψ ut0 dx n→∞ d d t0 R R n for every t0 ∈ Υ and every m ∈ N. Given t0 ∈ Υ, take the new sequence χt n 1 n defined (at least for n large enough) as χt = 0 for t > t0 + n , χt = −1 for t < t0, n 1 χt = −1 + n (t − t0) for t ∈ t0, t0 + n . We have n,m 1 Z T Z Z t0+ n Z Z ∂φt m m utdxdt = n ψ utdxdt → ψ ut0 dx. d ∂t d d 0 R t0 R R We deduce, from Ψχnψm (u) = 0, ∀n, m ∈ N, that identity (18) holds at time t0, for all φ = ψm, m ∈ N. Therefore it is true, for each φ = ψm, a.s. in time. By density MEAN FIELD LIMIT WITH PROLIFERATION 3049 of {ψm} and the regularity properties of u it is easy to deduce that (18) holds for 1,2 d every φ ∈ W R . Step 3. By the Portmanteau Theorem, the weak convergence of QNk to Q implies that Q (A) ≥ lim supQNk (A) if A is a closed set. Note, that {u: u ≥ 0} is closed in k→∞ Y, hence Q (u: u ≥ 0) ≥ lim sup QNk (u: u ≥ 0) = lim sup P hNk ≥ 0 = 1. k→∞ k→∞ Thus, Q is concentrated on non-negative functions, which completes the proof. In the next three lemmata we denote by C any constant depending only on σ2, 2 2 2 −d−2 N T , CF , D V ∞, kθkL2 , k∇θkL2 , supN εN /N, kφk∞, k∇φk∞, E ST and the diameter of the support of θ. Lemma 4.2 below treats the convergence of the divergence terms. Lemma 4.2. Z T Z N N N N N 2 θN ∗ ∇V ∗ Ss Ss − ∇V ∗ ht ht · ∇φtdxdt ≤ CN ST . d 0 R Proof. The left-hand-side is bounded above by Z T Z N N N N ≤ k∇φk∞ θN ∗ ∇V ∗ St St − ∇V ∗ St ht dxdt 0 d R Z T Z N N N N + k∇φk∞ ∇V ∗ St ht − ∇V ∗ ht ht dxdt . d 0 R Let us separately bound the two terms. The first one is a sort of commutation lemma estimate. We have N N N N θN ∗ ∇V ∗ St St (x) − ∇V ∗ St θN ∗ St (x) Z N N N ≤ θN (x − y) ∇V ∗ St (y) − ∇V ∗ St (x) St (dy) Z 2 N N ≤ D V ∞ St θN (x − y) |x − y| St (dy) . −d −1 Since θN (x) = N θ N x with N → 0, θ smooth non-negative compact support with diameter K, we have Z Z N N N θN (x − y) |x − y| Sr (dy) ≤ KN θN (x − y) Sr (dy) = KN hr (x) . Summarizing, the first term above is bounded by Z T Z 2 N N ≤ k∇φk∞ D V ∞ KN Sr hr (x) dxdr d 0 R 2 N 2 ≤ D V ∞ k∇φk∞ KN T ST R N N where in the last inequality we have used hr (x) dx = Sr . The second term above is bounded above by Z t Z N N N ≤ k∇φk∞ ∇V ∗ Sr − ∇V ∗ hr hr dxdr 0 Z t Z 2 N N 2 N 2 ≤ D V ∞ k∇φk∞ KN Sr hr dxdr ≤ D V ∞ k∇φk∞ KN T ST 0 3050 FRANCO FLANDOLI AND MATTI LEIMBACH where we have used the fact that N N ∇V ∗ Sr (x) − ∇V ∗ ht (x) ZZ N = [∇V (x − y) − ∇V (x − z)] θN (z − y) dzS (dy) r ZZ 2 N 2 N ≤ D V ∞ |y − z| θN (z − y) dzSr (dy) ≤ D V ∞ KN Sr .

Lemma 4.3 proves the convergence of the proliferation terms. Lemma 4.3. Z T Z N N N N F ht ht − θN ∗ FN St , · St φtdxdt d 0 R Z T ! N β N 2 N 2 ≤ C ST N ht W 1,2 dt + 1 + CαN ST . 0 Proof. We have, using assumptions (3)-(5),

|F (θN ∗ µ)(x) − FN (µ, y)|

≤ |F (θN ∗ µ)(x) − F (θN ∗ µ)(y)| + |F (θN ∗ µ)(y) − FN (µ, y)| β ≤C (kθN ∗ µkW 1,2 + 1) |x − y| + αN [µ] hence

