Three examples at the boundary of mean field theory

Three examples at the boundary of mean …eld theory

Franco Flandoli, Scuola Normale Superiore

Plan of the lecture

0: classical mean …eld 1: towards local interaction

2: Example 1: individual based SIR 3: Example 2: cluster of cells 4: environmental noise

5: scaling limit to independent noise

Classical mean …eld (partial exposition; e.g. Sznitman lect.

notes)

dX_ti = 1 N N

∑

j=1 K X_ti X_tj dt+σdWti i =1, ..., N X₀i i.i.d. µ₀ empirical measure: µN_t = 1 N N

∑

i=1 δ_Xi ,N t

K bounded Lipschitz: the family of laws of µN is tight on

Identity for the empirical measure

Local interactions

X₀i i.i.d. µ₀ )X_ti ,N almost ₂ compact set hence typical distance of nearest neighbor is

X_ti ,N X_tj ,N N 1/d.

Can we consider only local interactions, namely

dX_ti = 1 N N

∑

j=1 Nγ_{K N}β _Xi t X j t dt+σdWti

where K is compact support (for suitable β, γ)? Problem: tightness of µN

Local interactions

It is a very di¢ cult but central problem. Di¤erent levels: local interaction in the motion of particles (as above)

local interaction in mechanisms of change of species, proliferation and death.

Let us start from the second ones, perhaps easier.

Example 1: individual based SIR

S = Susceptible I = Infected R = Recovered. Classical SIR model :

dS dt = IS τinf dI dt = IS τinf I τrec dR dt = I τrec .

Example 1: individual based SIR

Individuals have a location X1, ..., XN which doesn’t change (their home).

They can meet and infect each other. State V_t1, ..., V_tN V_ti _{2 f}S, I , R_g.

Con…guration: η_t 2 fS, I , R_gN .

Example 1: individual based SIR

emp. meas.: µV ,N_t = 1 N N

∑

i=1 δ V_ti =V δXi, V =S, I , R dDµV ,N_t , φ E = L 0 @1 N

∑

i=1,Vi t=V φ Xi 1 A dt+dM_tN D µS ,N_t , φ E = DµS ,N₀ , φ E Z t 0 Z Z φ(x)λN(x, y)µI ,N_s (dy)µS ,N_s (dx)ds+MtN

Example 1: individual based SIR, mean …eld regime

λN(x, y) = K (x, y) τinf D µS ,N_t , φ E = DµS ,N₀ , φ E Z t 0 Z Z φ(x)K(x, y) τinf µI ,N_s (dy)µS ,N_s (dx)ds+MtN

The family of laws of µS ,N, µI ,N, µR ,N is tight on D [0, T]; M Rd 3. Unique limit point µS_{, µ}I_{, µ}R _{, solving}

Example 1: individual based SIR, mean …eld regime

Di¤erential form, for densities

µS_t (dx) = fS(t, x)dx etc. : ∂fS(t, x) ∂t = fS(t, x) Z _{K(x, y)} τinf fI(t, y)dy ∂fI(t, x) ∂t = fS(t, x) Z _{K (x, y)} τinf fI (t, y)dy 1 τrec fI (t, x) ∂fR(t, x) ∂t = 1 τrec fI (t, x)

Example 1: individual based SIR, local interaction

λN (x, y) = 1 τinf N λ N1/d (x y) D µS ,N_t , φ E = DµS ,N₀ , φ E Z t 0 Z Z φ(x) 1 τinf N λ N1/d (x y) µI ,N_s (dy)µS ,N_s (dx)ds+MtN

Intuition: if the limit measures have densities, D µS_t, φ E = DµS₀, φ E + Z t 0 Z φ(x) 1 τinf µI_s(x)µS_s (x)dxds ∂fS(t, x) ∂t = 1 τinf fS(t, x)fI (t, x)

Other local zero-order interactions

The previous example is paradigmatic of local zero-order interactions. Other examples (among many others):

rate of proliferation (e.g. cells) depending on the state of nearest

particles (->FKKP equations; see for instance F. Leimbach

-Olivera ’19)

change of species as above (for general models, see several works of Karl Oelschläger)

Smoluchowski coagulation equation

(Großkinsky-Klingenberg-Oelschläger ’04, Hammond-Rezakhanlou ’07

and many others)

Scheme of a general approach

d µS ,N_t = λN µ_tI ,N µS ,N_t dt+dMtN

where λN are classical molli…ers. The intuition is that

µS ,N_t (dx) * fS(t, x)dx

µI ,N_t (dx) * fI (t, x)dx

∂tfS(t, x) = fI (t, x)fS(t, x).

