Dipartimento di Matematica
Corso di Dottorato in Matematica
Ph.D. Thesis
Some perspectives on mathematical
modeling for aggregation phenomena
Advisor:
Prof. Franco Flandoli
Candidate:
Marta Leocata
XXXI Ph.D. cycle
Pisa, 2018
1 Introduction 1
1.1 Scaling limit for a non-local model to cell-cell adhesion . . . . 3
1.1.1 Main result of the Chapter . . . 4
1.2 A particle system approach to Vlasov-Fokker-Planck-Navier-Stokes Equations . . . 6
1.2.1 Main result of the Chapter . . . 6
1.3 A new averaging principle to investigate some aspects of tumor progression . . . 10
1.3.1 Main result of the Chapter . . . 11
1.4 Open Problems . . . 12
2 Scaling limit for a non-local model to cell-cell adhesion 16 2.1 A PDE model for cell-cell adhesion . . . 16
2.2 A microscopic Approach . . . 19
2.3 Cell-Cell Adhesion: local or non local phenomenon? . . . 20
2.4 Main assumptions and Basic results . . . 22
2.4.1 Basic results on particle system . . . 22
2.4.2 Basic results on the PDE system . . . 26
2.4.3 Some useful properties on the Fractional Sobolev space 27 2.4.4 Preliminaries results for the empirical measure and Main Result . . . 29
2.5 Tightness . . . 33
2.5.1 Compactness of function spaces . . . 33
2.5.2 Main estimate on the empirical density uN . . . 35
2.5.3 Tightness of uNt , mNt . . . 41
2.6 Passage to the limit . . . 43
2.7 Uniqueness . . . 46
2.8 Simulations . . . 48
2.8.1 Degenerate aggregation . . . 48 ii
2.8.2 Moderate aggregation . . . 49
2.8.3 Aggregation in clusters . . . 51
2.8.4 Moderate aggregation in clusters . . . 51
3 A particle system approach to Vlasov-Fokker-Planck-Navier-Stokes Equations 53 3.1 Overview on the PDE system . . . 53
3.2 A microscopic approach to (V F P N S) . . . 55
3.2.1 A truncated version of the Particle system . . . 58
3.3 Main Assumptions and Basic Definitions . . . 60
3.4 Preliminary results. . . 65
3.5 Some new result about the (V F P N S) . . . 70
3.5.1 Maximum principle for weak solutions of the linear Vlasov-Fokker-Plank equation . . . 71
3.5.2 Uniqueness of (V F P N S) . . . 72
3.6 Scaling limit for the truncated system . . . 77
3.6.1 Tightness . . . 78
3.6.2 Convergence of (P SR− N SR) to (V F P N SR). . . . . 92
3.7 Cancelation of cut-off at the PDE level . . . 96
3.8 Scaling limit for the full system . . . 99
3.8.1 Convergence criterion . . . 99
3.8.2 Path by Path solutions for (P S − N S) . . . 101
3.8.3 Cancelation of the cut off at the particles level . . . . 106
4 A new averaging principle to investigate some aspects of tumor progression 114 4.1 Biological Origin of the Problem . . . 116
4.2 A new averaging Principle . . . 118
4.2.1 A White Noise approximation . . . 119
4.2.2 Main results . . . 121
4.3 ‘Go or Grow’: a new explanation for tumour progression . . . 131
4.3.1 A PDE model for “Go or Grow” (GoG) . . . 134
4.3.2 Main Result on the Deterministic model for GoG . . . 138
4.3.3 Intrinsic Heterogeneity as White Noise . . . 141
Introduction
The main theme of this work is given by the bio-physical phenomenon of aggregation: it has driven my research and it is the leitmotiv of the three chapters of this thesis. These phenomena have recently caught the attention of scientific community from different different point of views (biology, physics, mathematics..). The relevance of this topic is due to its generality, detectable on different scales and in a wide spectrum of contexts. Between some of the most studied phenomena in Biology, we mention animal swarming, flocks formation, embryo development, tissue homeostasis or tumor growth. Systems in which we are most interested are systems of self propelled particles. More specifically we refer to those composed by individuals whose motion is conditioned by other particles located in their neighborhood. Many questions can arise on this type of systems. Let us consider for instance a finite number of agents, acting in an environment and following the same rules. It is possible to notice the presence of a complex behavior as a collective phenomenon, appreciating when the whole is greater then the sum of the parts. A possible question to investigate, is the detection of such emergence behavior. A related problem which arises is the multiscale modeling, namely the linking between microscopic and macroscopic scale.
Due to the spatial structure of these systems, a natural tool are Partial Differential Equation. However in more applied contexts it is very convenient to simplify the model, to eliminate degrees of freedom and to work with Ordinary Differential Equations. Indeed in these contexts the need to explore the role of different parameters is very common, and it is easier to do that dealing with ODEs. The results presented in this thesis fall in the two last mentioned classes. In the first two Chapters we present two results of scaling limit for particle systems, linking the microscopic viewpoint with a system
of PDE. In the last Chapter, we investigate a simplified system based on ordinary differential equations, perturbed by noise in a very special way.
A first phenomenon that we aim to treat is the contact force between single agents. So in Chapter 2 , we develop a scaling limit result, based on our work [19] on cellular adhesion. In existing literature, see in particular [3], a non-local PDE model for cell-cell adhesion is presented, nowadays called the Amstrong-Painter-Sherratt model. For this PDE we propose a microscopic description based on a system of stochastic differential equations and we explain, how, thanks also to the non linearity of the system , one can use such model to describe some aggregative behavior.
In Chapter 3, we focus on a system of particles, interacting between themselves and interacting with an incompressible fluid. A discrete-continuous system, based on stochastic ordinary differential equations for the particles and a PDE for the fluid is presented. The PDE is the classical Navier-Stokes equations for an incompressible Newtonian fluid, with an additive term of interaction with the particles. In this framework we prove a scaling limit result for the PDE system called Vlasov-Fokker-Planck-Navier-Stokes. This chapter is based on our two works, [21] and [20].
In both microscopic descriptions each individual’s dynamic is given by a SDE. So the total system is described by a system of SDEs. In the chapter 2 we deal with first order SDE, while in Chapter 3 we deal with second order SDE. To introduce the basic difficulties related to the scaling limit of such systems, for simplicity we refer to a system of first order. In abstract terms the global system has the form,
dXti,N = µ(t, Xti,N, ρNt , uNt )dt + √
2D · dBi,Nt i = 1, . . . , N t ∈ [0, T ]. In this expression, Btit∈[0,T ] are independent d-dimensional Brownian Mo-tions, uNt is an external field described by a specific PDE, that influences and it is influenced by particles and ρNt is a quantity that represents particle den-sity. It can be the empirical measure StN :=PN
i=1δXti,N or a mollified version
of the empirical measure. The classical way to proceed can be summarized in three steps:
1. Tightness of (ρNt , uNt );
2. Uniqueness of the solution of the limit PDE; 3. Convergence to the solution of the PDE.
A fundamental passage precedes the three mentioned steps: the choice of topology of convergence. To approach this problem one can write down the
identity satisfied by the empirical measure,
dhStN, ϕi = hStN, ∆ϕidt + hStN, ∇ϕ · µ(t, Xti,N, ρNt , uNt )i + MtN,ϕ and then, looking at each term, one can figure out what is the right space to work in.
The purpose of the Chapter 4 is a bit different. There we aim to answer a specific biological question, related to tumor progression. This question allows us to give meaning to an ODE, that presents a non linearity of the white noise in the drift. The solution come from a new Averaging Principle. This Chapter is based on our work, [30].
1.1
Scaling limit for a non-local model to cell-cell
adhesion
Let us focus on the problem we are going to discuss in Chapter 2. In [3] the following transport diffusion PDE-ODE system to describe cellular adhesion, has been introduced:
∂u
∂t = ∆u − div(ub(u, m)) ∂m
∂t = −λum
ζ ζ ≥ 1, (AP S)
where u represents cellular density and m : [0, T ] × Rd→ R+, represents a
collection of extracellular molecules that provides nutrients to the embedded cells, called extracellular matrix, (ECM). The peculiarity of the PDE is given by the non local transport term b(u, m), where
b(u, m)(x) := Z
Rd
y − x
|y − x|g(|y − x|, u(y), m(y))dy,
with g : R+× R+× R+→ R, g = g(r, u, m), is differentiable, bounded with
bounded derivatives, and satisfies
|g(r, u, m)| + |∇g(r, u, m)| ≤ C · exp(−r). The particle system that we propose is the following
dXti,N = b(uNt , mNt )(Xti,N)dt + σ · dBti i=1,. . . ,N ∂mNt (x)
∂t = −λu
N
where uNt is a function describing particles density. The microscopic picture is given by a discrete-continuos description. It is discrete because particles are given by a finite number of equations, it is also continuous because also at the microscopic level, we idealize extracellular matrix as a continuous quantity (even if it is still interacting with a finite number of particles).