β β ≤ K (kθN ∗ µkW 1,2 + 1) N + αN [µ] (θN ∗ µ)(x) therefore the left-hand-side in the statement of the lemma is bounded above by Z T Z β N β N N ≤ kφk∞ K ht W 1,2 + 1 N + αN St ht (x) dxdt d 0 R Z T β N β N N ≤ kφk∞ K ht W 1,2 + 1 N + αN St St dt. 0

Finally, Lemma 4.4 provides the needed control over the martingale terms. Lemma 4.4. For i = 1, 2  2 Z Z T i,N 2 E  φt(x)dMt (x)dx  ≤ CN . d R 0 Proof. The proof is very similar to the one of Lemma 3.6, but taking advantage of the smoothness of φ which did not appear there. We only sketch the computations. Denoting MEAN FIELD LIMIT WITH PROLIFERATION 3051

Z Z N gt (y) := − φt(x)∇θN (x − y) dx = ∇φt(x)θN (x − y) dx d d R R for the ﬁrst martingale term we have  2 Z Z T 2 "Z T # 1,N σ X N a 2 a E  φt(x)dMt (x)dx  = 2 E 1t∈I gt (Xt ) dt d N R 0 a∈AN 0 2 Z T σ 2 2 N ≤ kθN kL2 k∇φk∞ E St dt N 0 N R and then we use (15) and Lemma 3.11. Set g (y) := − d φt(x)θN (x − y) dx, then et R for the second martingale term we have  2 Z Z T "Z T # 2,N 1 X N a 2 a a E  φt(x)dMt (x)dx  = 2 E 1t∈I gt (Xt ) λt dt d N e R 0 a∈AN 0 Z T CF 2 2 N ≤ kθN kL2 kφkL∞ E St dt N 0 and we conclude by the same argument. This completes the proof. With the preceding three lemmata, which are used in the proof of Lemma 4.1, the proof of Theorem 1.1 is complete.

REFERENCES

[1] R. E. Baker and M. J. Simpson, Correcting mean-field approximations for birth-death- movement processes, Phys. Rev. E, 82 (2010), 041905, 12 pp. [2] L. Banas, Z. Brzezniak, M. Neklyudov and A. Prohl, Stochastic Ferromagnetism: Analysis and Numerics, De Gruyter Studies in Mathematics, 58. De Gruyter, Berlin, 2014. [3] M. Bodnar and J. J. L. Velazquez, Derivation of macroscopic equations for individual cell- based models: A formal approach, Math. Meth. Appl. Sci., 28 (2005), 1757–1779. [4] Z. Brzezniak, M. Ondrejat and E. Motyl, Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains, preprint, arXiv:1502.02637. [5] Z. Brzezniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains, J. Diff. Eq., 254 (2013), 1627–1685. [6] F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, preprint, Probab. Theory Relat. Fields, 102 (1995), 367–391. [7] J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires, Dunod, Gauthier-Villars, Paris, 1969. [8] S. Meleard and V. Bansaye, Some Stochastic Models for Structured Populations: Scal- ing Limits and Long Time Behavior, Stochastics in Biological Systems, 1.4. Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 2015, arXiv:1506.04165v1. [9] M. Metivier, Quelques problèmesliésaux systèmesinfinis de particules et leurs limites, Séminaire de Probabilités, (Strasbourg) 20 (1986), 426–446. [10] G. Nappo and E. Orlandi, Limit laws for a coagulation model of interacting random particles, Annales de l’I.H.P. section B, 24 (1988), 319–344. [11] K. Oelschläger, A law of large numbers for moderately interacting diffusion processes, Zeitschrift fur Wahrsch. Verwandte Gebiete, 69 (1985), 279–322. [12] K. Oelschläger, On the Derivation of Reaction-Diffusion Equations as Limit Dynamics of Systems of Moderately Interacting Stochastic Processes, Probab. Th. Rel. Fields, 82 (1989), 565–586. [13] R. Philipowski, Interacting diffusions approximating the porous medium equation and prop- agation of chaos, Stoch. Proc. Appl., 117 (2007), 526–538. [14] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183–212. 3052 FRANCO FLANDOLI AND MATTI LEIMBACH

[15] S. He, J. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Beijing: Science Press, 1992. [16] P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K. R. Swanson, M. Loveless and A. R. A. Anderson, A spatial model of tumor-host interaction: Application of chemoterapy, Math. Biosc. Engin., 6 (2009), 521–546. [17] K. Uchiyama, Pressure in classical Statistical Mechanics and interacting Brownian particles in multi-dimensions, Ann. H. Poincar´e, 1 (2000), 1159–1202. [18] S. R. S. Varadhan, Scaling limit for interacting diﬀusions, Comm. Math. Phys., 135 (1991), 313–353. Received October 2015; revised March 2016. E-mail address: [email protected] E-mail address: [email protected]