Assume we prove tightness of µS ,N, µI ,N and existence of densities for

the limit measures. How to prove that D

λN µI ,N_t µS ,N_t , φ

E

Auxiliary molli…ers

Introduce auxiliary molli…ers θδ (smooth, symmetric, compact support)

converging to delta Dirac as δ_!0. rewrite

λN µI ,N_t µS ,N_t = λN θδ µ I ,N t µ

S ,N

t +RtN ,δ

so that the original equation

d µS ,N_t = λN µ_tI ,N µS ,N_t dt+dMtN becomes d µS ,N_t = λN θδ µ I ,N t µ S ,N t dt+RtN ,δdt+dMtN.

Lemma in the spirit of defect measures

From d µS ,N_t = λN θδ µ I ,N t µ S ,N t dt+RtN ,δdt+dMtN.

under suitable assumptions, which include tightness of µS ,N, µI ,N and

existence of L2 _{densities for the limit measures, we deduce:}

Lemma

The limit exists

Analysis of the remainder

How to prove that R_tN ,δ, de…ned as

R_tN ,δ = λN µ_tI ,N µS ,N_t λN θδ µ

I ,N t µ

S ,N t

goes to zero in distribution? Two interpretations:

local interaction with infected individuals λN µI ,N_t is similar to

interaction with a macroscopic average of them: λN θδ µI ,Nt (local

equilibrium)

the family of random …elds (empirical densities)

ρI ,N_t (x):= λN µI ,N_t (x)

Other interpretations

Recall R_tN ,δ = λN µ_tI ,N µS ,N_t λN θδ µ I ,N t µ S ,N t

Commutator estimates: the problem is similar to the smallness of

θδ (b uN) b (θδ uN)

Closure problem: similar also to the smallness of huNvNi huNi hvNi

Oelschläger solution

Without boring with other details, let us say that in F.-Grotto-Papini-Ricci we have solved the problem by following ideas of Karl Oelschläger:

write an identity for the empirical density ρI ,N

t (x):= λN µI ,Nt (x)

use PDE ideas to prove its regularity (in particular its hölderianity uniform in N).

The price is rescaling as Nβ

λ Nβ/d(x y) for some β <β₀ 1

instead of N λ N1/d (x y) . It means interaction with in…nitely many

Relevant literature for Examples 1 and 2

Oelschläger, K., A law of large numbers for moderately interacting di¤usion processes Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete(1985) 279–322.

Oelschläger, K., Large systems of interacting particles and the porous medium equation. Journal of di¤erential equations(1990) 294-346. Varadhan, SRS, Scaling limits for interacting di¤usions

Example 2: Brownian particles with local interaction

Even more di¢ cult is the case of local interaction in the dynamics. In order to appreciate the di¢ culty it is necessary to start from particles at unitary distance in Rd: dY_ti = N

∑

j=1 rV Y_ti Y_tj dt+σdBti

with compact support potential V :Rd _!R. Assume Y₀i i.i.d. in or near

Example 2: cluster of cells

Our interest (F.-Leocata-Ricci ’19) started looking at biological cell-dynamics

Brownian particles with local interaction

Let us see in more detail the rescaling of the potential:

dX_ti = 1 N N

∑

j=1 N1/dN(rV) N1/d X_ti X_tj dt+σdWti VN (x) : =NV N1/dx (classical molli…ers) rVN (x) = N1/dN(rV) N1/dx dX_ti = 1 N N

∑

j=1 rVN Xti X j t dt+σdWti.