At the microscopic level one can appreciate much better the effect of the transport term. Interaction between cells is not described as interaction between individuals, but between individuals, Xti,N, and collectivity uN.
Heuristically speaking, cell Xi,N “looks” at point y, from that point it is subject to a force that depends on the density of particles through the function g(u(y)), rescaled by the distance |y − Xi,N|. Function b is obtained,
integrating on the whole space.
1.1.1 Main result of the Chapter
The interaction cell-density introduces a mathematical difficulty, that we now explain. Dealing with scaling limit of particle system, one usually looks at the empirical measure StN = N1 PN
i=1δXti,N, and try to control it in a suitable
way.
In our framework this type of control is not enough: the nature of the problem, impose an interaction particle-density. Being b a non linear function of density, density must be described by function and not by a measure. To deal with a functional representation for the density, inspired by Oelschläger works, see [36], we introduce a mollified empirical measure,
uNt (x) := WN ∗ SNt (x) = N X i=1 WN(x − Xi,N), where WN(x) := NβW (Nβ/dx),
for some β ∈ [0, 1]. Hence, uN has to be estimated. To choose the topology for the scaling limit, we look at the transport term in the identity for the empirical measure. Let us focus on a simpler situation, where b depends only on uN. We aim to prove that
lim
N →∞S N
s , ∇ϕs· b(uNs ) = hus, ∇ϕs· b(us), i
under the assumption that SsN → us weakly. A possibility is to prove that
have that uN → u in L2
loc. We gain the desired result as a consequence of a
stronger result, namely convergence in C [0, T ] ; Wlocα,2 Rd , for some 0 < α < 1.
The kernel smoothing parameter β plays an important role in the estimate proposed. One of our technical aims is to achieve the maximum possible generality on β. Following a semigroup approach we managed to obtain the desired convergence for β ∈ [0, 1). This generality on β, compared to the existing literature on Oelschlaeger approach, is uncommon. The price to avoid restrictions on β is to loose some regularity on the spatial derivative. Indeed the parameter α of the fractional Sobolev space, depends on β: the larger is β the smaller is α. See Lemma 2.5.1 for a detailed explanation.
In the previous argument we made a simplification: the omission of mNt . The introduction of an external field in some problems may cause considerable technical difficulties. It is not the case for us, for the following two reasons. First, thanks to the boundness of g, we are able to prove the tightness result for uN independently on mN. Then we simply use the fact that mN can be expressed as a continuous function of uN, so that it naturally inherits the tightness property proved on uN.
Theorem A (see Theorem 2.4.3). The pair uN, mN converges to (u, m) in C [0, T ] ; Wlocα,2 Rd × C[0, T ] ; Wlocα,2 Rd
for some 0 < α < 1 in probability.
At this point a natural question that emerges is whether it is correct to think of cellular adhesion as a non local phenomenon or if the interaction should be rescaled. We have spent long time on this issue but unfortunately, in this thesis, we are not be able to answer this very difficult question. Despite this doubt, in [3], [9] and [39] it has been verified through numerical simulations that the non local model, with a proper choice of the radius of interaction, is able to replicate adhesion phenomena. Moreover we show how, thanks to the introduction of function g and to the non-locality of the model, we are able to catch different phenomena, namely different type of aggregation behavior. See figure 1.1, for some examples with different choice of g.
(a) (b) T=50
Figure 1.1: Simulation of N = 100 particles with different choice of g. In (a) gR(r, u) := 1+uu · exp −rR2and R = 0.3. In (b) g(r, u) := u·log( r u) 1−u·log(r u) .
1.2
A particle system approach to
Vlasov-Fokker-Planck-Navier-Stokes Equations
In the next Chapter, we focus on a different system of SDEs and we obtain an analogous scaling limit type of result. Here the main object of investigation is the Vlasov-Fokker-Planck-Navier-Stokes equation:
∂tu = ∆u − u · ∇u − ∇π − R R2(u − v)F (x, v) dv div(u) = 0, ∂tF = σ 2
2 ∆vF − v · ∇xF − divv((u − v)F ) − divv(F (K ∗ ρ))
(V F P N S) where (t, x, v) ∈ [0, T ] × T2× R2. For technical reasons, we work in the two
dimensional case, d = 2, with the spatial components reduced to torus T2. The first equation for the function (u, π) is the classical Navier-Stokes equation for the incompressible Newtonian fluid, coupled with a density field F through the so called Stokes drag force. The second equation is the Vlasov-Fokker-Planck equation for the density of particles, influenced by the velocity field u by the corresponding reaction drag force.
1.2.1 Main result of the Chapter
Our purpose is to prove that this PDE system is the scaling limit for the par-ticle system presented in the next lines. We propose the following
continuous-discrete system, where particles are described by a finite number of second order SDE, while the fluid is described as a continuous quantity,
∂tuN = ∆uN − uN· ∇uN− ∇πN −N1 PNi=1(uNεN(X
i,N t ) − V i,N t )δ εN Xti,N div(uN) = 0, ( dXti,N = Vti,Ndt i = 1, . . . , N dVti,N = N1 PN i=1K(X i,N t − X j,N t ) dt + (uNεN(X i,N t ) − V i,N t ) dt + σdBit. (P S − N S) In the previous we can identify two type of interactions (and consequently two sources of technical difficulties): particle interaction with themselves and particle-fluid interaction.
To take in account aggregation phenomena, we consider a classical mean-field interaction with a proper choice on the Kernel K. To fix the ideas, we consider an attractive/repulsive kernel, as the Lennard-Jones potential. So K(r) = ∇U (r), where U (r) = εh rm r 12 − 2 rm r 6i .
Particle-fluid interaction is more intricate. The problem has been ap-proached in literature on different scales of complexity. Correct boundary conditions for the interaction between the particle and the fluid have been investigated, but with strong restriction on the fluid regime, see [5], [1], [2], [18], [17] and [16]. What we propose here is a phenomenological de-scription, but without any restrictions on the fluid. Here we consider point particles with with a soft radius of interaction εN, a sort of boundary layer.
Indeed each particle acts on the fluid through a smoothed delta Dirac, δεN
Xti,N(x) := θ
0,εN(x − Xi,N
t ) and the velocity difference is computed between
particle velocity Vti,N and a local average of the fluid, centered at the particle position, uNε N(X i,N t ) := u ∗ θ0,εN (X i,N t ). θ0,εN(x) is a sequence of mollifiers
on the thorus, rescaled by the factor εN := N−β with β ≤ 1/4 (the restriction
on β is due to technical reasons).
The assumption of this phenomenological description has a fundamental technical consequence. To get estimate on uN, estimates on a mollified version of empirical measure are required. We denote with FtN(x, v) :=
θεN ∗ SN
t (x, v), the mollified empirical measure, where θεN := ε −2 N θ(ε
−1 N (x, v))
sequence of mollifiers, rescaled by the factor εN.
To understand which topology we have to work with, we focus on the particles-fluid interaction term in the identity for the empirical measure StN := N1 PN i=1δ(Xti,N,V i,N t ) , hSN t , (uNεN − v)∇ϕi.
For this term, we expect that the following limit holds, lim
N →∞hS N
t , (uNεN− v) · ∇ϕi = hFt, (ut− v)∇ϕi.
Assuming that SsN → Fs only as a probability measure, the limit requires
that uNs → us uniformly.
By Sobolev embedding in d = 2, uniform estimates are given by estimates in W1+ε,2, in particular in W2,2. We work on the vorticity ωN, to avoid intricate computations on the derivative of uN. Energy estimates on the vorticity reveals many obstacles. The trickiest term to manage is
1 N N X i=1 (uNεN(Xti,N) − Vti,N)δεN Xi,Nt (x) L2 .
At first sight one could think that there is no possibility to control such term. In principle it could be reduced to the sum of the L2-norm of δεN
Xti,N
(x). However unlike the L1-norm, the L2-norm of the mollified Delta dirac is
not uniformly bounded with respect to N . So there is no possibility to proceed in this direction. To obtain L2-estimate we need to consider the sum in its totality and not to reduce its study to the study of its summands. Following this idea two hidden terms related to the zero and the first moment of particles are revealed:
≤ uN N(t, ·) ∞ Z R2 FtN(x, v) dv + Z R2 vFtN(x, v) dv . Still, these remarks are not enough to conclude tightness estimates.
So a new system, (P S − N S)R, is introduced. It is characterized by a cut-off on the velocity of the fluid, see χR(u) defined in Chapter 3. On this
new system we are able to use all the comments mentioned above. Through a combination of semigroup approach and energy estimates we get tightness estimates and then the convergence of the truncated system to a truncated version of the (V F P N S), (V F P N S)R. Our first work [21] is based on this
idea. In [21], a bounded variant of the Stokes Drag force is introduced. Thanks to this variation we are able to close tightness estimates and to prove a partial convergence result. The partiality was due to lack on the uniqueness of PDE system.
To get the scaling limit result on the original system, the idea is to use the truncated system as a bridge for the original system, see figure 1.2 for synthetic scheme of the proof. Obtained the convergence result for
(P S − N S)R (V F P N S)R
(P S − N S) (V F P N S)
(I)
(II) (III)
Figure 1.2: A synthetic scheme of the proof
the truncated system, we “remove" the truncation at the PDE level. In Proposition 3.7.1 uniform bound with respect to the cut off parameter R, are proved on uR i.e. the Navier Stokes component of (V F P N S)R,
uR ∞≤ C.
Through this estimate, we get that uniformly in R, a solution of the truncated system is also solution of the original (V F P N S). So a peculiar result is obtained: for some R, a truncated system (P S − N S)R converges to the
original (V F P N S).
Removal of the cut-off at the particle level has been a very difficult issue. Intuitively we expect the converging object, (P S − N S)R, inherit the boundness property that holds on the limit object. A first obstacle is the type of converge which we are dealing with: convergence in law. We overcame this difficulty through Skohorokod theorem: new random variables with the same law and converging almost surely, are obtained. We denote with uN,R the Navier Stokes component of the system (P S − N S)R. We can state that
there exists N (ω) > 0 such that for each N > N (ω) uN,R ∞≤ C.
Notice that this estimates holds randomly. The bound it is not uniform on N , because we cannot bound the other finitely many terms, since they are a random number of terms, potentially unbounded in cardinality. So we propose an approach that seems to be new, and it is strongly based on the path by path uniqueness. Despite all these obstacles, we prove the following result
Theorem B (see Theorem 3.8.3). The family of lawsQN N ∈N of the couple (uN, SN) N ∈N is tight on C([0, T ] × T2) × C([0, T ]; P1(T2× R2)). MoreoverQN
N ∈N converges weakly to δ(u,F ), where the couple (u, F ) is the
unique bounded weak solution of system of equation (V F P N S).
As a final remark, recall that our main interest is to understand aggrega-tion and our investigaaggrega-tion of Vlasov-Navier-Stokes system has the purpose of highlight the influence of a fluid on such phenomena. With respect to the original purpouse, the results of Chapter 3 are only preliminary, they just claim a compatibility between a microscopic and a macroscopic description. In future researches we hope to characterize specific properties arising from the interaction between particles and fluid.
1.3
A new averaging principle to investigate some
aspects of tumor progression
The purpose of Chapter 4 is different from the two previous ones. Until now, biology and physics have been a a general motivation. The objective was to support the validity of certain PDE systems by a microscopic modelling and a scaling limit result, but we did not aim to solve specific biological questions. The mathematical problem argued in Chapter 4 comes from the need to give answer to a precise biological question, related to tumor progression (and indirectly to cancerous cells aggregation). It is well known that tumor heterogeneity plays a fundamental role in tumor progression and tumor resistance. There are two major sources of heterogeneity: the first is due to phenotypic plasticity, where cells react differently in varying microenvironmental conditions and the second is an intrinsic cell variability attributed to for instance genetic variations. On the Gliobastoma Multiforme, a particular type of brain cancer, the phenotypic plasticity has been deeply investigated, but the role of intrinsic heterogeneity remains unclear, see [31]. We investigate the role of this type of heterogeneity to understand if it contributes positively or negatively to tumor progression.
In the attempt to answer specific biological questions, it is very common to reduce some degrees of freedom and then to use Ordinary Differential Equations (ODE) instead of PDE. This is exactly what is done in the Chapter 4.
After the reduction from PDEs to ODEs as described in Chapter 4, the question regarding the ODE is the following one. Assume the drift F of the ODE depends on a set of parameters k, that in order to simplify the exposition we assume one dimensional, as well as the ODE itself:
(dρ
dt = F (ρ, k) t ∈ (0, T ]
ρ(0) = ρ0.
Assume now that the parameter k is not constant but varies in time; consider the case when it varies very strongly and at very short time scale. When the dependence on the parameter is linear, such parameter can be approximated by the white noise and one can deal with a stochastic differential equation. But it is not very clear how to treat the non linear case: a priori, the equation
(dρ
dt = F (ρ, ξ) t ∈ (0, T ]
ρ(0) = ρ0,
is meaningless when ξ is a white noise, because we do not know how to give meaning to a non linear function of the white noise.
1.3.1 Main result of the Chapter
The purpose of the mathematical theory developed in this Chapter is to give meaning to the previous equation. The strategy we follow is to consider a sequence (ξN)N ∈N approximating the white noise, i.e. to study
(dρN
dt = F (ρN, ξN) t ∈ [0, T ]
ρN(0) = ρ0,
and to investigate whether the sequence ρN converges and which equation is satisfied by its limit points. Under suitable hypotheses on the drift F , we shall prove that in the limit equation the randomness is completely lost and we get a deterministic ODE.
Let us see some details. Assume our drift satisfies the following condition, lim
k→±∞F (x, k) = β±(x).
We expect that for large N , ξN takes large positive value (and so F (x, ξtN) ≈ F (x, β+) ) or large negative value (and so F (x, ξNt ) ≈ F (x, β−)). So we expect
be β+(ρ) or β+(ρ) with equal probability. In summary α(ρ) : [0, T ] × Ω → R is such that αt(ρ) = ( β+(ρ) p = 12 β−(ρ) p = 12.
Surprisingly we prove that randomness is lost, and in the limit equation what we obtain is the average between the two extremes β+(ρ) and β−(ρ),
see Proposition 4.2.1. (dρ dt = β+(ρ)+β−(ρ) 2 ρ(0) = ρ0
Theorem C (see Theorem 4.2.5). Let ξtN be a suitable approximation of the white noise.Then the trajectories ρNt converges to ρt, solution of the averaged equation, i.e.
lim
N →∞t∈[0,T ]sup |ρ N
t − ρt| = 0 a.s.
Obtained this result, we have a useful tool to answer to a specific bio-logical question. We focus on a model for the progression of Glioblastoma. Main peculiarity of this tumor progression is the dichotomic cell behavior. In practice, cells move or proliferate in an exclusive way, they do not do it simul-taneously. In this context two different global behavior have been observed: an aggregative behavior and a repulsive one. This difference depends mainly on how individual cellular choice depends on local density. When the attitude to proliferate increases with local density a global aggregative behavior is ob-served, otherwise the repulsive one is observed. This dependence is calibrated by parameter k. We would include in the model the intrinsic heterogeneity, namely the fact that in the population k is not homogeneous, and moreover subjected to huge fluctuation. Applying the theorem presented we were able to include also this aspect in the modelization, and we investigate its consequences.
1.4
Open Problems
The problems presented in this thesis have induced many other questions, which remain open. In the present subsection we present those that we consider more significant.
• The problem discussed in the Chapter 2, arises a deep question. How to describe local interaction at the macroscopic level? By local we mean
"contact" interactions, namely interaction with very close particles, nearest neighbourhood in a sense.