Brownian particles with local interaction

Summarizing, we study a system in a "bounded" region, with interaction

dX_ti = 1 N N

∑

j=1 rVN Xti X j t dt+σdWti VN(x):=NV N1/dx (classical molli…ers).

Heuristically the mean …eld equation (VN (x) =VN0 (x)) would be

Brownian particles with local interaction

∂tf = σ2 2 ∆f +CV div(frf) = σ 2 2 ∆f + CV 2 ∆f 2_.

Rigorously proved by Oelschläger for repulsive potentials when

VN (x) =NβV Nβ/dx for some β <β₀ 1

instead of VN (x):=NV N1/dx

For VN(x):=NV N1/dx , d =1, repulsive potential, Varadhan ’91

obtains

∂tf = σ 2

2 ∆f +∆QV (f)

Brownian particles with local interaction

∂tf =

σ2

2 ∆f +∆QV (f)

For VN(x):=NV N1/dx , d =1, repulsive potential: Varadhan

’91.

Extended by Uchyama ’00 to d 1 and some attracting-repulsive

potentials, still QV (ft) unknown.

Remark: for many lattice systems like the zero-range process, QV (f)

Conjectures on the nearest neighbour case

In F.-Leocata-Ricci 2019 we have discussed numerically and heuristically

the shape of QV (f)in the limit equation

∂tf =

σ2

2 ∆f +∆QV (f)

CV =

R

V(x)dx _{2 (}0,∞): our investigation con…rms

∂tf = σ2 2 ∆f + CV 2 ∆f 2

also in repulsive+attracting cases. Uchiyama ’00 has a rigorous result

of type QV (f) C₂Vf2 for f !∞.

CV <0: strongly unclear.

Attractive-repulsive, not integrable: our conjecture

1 2 3 4 -1 0 1 2

x

y

V (x) = R α 0 αjxjα R₀β βjxjβ

with a cut-o¤ at R1 (min in R0)

with α> β. Conjecture in d =1: large f : QV(f)

Final subject: Environmental noise

First: classical mean …eld. Clearly if the system is

dX_ti =bt Xti dt+ 1 N N

∑

j=1 K X_ti X_tj dt+σdWti

the limit equation is

Final subject: Environmental noise

Therefore it is not surprising that in the case when bt(x)is a

space-dependent white-noise-in-time

bt(x)dt =

∑

k

σk(x) d βk_t

the limit equation is the SPDE (σ =0 for simplicity)

hµ_t, φi = hµ₀, φi + Z t 0 hµs ,(K µ_s) rφids +

∑

k Z t 0 hµs, σk rφi d β k s.

Environmental noise, almost uncorrelated in space

Assume the noise is invariant by space-translations, with covariance depending on a parameter e: Qe_{(x y) =}

∑

k σek(x) σek(y). If Qe_{(x)!I δ} 0(dx)

Small correlation scaling limit

In other words we conjecture that, under suitable link N _!eN and

suitable assumptions on the noise coe¢ cients, the empirical measure of the particle system

dX_ti = 1 N N

∑

j=1 K X_ti X_tj dt+

∑

k σe_kN Xti d βkt

converges to the deterministic parabolic equation hµ_t, φi = hµ₀, φi + Z t 0 hµs ,(K µ_s) rφids +C Z t 0 h µ_s,∆φids.

Example 3: rain drop formation

Example 3: rain drop formation

This topic mixes several ideas illustrated above: particles are cloud droplets (0.02-0.5 mm) di¤erent species means di¤erent size

transition of size has a rate depending on relative position (collision) thus interaction of zero order terms, local in space

E¤ect of turbulence on rain drop formation

The limit model seems to be a Smoluchowski equation of the form

∂tfm(t, x) =Q+ Q +∆fm(t, x) where Q+ = 1 2 Z m 0 K m0, m m0 fm0(t, x)f_{m m}0(t, x)dm0dt Q = fm(t, x) Z _∞ 0 K m, m0 fm0(t, x)dm0dt.

The key aspect is that a Brownian displacement of particles should take place due to environmental noise with small correlation. Following Hammond-Rezakhanlou ’07, this modi…es the interaction kernel.