Let us consider a system of interacting particles, described by a stochas-tic differential equations of the following form.
dXti,N = −N−1/d
N
X
j=1,j6=i
∇UN1/d(Xti− Xtj)dt+dBit i = 1, . . . , N. Assuming that U is compactly supported, with such rescaling we have that each particle interacts only with particles in their neighborhood of radius N−1/d, namely only with particles in contact with it. The open question is the PDE description associated to such particle system. In the case when U is both attractive and repulsive in the sense of the above mentioned Lennard-Jones potentials, there is no answer to this question yet. As a introductory investigation of this PDE, we have done a few numerical simulations based on formula taken from the few works in literature on this topic. Under suitable assumption on U (see [46] in d = 1 , see [45] in any dimension) it is proved that the empirical measure StN := N1 PN
i=1δXi,Nt
converges weakly to the measure valued function u(t, dx), that satisfies the following PDE:
∂tρ = ∆ (P (ρ)) t ≥ 0.
The function P (ρ) is not explicitly known, in a simple analytic form. P (ρ) is given by a very complicated formula, called the Virial Formula. From our numerical simulations we have obtained, in a purely repulsive case relatively stable and clear results, as illustrated in Figure 1.3. For instance when U (x) = V (|x|) = |x|12, then P (ρ) = ρ + αρ2.4, with
α = 3.5. These numerical results may have a relevant consequence. In [37] similar situation has been investigated by Oeschlager. One of the novelty introduced by Oeschlager consists in the definition of interaction neighbourhood. The notion of intermediate regime is introduced: cells interact not locally, i.e. in a neighbourhood of radius N−1/d, but in a larger neighborhood of radius N−β/d, with β < 1. In a purely repulsive case, the intermediate regime lead to the Porous media equation
∂ρ = ∆(ρ + ρ2),
independently on the drift V . Through numerical simulations we understand that the power 2 could not be the correct power to the describe local attraction. Further developments are need to understand the nature of such PDE.
Figure 1.3: Plot of P (ρ) when V (|x|) = |x|12. In dots, the numerical result
obtained for some values of ρ, in red the function ρ + 3, 5ρ2.4
• In the attempt of deriving different models for describing aggregation, a different paradigm is to assume that the diffusion is density-dependent. Consider
dXti= σ(ρNt ) · dBit i = 1, . . . , N
where σ : R → R is a monotone decreasing function and ρN is a function describing particle density. We expect that when the local density is high, cell motility is reduced. Again, the question on this particle system is about the scaling limit. We conjecture that the corresponding PDE limit is
∂ρ = 1
2∆ σ(ρ)
2ρ .
We did not find this case rigorously treated in the literature and a first attempt to generalize previous results meets the following main difficulty. By looking at the identity for the empirical measure, we identify the term from which one can understand in which topology to work in. We expect that
lim
N →∞h∆ϕ · σ 2(ρN
t ), StNi = h∆ϕ · σ2(ρt), ρti.
Assuming that SsN → ρs as a probability measure, the limit holds if
ρNt converges uniformly to ρt. Investigation of the scaling limit for this system seems a difficult question.
• The last problem, again related to the investigation of different model for aggregation, is inspired by the work presented in Chapter 4. Let
us consider a particle system composed by two different populations. Motile cells (M ) move following a brownian motion and static cells (S) simply do not move:
(
dXta,i= dBti a = M
dXta,i= 0 a = S i = 1, . . . , N.
Assume that a switch between the two species may occur. In particular, motile cells becomes static when local density is high and viceversa static cells became motile when local density is low. We denote with rM →S(ρ) the switch rate from motile to static type and with 1 − rS→M(ρ) the
switch rate from static to motile type, where ρ is a just a function representing local density. To replicate an aggregative behavior, we assume that rM →S(ρ) is monotone increasing function, while rS→M(ρ)
is monotone decreasing. Recalling a work by Oelschleger [36], it is possible to derive a system of PDEs for such system:
(
∂tρM = ∆ρM − rM →S(ρ)ρM + rS→M(ρ)(ρ)ρS
∂tρS= −rS→M(ρ)ρS+ rM →S(ρ)ρM.
Our conjecture on this system is that the system of two species is strictly correlated to the system presented in the previous point (the one with the diffusion density-dependent). Assuming that the detailed balance condition between the two species holds,
ρSrS→M(ρ) = ρMrM →S(ρ),
then it is possible to derive a closed equation for the total population ρ := ρM+ ρS,
∂tρ = ∆(rS→M(ρ))ρ).
In this way we found out the same equation on which we focus before, namely the equation derived from particles, driven by Brownian motion, modulated by the function prS→M(ρ). We would like to prove the
detailed balance condition, at least in a Large time regime. In this way we could assert that for large N and asymptotically in time, the two systems describe the same phenomenon.
Scaling limit for a non-local
model to cell-cell adhesion
This chapter is devoted to the derivation of the scaling limit for a non local model for cell-cell adhesion. The model, proposed by Amstrong, Painter and Sherrat in [3], has received attention into the biomathematical community. The biological phenomena captured by the authors is cell-cell adhesion, namely cells adhere to each other through cell surface proteins, called cell-adhesion molecules (CAMs). Such phenomenon is of great interest, because it rises up in different areas of biology as early embryo development, tissue homeostasis and tumour growth. In [3], the authors prove, also through simulations, that the model is capable of predicting an adhesive behaviour. Moreover they propose a generalization at the level of different populations: they demonstrate that their model is able to replicate different types of cell sorting behavior observed experimentally, see figure 2.1. Adhesive behaviour and cell sorting have been caught in the past also by other models, for instance through discrete systems. In [3], for the first time these phenomena are caught through a continuous model (PDE). The theory developed in this Chapter is not explicitly devoted to cell sorting but it can be extended to such case. In this chapter we focus on the simplest case proposed by [3]: the case of one population.
2.1
A PDE model for cell-cell adhesion
Before going into all technicalities, let us discuss the bio-physical motivations behind the model, see [3] for a more detailed derivation. Cells movement is driven by various factors: cells move in response to chemical stimuli
Figure 2.1: Illustrations of cell adhesion and sorting that show aggregations in experiments.
taxis), in response to fixed environmental factors such as the extracellular matrix (haptotaxis) or in response to adhesive forces between the cells. The latter is the one that we are going to describe. For simplicity, we derive the model in the monodimensional case, d = 1, neglecting cell birth and cell death. Let us assume that cells act in a conservative system, then mass conservation implies
∂u ∂t = −
∂J ∂x,
where u is population density and J is the flux of cell. Let us suppose that cells are subjected to random motion as well as to the movement due to adhesive forces. Then the total flux is given by:
J = JD+ JA
where JD is the flux due to diffusion and JAis the flux due to adhesion. In [3] a Fickian Diffusion is assumed, namely:
JD = −
∂u ∂x.
This assumption could appear a bit rough. It does not take into account the fact that cells motion is reduced in regions of high cellular density. In order to include this fact, a more accurate choice would be given by a non linear diffusive flux. However, we will not generalize this aspect of [3] here, also because we shall make extensive use of the linear diffusion operator in the Mathematical analysis. Movement due to adhesion occurs as a result of the forces produced when adhesive bonds between cells are formed. Then it is
X g(u(y)) R −R
x y
Figure 2.2: A schematic illustration of cell movement under an attractive force due to cell adhesion molecules
natural to expect that the adhesive flux is proportional to the density of the cells and the forces between them and inversely proportional to cell size:
JA= c
F u R
Here c is a constant of proportionality related to viscosity, F is the total force acting on the cells and R is the sensing radius of the cells. The sensing radius R represents the range over which cells detect their surrounding cells. Considering cellular extension, protrusions such as filopodia, R is larger then the radius of cell. However, the notion of sensing radius is a very delicate issue, we will give our personal view on the topic in subsection 2.3.
The intensity of the force at the point x, F (x), depends by cell density in its surrounding. Heuristically, let us imagine to observe at the cellular level from position x. Then we look at the position y, we evaluate density in the point y, u(y), and then the intensity of the force as a function of density, g(u(y)). We are pushed in the direction of y − x, with a push that depends on the distance, with intensity g(u(y)) (see figure 2.2). In summary the force that act from y to x is of the following form
f (x, y) = αg(u(y))θ(x − y).
Here, g represents the nature of the force, while θ describes how the magnitude of the forces changes according to distance. Finally α is just a parameter that calibrates the intensity of the force. In next pages we discuss, which assumptions occur on g. To compute the total force F , we just integrate on space: In conclusion the adhesive flux is given by the non-local formula:
JA(x) =
c Ru(x)
Z R
−R
αg(u(y))θ(x − y)dx0 = c(g(u) ∗ θ)(x).
After a normalization step, the final equation for u in dimension d = 1 is: ∂u ∂t = ∂2u ∂x2 − ∂ ∂x(u(g(u) ∗ θ)).
Let us discuss some reasonable assumption on g and θ. A natural and basic assumption is that g increase with local density, for instance g(u) = u. This basic case has been deeply investigated already investigated in literature, we refer to the Mean Field case. So we do not focus on that. The cases in which we are interested are those where g depends both on density and distance not linearly, see for instance Subsection 2.8.2. For technical reasons (see tightness estimates in Lemma 2.5.1) we assume that g is bounded.
Regarding θ, we assume that θ(y) has the same direction of y and its intensity decrease with the distance. One could choose:
θ(x − y) = y − x
|y − x|1{|x−y|<R},
or one can also work also with a function not compactly supported: θ(x − y) = y − x
|y − x|.
In summary the PDE, in a generic dimension d, is the following: ∂u
∂t = ∆u − div(ub(u)). where b(u) = (g(u) ∗ θ).
2.2
A microscopic Approach
In this chapter we treat an enriched version of the PDE presented above. In particular we consider the case where the Extracellular Matrix (ECM) is included in the model, as a contribute in the transport term,
∂ut
∂t = ∆ut− div(utb(ut, mt)) ∂mt ∂t = −λutm ζ t ζ ≥ 1 (AP S) where b(u, m)(x) := Z Rd y − x
|y − x|g(|y − x|, u(y), m(y))dy.
The microscopic description provided is continuous-discrete. It is discrete because at the microscopic level we will associate to u a system of particles whose dynamic is given by a SDE, and it is continuous because the ECM remains a continuous quantity at the microscopic level. Let us focus on
particles dynamic. The PDE for u is a transport-diffusion equation, so we expect that each particle is driven by a Brownian Motion and its drift is the corresponding transport term b(u, m). Here the first problem arises: b is defined on a functional space, it is not defined on a space of measures. So a function to describe cellular density is needed. To do that, we follow Oelschlager’s idea: we introduce the empirical measure mollified,
uNt (x) := WN ∗ SNt (x) = N
X
i=1
WN(x − Xi,N).
where WN(x) := NβW (Nβ/dx) for some β ∈ [0, 1). We use this definition
also to define the microscopic version of the ECM.
dXti,N = Z Rd y − Xti,N |y − Xti,N|g(|y−X i,N t |, u N(y), mN(y))dydt+σ·dBi t i = 1, . . . , N ∂mNt (x) ∂t = −λu N t (x)(mNt (x))ζ on [0, T ] × Rd
One of the main feature of the non local interaction here described is that it is not a pairwise interaction, particle-particle interaction, but each particle interact with the density. Heuristically speaking, cell Xi,N “look” at the point y, from that point it is subject to a force that depends on density g(|y − Xi,N|, uN(y), mN(y)), rescaled by the distance |y − Xi,N|. Then
summing up on the whole space, function b(Xi,N) is obtained. This peculiar type of interaction make the model applicable in different biological fields. Indeed there are situations in reality where it is more natural to assume that aggregation is due to the interaction between each single individual and the density field of all other particles. For instance, let us consider the motion of animals in a swarm or a flock; presumably each animal moves driven by a general overview of the others, not computing several pairwise interactions. Moreover using different functions g we may describe different kinds of attraction; presumably each animal moves driven by a general overview of the others, not computing several pairwise interactions. A wide discussion is presented in the last section of the chapter, see section 2.8.
2.3
Cell-Cell Adhesion: local or non local phenomenon?
Non local interaction is one of the features of the model presented. In our view this seems a very delicate point: is the Biology of adhesion a non localphenomenon? Some literatures indicates the presence of non local features. Cells have developed many mechanisms of translocation, with at least four different kinds of membrane protrusions: lamellipodia, filopodia, blebs and invadopodia [41]. In simpler terms, cells develop a sort of long arm to contact cells (or extracellular matrix) in their surrounding. The first question from the Mathematical viewpoint is: are these arms so big to induce us to consider the phenomenon as non local? In the PDE presented above the sensing radius, R, is a macroscopic variable. In the limit, in the particle system, infinitely many particles interact with each one. Should it be more realistic to model the interaction radius by a microscopic variable, properly rescaled in the scaling limit? If we take in account protrusions, and we do not rescale R with N , increasing the number of cell, each cell will interact with an increasing number of cells and for N → ∞ it will interact with infinite cells. This last assertion seems unrealistic: then we could assume ro rescale with N the sensing radius, R = RN → 0 for N → ∞. In this case θ would become a function of N , θN.
θN(x − y) =
x − y
|y − x|1|x−y|<RN. Then the drift would become
G(uN)(Xti,N) = Z
Rd
y − Xti,N
|y − Xti,N|I|x−y|<RNg(u
N t (y))dy = Z |x−y|<RN y − Xti,N |y − Xti,N|g(u N t (y))dy
Being g(uNt ) bounded, we get that: lim N →∞G(u N t )(X i,N t ) = 0
This would lead to the Heat Equation, instead of the PDE, proposed by Armstrong Painter and Sherratt.
Despite these doubts, very realistic simulations are presented in literature, see for instance [3], [39] and [9]. Indeed, choosing very small R the model is able to predict the formation of compact bodies. Moreover, as discussed in the Introduction, the problem of local interaction seems extremely difficult. One could think to work with a system like
dXti,N = N X j=1 Xtj,N− Xti,N |Xtj,N− Xti,N|θN(|X j,N t − X i,N t |)dt + √ 2 · dBti i=1,. . . ,N
but for this type of system, scaling limit results are fragmentary. There are some works in this direction, see [45], [46], but still a PDE limit is still not known. Being literature on this topic fragmentary, the proposal of Armstrong Painter Sherrat appears a good compromise.
2.4
Main assumptions and Basic results
2.4.1 Basic results on particle system
For every positive integer N , we consider a particle system described by the following stochastic differential equations:
dXti,N = b(uNt , mtN)(Xti,N)dt + σ · dBti i=1,. . . ,N (2.1) ∂mNt (x)
∂t = −λu
N
t (x)(mNt (x))ζ on [0, T ] × Rd (2.2)
with initial conditions X0i,N = X0i, i = 1, ..., N . X0i, i ∈ N, is a sequence of F0-measurable independent random variables with values in Rd, identically
distributed with density u0,
X0i ∼ u0.
where uNt is a function describing cellular density, that will be described in the next lines. Regarding initial condition for (2.2) mN0 (x) = m0(x) for
some integer ζ ≥ 1, where m0 : Rd → R is a measurable function with 0 ≤ m0 ≤ M . Bit, i ∈ N, is a sequence of independent Brownian motions on
a filtered probability space (Ω, F , Ft, P ); σ ∈ R is the diffusion coefficient.
Let us finally discuss the concept of density uNt (x). Given the particles Xti,N, one first associates to them the classical concept of empirical measure:
StN(dx) := 1 N N X i=1 δXi,N t (dx) .
Its direct use, however, in the previous modelling would oblige us to choose functions g depending on measures, instead of functions, which are less easy to formulate in examples. And, more importantly, we could not speak of uNt (y), the density at position y. In numerics it is common to overcome this difficulty by the so called kernel smoothing, which consists in mollifying the measure by convolution with a smooth kernel. We adopt this procedure. We choose a smooth (C2(Rd)), compactly supported, probability density W (the kernel) and rescale it with N in a suitable way. A general form of rescaling is
for some β ∈ [0, 1), as suggested by K. Oelschlager [36]. The density uNt (x) is thus given by uNt (x) := WN∗ StN (x) = N X i=1 WN(x − Xi,N).
for some β ∈ [0, 1). The functional
b : L2(Rd) × L2(Rd) → L∞(Rd) is given by b(u, m)(x) := Z Rd y − x
|y − x|g(|y − x|, u(y), m(y))dy
where g : R+× R+× R+→ R, g = g(r, u, m), is differentiable, bounded with
bounded derivatives, and satisfies
|g(r, u, m)| + |∇g(r, u, m)| ≤ C · exp(−r)
for some constant C > 0 (where ∇g = (∂rg, ∂ug, ∂mg) denotes the gradient
in all variables). It follows that, for every pair of measurable functions u(x), m(x), the aggregation force is bounded:
|b(u, m)(x)| ≤ Z
Rd
|g(|y − x|, u(y), m(y))| dy ≤ C Z Rd e−|x−y|dy := C0 < ∞. (2.3) We also have |b(u, m)(x)−b(u0, m0)(x)| ≤ C· Z Rd
e−|x−y| u (y) − u0(y) + m (y) − m0(y) dy (2.4) and with more tedious computations
|∇x· b(u, m)(x)| ≤ Z Rd ∇x· y − x
|y − x|g(|y − x|, u(y), m(y)) dy ≤ ≤ Z Rd
∇x(g(|y − x|, u(y), m(y))) ·
y − x |y − x|dy ≤ Z Rd
Under these assumptions, existence and uniqueness of a solution, for finite N , of system (2.1)-(2.2) can be proved by classical methods. Let us explain some details. Let us denote by C L2, C+ L2, C0,M L2 the spaces
C L2 := C([0, T ]; L2(Rd))
C+ L2 := u ∈ C L2 ; ut≥ 0 for all t ∈ [0, T ]
C0,M L2 = m ∈ C L2 ; 0 ≤ mt≤ M for all t ∈ [0, T ] .
We say that a random field mNt (x), t ∈ [0, T ], x ∈ Rddefined on (Ω, F , Ft, P ), is adapted of class C0,M L2 if P -a.s. the functions (t, x) 7→ mN
t (x) belong
to C0,M L2 and for every t ∈ [0, T ] the function (x, ω) 7→ mNt (x, ω) is
B Rd × F
t-measurable. Let us recall the difinition of strong solution
Definition 2.4.1. X1,N, ..., XN,N, mN is a strong solution of system (2.1)-(2.2) if Xt1,N, ..., XtN,N are continuous adapted processes on a fixed probability space (Ω, F , Ft, P ), mNt (x) is adapted of class C0,M L2.Ando moreover
iden-tities (2.1)-(2.2) hold, with the equations understood integrated in time. We say that pathwise uniqueness hold if two such solutions are indistinguishable processes.
Proposition 2.4.1. Given any positive integer N and any function m0 ∈
L2(Rd) such that 0 ≤ m0 ≤ M , there exists a strong solution of system
(2.1)-(2.2) and pathwise uniqueness hold.
Proof. The proof is classical, we explain only the idea. Given an integer ζ ≥ 1, uN and a.e. x ∈ Rd, the solution of equation (2.2) is global, unique and explicit: mNt (x) = Fζ m0(x), Z t 0 uNs (x)ds Fζ(a, b) = a · eFζ(a, b) e Fζ(a, b) = exp (−λb) if ζ = 1 1 [aζ−1(ζ−1)λb+1]ζ−11 if ζ ≥ 2.
The function eFζ : [0, M ] × [0, ∞) → R is bounded and the function Fζ :
[0, M ] × [0, ∞) → R is Lipschitz continuous, with at most linear growth in a, uniformly in b. Then one may consider the system of integral equations Xti,N = X0i+ Z t 0 b(uNs , Fζ m0(·), Z s 0 uNr (·)dr )(Xsi,N)ds+√2Bti i=1,. . . ,N (2.6)
as a closed system, with only the variables Xt1,N, ..., XtN,N. It is a path-dependent equation: the past appears in the drift; but this does not change the way contraction principle applies. One can check that strong exis-tence and pathwise uniqueness for the original system for the variables
X1,N, ..., XN,N, mN is equivalent to strong existence and pathwise unique-ness for this reduced path-dependent system in the variables X1,N, ..., XN,N only; property mN ∈ C0,M L2 is deduced from the explicit formula. Let us
say how to prove existence and uniqueness for (2.6). Thanks to the property (2.5) the drift of equation (2.6) is globally Lipschitz continuous. We define
the family of maps Ji as
Ji : E → R Ji(Y ) := X0i+ Z t 0 b(uNs , Fζ m0(·), Z s 0 uNr (·)dr )(Y )ds +√2Bti i=1,. . . ,N where E = L2F(Ω × [0, T0]), with T0 < T . Then with classical computation we get that Ji is a contraction on the space E:
Ji(Y ) − Ji(Y0) E ≤ CT 0 Y − Y0 E
choosing CT0 < 1. Hence local existence and uniqueness of strong solutions is proved. Iterating this argument one can get the global existence result, because the amplitude of the interval of iteration depends only on CT0, namely it is fixed for each iteration.
Remark 2.4.1. Existence and uniqueness of solution of the system (2.1)-(2.2) could be obtained following another approach. With less effort could be possible to obtain just weak existence and uniqueness in law for the system (2.1)-(2.2): the method of creating weak solution to SDEs is trasformation of drift via Girsanov theorem, see [34]. Thanks to the bound on b, see condition (2.3), the Novikov Condition is verified:
E " exp 1 2 Z T 0 b(uNs , Fζ m0(·), Z s 0 uNr (·)dr )(Xsi,N) 2! ds # ≤ ∞.
Then Proposition 5.3.6 and Proposition 5.3.10 of [34] can be used to obtain respectively existence and uniqueness of the system. Then Xti,N is solution of (2.1). Thus also mNt exists, is unique and explicit. This kind of existence would be enough for our purpose, but we still to decide to emphasize in Proposition 2.4.1 that a stronger result is achievable.
(H1) g : R+× R+× R+→ R, g = g(r, u, m), is differentiable, bounded with
bounded derivatives, and satisfies
|g(r, u, m)| + |∇g(r, u, m)| ≤ C · exp(−r); (H2) X0i,N = Xi
0, i = 1, ..., N . X0i, i ∈ N, is a sequence of F0-measurable
independent random variables with values in Rd, iid such that: X0i ∼ u0,
with u0 ∈ C2(Rd);
(H3) mN0 (x) = m0(x) for some integer ζ ≥ 1, where m0 : Rd → R is a
measurable function with 0 ≤ m0≤ M ;
(H4) Rescaling on WN: N−β with β ∈ [0, 1).
2.4.2 Basic results on the PDE system
After the indentity of Lemma 2.4.3 below for the empirical measure is proved, it is natural to conjecture that the limit of the pair uNt (x) , mNt (x) solves the system (AP S) with initial condition (u0, m0), where u0 is the density
of the r.v.’s X0i and m0 is the limit of mN0 . We interpret the first equation of this system in the so called mild form and the second one in integral form. Concerning the initial conditions, we make a choice of simplicity. We assume that u0 : Rd → R (the initial distribution of individuals) is a probability density of class C1 with compact support, see Lemma 2.4.6. About m0: Rd→ R, we assume it is of class L2(Rd) and 0 ≤ m0 ≤ M . We
have avoided technical assumptions with fractional Sobolev norms that would be of poor interpretation from the biological viewpoint.
Definition 2.4.2. By mild solution of system (AP S) we mean a pair (u, m) belonging to C+ L2 × C0,M L2 such that
ut(x) = et∆u0+ Z t 0 ∇ · e(t−s)∆(u sb(us, ms))ds mt(x) = m0(x) − Z t 0 λus(x)mζs(x)ds.
Where etA denote the heat semigroup. Notice that the L2(Rd)-norm of usb(us, ms) is bounded, since b is bounded and u ∈ C L2. Hence
∇ · e(t−s)∆(u
sb(us, ms)) is integrable by property (2.5.4) below.
It will be useful to know the following result on the improvement of regularity.
Lemma 2.4.1. If u ∈ Lp 0, T ; L2(Rd) for some p > 2 and satisfies ut(x) = etAu0+
Z t
0
∇ · e(t−s)A(usbs)ds
for some bounded measurable function b, then u ∈ C([0, T ], L2(Rd)). Proof. The product ub is in Lp 0, T ; L2(Rd). Using the bound ||∇ · e(t−s)A||L2→L2 = ||∇ · (I − A)−1/2(I − A)1/2e(t−s)A||L2→L2 ≤
C (t − s)1/2 we deduce that t 7→R0t∇ · e(t−s)A(u
sbs)ds is of class C([0, T ], L2(Rd)); the
same is true for t 7→ etAu0 because u ∈ L2(Rd) as a byproduct of our
assumptions. Hence u ∈ C([0, T ], L2(Rd)).
This regularity result will be very useful in next steps. The macroscopic limit presented in next sections we will give us, for free, the existence of a couple of (uN, mN) for the system (AP S) in the space C([0, T ], L2loc(Rd)). To prove that this solution is also a mild solution for the system (AP S), we will need to improve the regularity using this Lemma. In next sections for simplicity we will denote
C L2loc := C([0, T ], L2loc(Rd))
Moreover we recall that the topology on L2loc(Rd) is given by the metric dL2 loc(f, g) = ∞ X n=1 2−nkf − gkL2(B(0,n))∧ 1 .
2.4.3 Some useful properties on the Fractional Sobolev space
We denote with Wα,2(Rd) the fractional sobolev space, which is a Banach space for the norm
||f ||Wα,2(Rd)||f ||L2(Rd)+ [f ]α,2,Rd where [f ]α,2,Rd = Z Rd Z Rd |f (x) − f (y)|2 |x − y|2α+d dxdy. Or equivalently uN0 Wα,2 = uN0 L2( Rd)+ (−∆)αuN0 L2( Rd).
Lemma 2.4.2. Let s ∈ (0, 1). Then there exists a constant C (s, d) such that (−∆)sf (x) = C (s, d) Z Rd f (x + y) + f (x − y) − 2f (x) |y|d+2s dy, x ∈ R d
for every compact support twice differentiable function f .
Notice that boundedness of f guarantees integrability at infinity, while twice differentiability implies that the numerator is, for small |y|, infinitesimal of order two, which compensates the singularity of the denominator. Another very useful property on the fractional laplacian is that it is a local operator, namely it preserves compact support of functions.
Corollary 2.4.1. Assume f (x) = 0 for |x| > R. Then also (−∆)sf (x) = 0 for |x| > R.
Proof. If f (x) = 0 for |x| > R and we take a point x such that |x| > R, then for sufficiently small values of |y| the numerator is zero, hence the singularity at y = 0 does not exist and we can split the integral in three converging terms. In this case, the term f (x) in the integral of the previous formula is zero; the term f (x + y) is (possibly) not zero only for values y of the form −x + z with |z| ≤ R; the term f (x − y) is (possibly) not zero only for values y of the form x − z with |z| ≤ R. Therefore
(−∆)sf (x) = C (s, d) Z Rd f (x + y) |y|d+2s dy + C (s, d) Z Rd f (x − y) |y|d+2s dy = C (s, d) Z |z|≤R f (z) |−x + z|d+2sdz − C (s, d) Z |z|≤R f (z) |x − z|d+2sdz where we have used the change of variable y = −x + z in the first integral and y = x − z in the second one. We deduce (−∆)sf (x) = 0.
Let us recall some well known properties of analytical semigroups . The family of operators etAf (x) := Z Rd 1 (2πσ2t)d/2e −|x−y|2 2σ2t f (y)dy
for t ≥ 0, defines an analytic semigroup (the heat semigroup) on the space Wα,2(Rd), for every α ≥ 0. The infinitesimal generator in L2(Rd) is the operator A : D(A) ⊂ L2(Rd) −→ L2(Rd), D(A) = W2,2(Rd), given by
Af = σ22∆f . It is possible to define fractional power of the operator (I − A)δ for δ ∈ R and a well known fact is the equivalence of norms:
||(I − A)δ/2f ||L2 ∼ ||f ||Wδ,2 (2.7)
Another property, often used in the sequel, is that for every δ, T > 0 there is a constant Cδ,T such that for t ∈ (0, T ]
||(I − A)δetA||L2→L2 ≤
Cδ,T
tδ . (2.8)
Finally, we remark that the operator ∇(I − A)−1/2 is bounded in L2 ||∇(I − A)−1/2||L2→L2 ≤ C (2.9)
where, here and below, we continue to write simply L2also when the functions are vector valued, as in the case of ∇(I − A)−1/2f .
2.4.4 Preliminaries results for the empirical measure and
Main Result
Lemma 2.4.3. For every ϕ ∈ C2([0, T ] × Rd), StN satisfies the following identity: SN t , ϕt − S0N, ϕ0 = Z t 0 SsN,∂ϕs ∂s ds +σ 2 2 Z t 0 SN s , ∆ϕs ds+ + Z t 0 SN s , ∇ϕs· b(uNs , mNs ) ds + M N,ϕ t where MtN,ϕ= σ N N X i=1 Z t 0 ∇ϕ(Xsi,N) · dBsi.
In particular, choosing ϕ (·) = ϕx(·) = WN(· − x), for x ∈ Rd, we get
uNt (x)−uN0 (x) = σ 2 2 Z t 0 ∆uNs (x)ds+ Z t 0 div(WN∗ b(uNs , mNs )SsN)(x)ds+MtN(x) where MtN(x) = σ N N X i=1 Z t 0 ∇WN(x − Xsi,N) · dBsi. Proof. The proof follows by Itô formula and Gauss Green formula.
Concerning the family of mollifiers, we have the following useful properties, whose proof is an elementary computation, see for instance [flandoli2016uniform]. Lemma 2.4.4. Recall that WN(x) = NβW (Nβ/dx). Then
||WN||2
L2 ≤ CNβ
||WN||2Wγ,2 ≤ CNγ ∗
with γ∗ = βd(2γ + d).
We shall use also the following tightness result.
Lemma 2.4.5. Let X1 and X2 be two metric spaces with their Borel σ-fields
B1, B1 and let ϕ : X1 → X2 be a continuous function. Let G1 be a family
of probability measures on (X1, B1). Denote by G2 the family of probability
measures on (X2, B2) obtained as image laws of the measures in G1 under the
map ϕ. If G1 is tight, then G2 is tight.
Proof. Given > 0, let K1⊂ X1 be a compact set such that µ (K1) > 1 −
for every µ ∈ G1. Set K2= ϕ (K1); it is a compact set of X2 and for every
ν ∈ G2, called µ a measure in G1 such that ν is the image of µ under ϕ, we
have
ν (K2) = µ (K1) > 1 − . This proves tightness of G2.
Regarding the initial condition, we state a result, that will be usefull in the proof of tightness
Lemma 2.4.6. Assume that X0i , i = 1, ..., N, are independent identically distributed r.v with common probability density u0 ∈ C2(Rd), then on uN0 ,
defined as uN0 (x) = (WN ∗ u0)(x), we get the following uniform bounds for p > 1: E h uN0 p Wα,2 i ≤ Cp,α
where C is a constant depending on p and α.
Proof. By the definition of the norm in the fractional Sobolev space, we need to estimate uniformly in N: E h uN0 p Wα,2 i = E h uN0 p L2( Rd) i + E h (−∆)αuN0 p L2( Rd) i . (2.10)
We recall that u0is compactly supported and moreover fractional laplacian
is a local operatore, namely it preserves compactness properties of functions. Then, for p ≥ 2, uN0 p L2( Rd) ≤ Z B1 uN0 (x) p dx (−∆)αuN0 p L2( Rd) ≤ Z B1 (−∆)αuN0 (x) p dx where B1, B2 are respectively compact supports of uN0 and (−∆)
α uN0 Assum-ing that Yi= Yi(x) = WN x − X0i , e Yi = eYi(x) = (−∆)αWN x − X0i ,
we can write the estimates for (2.4.3) in the following terms: ≤ E " Z B1 1 N N X i=1 Yi(x) p dx # + E " Z B2 1 N N X i=1 e Yi(x) p dx #
Then we need to estimate
E " 1 N N X i=1 Yi(x) p# + E " 1 N N X i=1 e Yi(x) p# . Being Yi ≥ 0 on the first summand we have:
E " 1 N N X i=1 Yi p# = Z ∞ 0 P 1 N N X i=1 Yi !p > t ! dt = Z ∞ 0 P 1 N N X i=1 Yi> t1/p ! dt = Z ∞ 0 P exp 1 N N X i=1 Yi ! > expt1/p ! dt ≤ Z ∞ 0 exp−t1/pEheN1 PN i=1Yi i dt = eN log E eYN Z ∞ 0 exp−t1/pdt
where Y has the same law of Yi. Notice that the equality E h eN1 PN i=1Yi i = eN log E eNY
follows easily from the fact that Yi are iid. Because also ˜Yi ≥ 0, the same result holds for the second term. Then
E " 1 N N X i=1 Yi(x) p# + E " 1 N N X i=1 e Yi(x) p# ≤ eN log E eNY + eN log E e ˜ Y N ! Z ∞ 0 exp−t1/pdt
Let us estimate the first term (the same result will hold for the second term). We recall some basics inequalities log (1 + x) ≤ x, ex− 1 ≤ xex per
x ≥ 0. Then log EheY (x)N i = log1 + EheY (x)N − 1 i ≤ EheY (x)N − 1 i ≤ E Y (x) N e Y (x) N We have to estimate: E h Y (x) eY (x)N i and E ˜ Y (x) e ˜ Y (x) N
Recalling the definition of Yi, Y (x) N = N −1W N(x − X0) = Nβ−1W Nβ/d(x − X0) ≤ C.
being W bounded, we get that Y (x)N is bounded. Now we just need to estimate E [Y (x)].
E [Y (x)] = WN ∗ u0 (x)
the last term is bounded because u0 is it. Let us analyze the second term, which is a bit more delicate. By the definition of ˜Yi,
e Y (x)
N = N
−1(−∆)αW
N(x − X0) ≤ CN−1+βN2αβ/d.
Choosing an α small enough the term Y (x)e
to prove a uniform estimate on Eh ˜Y (x) i . E h e Y (x)i= E [(−∆)αWN(x − X0)] = Z [(−∆)αWN] (x − x0) u0(x0) dx0 x0 0=x−x0 = − Z (−∆)αWN x00 u0 x − x00 dx 0 0 = − h(−∆)αWN, u0(x − ·)iL2 = − hWN, (−∆)αu0(x − ·)iL2 = − Z WN x00 [(−∆)αu0] x − x00 dx00 = [WN ∗ ((−∆)αu0)] (x) .
Being (−∆)αu0 compactly supported and continuous also WN ∗ ((−∆)αu0)
is uniformly bounded. In summary, E h uN0 p Wα,2 i ≤ CB1,B2,u0,α,p.
Now, we have all the main ingredients to state the main result of this chapter
Theorem 2.4.3. System (AP S) has one and only one mild solution (u, m) in C+ L2×C0,M L2; and the pair uN, mN converges to (u, m) in C L2loc×
C L2
loc, in probability.
The sketch of the proof is very classical. We proceed in three steps. 1. Tightness: we will prove that the law of family uN, mN is tight in
C+ L2 × C0,M L2;
2. Passage to the Limit: we will prove that the limit objects are supported on the set of mild solution of (AP S);
3. Uniqueness: we will prove uniqueness of the coupled system (AP S).
2.5
Tightness
2.5.1 Compactness of function spaces
We use Corollary 9 of J. Simon [42], using as far as possible the notations of that paper, for easiness of reference. Given a ball BR:= B (0, R) in Rd,
taken α > > 0, consider the spaces
X = Wα,2(BR) , B = Wα−,2(BR) , Y = W−2,2(BR) .
We have
X ⊂ B ⊂ Y
with compact dense embeddings. Moreover, we have the interpolation in-equality (see Theorem 6.4.5 in [4]),
kf kB ≤ CRkf k1−θX kf kθY for all f ∈ X, with
θ = 2 + α.
These are preliminary assumptions of Corollary 9 of [42]. The Corollary tells us that the embedding of
WR:= Lr0(0, T ; X) ∩ Ws1,r1(0, T ; Y )
is relatively compact in C ([0, T ] ; B), if sθ > r1θ where (always following the
notations of [42]) sθ= θs1, r1θ = 1−θr0 +rθ1. Below we shall choose for instance
s1 = 13 (any number smaller than 12) and r1= 4, so s1r1> 1 is fulfilled. Then
we need θs1> 1 − θ r0 + θ r1 .
Notice that we shall choose for instance s1 = 13 (any number smaller than 12)
and r1= 4, and r0 large enough to verify the condition. The logical sequence
of our choices is: given β ∈ (0, 1) (think to β close to 1, which is the most difficult choice), we shall choose α > 0 so small to satisfy a condition related to β which appears in the proof of Lemma 2.5.1 below (when β is close to 1, we have to choose α small). Given this small α, we choose ∈ (0, α) and then θ = 2+α is determined, typically very small. Now, we choose r0 so large
that θ3 > 1−θr
0 +
θ
4. Summarising we choose (α, s1, r1, r0, ), in the following
way: α : determined by β : < α arbitrarily small
(s1, r1) : determined (almost) a priori. See Proposition 2.5.2
In in Proposition 2.5.2 some constraints on (s1, r1) will emerge, indeed
(s1, r1) must be choosen such that s1r1−r21 < 0 The final step consists in
taking Rd instead of BR. We denote by Wlocα,2 Rd the space of functions f ∈ ∩R>0Wα,2(BR) and we endow this space with the metric
dWα,2 loc (f, g) = ∞ X n=1 2−nkf − gkWα,2(B n)∧ 1 .
Under the same conditions on the indexes, we have now that W := Lr0 0, T ; Wα,2Rd ∩ Ws1,r1 0, T ; W−2,2Rd
is compactly embedded into C
[0, T ] ; Wlocα−,2 Rd
.
2.5.2 Main estimate on the empirical density uN
Before looking into details for the derivation of main estimates for the empirical density, we state the mild formulation for uNt , see Lemma 2.4.3 for the identity for uNt :
uNt = etAuN0 + Z t
0
e(t−s)Adiv(WN∗ b(uNs , mNs )SsN)(x)ds +
Z t
0
e(t−s)AdMsN Lemma 2.5.1. Given β ∈ (0, 1), there exists α > 0 small enough such that the following holds: for every p > 1 there is a constant Cp> 0 such that
sup t∈[0,T ] E h uNt p Wα,2(Rd) i ≤ Cp independently of N .
Proof. Step 1 (preliminary estimates). We shall use the equivalence between norms (2.7):
||(I − A)α/2f ||L2(Rd)∼ ||f ||Wα,2(Rd).
Then, up to a constant, denoting with fsN(x) = div(WN∗ b(uNs , mNs )SsN)(x)
||uNt ||Wα,2(Rd)≤ ||(I−A)α/2etAuN0 ||L2(Rd)+ (I − A)α/2 Z t 0 e(t−s)AfsNds L2(Rd) + (I − A)α/2 Z t 0 e(t−s)AdMsN L2(Rd) .
On the first term, using (2.8) we prove the following estimate E h ||(I − A)α/2etAuN0 ||pL2(Rd) i ≤ ||etA||pL2→L2E h ||(I − A)α/2uN0 ||pL2(Rd) i ≤ CEh uN0 p Wα,2(Rd) i . The last expected value is bounded by the assumption that u0 is C2 and compactly support, see Lemma 2.4.6 for a complete proof. On the third term, we use the following fact. For every p > 1 there is a constant Cp > 0 such
that, if Φ1t, ..., ΦNt are adapted square integrable processes with values in a Hilbert space H, E " N X i=1 Z T 0 ΦitdBit p H # ≤ CpE N X i=1 Z T 0 Φit 2 Hdt !p/2 . Therefore E " Z t 0 (I − A)α/2e(t−s)AdMsN p L2(Rd) # = E σ N N X i=1 Z t 0 (I − A)α/2e(t−s)A∇WN(· − Xsi,N)dBsi p L2(Rd) ≤ CpE σ2 N2 N X i=1 Z T 0 (I − A) α/2e(t−s)A∇W N(· − Xsi,N) 2 L2(Rd)ds !p/2 = CpE σ2 N2 N X i=1 Z T 0 (I − A) α/2e(t−s)A∇W N 2 L2(Rd)ds !p/2 = Cp σ2 N Z T 0 (I − A) α/2e(t−s)A∇W N 2 L2(Rd)ds p/2 .
Moreover, the gradient commutes with the heat semigroup and the fractional powers of the Laplacian. Hence, using (2.8) and (2.9), the integrand can be estimated as follows (I − A) α/2e(t−s)A∇W N 2 L2(Rd) ≤h||∇(I − A)−1/2||L2→L2||(I − A) 1− 2 e(t−s)A|| L2→L2||(I − A) α+ 2 WN|| L2(Rd) i2 ≤ c (t − s)1−||WN|| 2 Wα+,2(Rd).
