Kinetic equations from stochastic dynamics in · PDF filefrom stochastic dynamics in continuum...

Kinetic equationsfrom stochastic dynamics in continuum

Yuri Kondratiev

Bielefeld University

Yuri Kondratiev (Bielefeld) Kinetic equations from stochastic dynamics in continuum 1 / 45

Setup

MACRO, MESO, and MICRO

Non-linear PDE and SPDE

as phenomenological models of complex systems.

Reaction-diffusion equations (RDE)

∂u

∂t= 4u+ f(u), u = u(t, x)

in combustion theory, bacterial growth, nerve propagation, epidemiology, geneticsetc.

RDE = Allen-Cahn = Ginzburg-Landau

Fisher equation: f(s) = s(1− s).


Setup

We observe an active recent development in the study ofnon-local versions of RDE:

∂u

∂t= J ? u− u+ f(u), u = u(t, x).

u(0, x) = ϕ(x), x ∈ Ω ⊆ Rd

0 ≤ J ∈ L1, ‖J‖1 = 1 jump kernel.

Just few references:Coville, Dupaigne (2008)Ignat, Rossi (2010)Berestycki, Nadin, Ryzhik (2009)Pan, Li, Lin (2009)Zhang, Li, Sun (2010)


Setup

Actually, such kind of equation was introduced byKolmogorov, Petrovsky and Piskunov (KPP) in 1937 (!)as a way to derive Fisher equation.

Local RDE (macroscopic description)are approximations tonon-local RDE (mesoscopic description)


Setup

MACRO, MESO, and MICRO

Problem: derivation of meso- and macroscopic equations(deterministic or stochastic)from microscopic models .

There are known different techniques:– scaling limits for dynamics (hydrodynamic, Vlasov, Landau etc.)

– scaling of fluctuations (equilibrium or non-equilibrium)

– closure of (infinite linear) moment systems

– hierarchical chains (BBGKY etc.)

– particle approximation method


Setup

Three levels in physics

In context of the theory of classical gases:

(Mi) is the level of particle dynamics (Newtons laws)

(Me) is the level of Boltzmann description

(Ma) is the level of continuum description.


Setup

Microscopic Stochastic Systems

In mathematical terms we are interested in the links between the followingmathematical structures:

(Mi) the micro–scale of stochastically interacting entities (cells, individuals,. . .),in terms of (linear) Markov semigroups (or corresponding processes)

(Me) the meso–scale of statistical entities, in terms of continuous nonlinearsemigroups related to the solutions of nonlinear Boltzmann–type non-local kineticequations

(Ma) the macro–scale of densities of interacting entities (in terms of dynamicalsystems related to reaction–diffusion type equations).


Setup General facts

Dynamics of configurations

Deterministic dynamics:

Hamiltonian dynamics

Interacting particle dynamical systems

Vlasov equation

due to Braun/Hepp, 1977 and Dobrushin, 1979:

we study asymptotic for N →∞ of the solution to

dxi(t)

dt= A(xi(t)) +N−1

N∑j=1

B(xi(t)− xj(t)),

xi ∈ Rd, i = 1, . . . , N.

Actually, shall be x(N)i notation.


Setup General facts

Empirical measure:

µNt =1

N

∑i

δxi(t)

We assume (initial density approx.)

µN0 (dx)→ ρ0(x)dx, N →∞.

VE for the limiting density (sic!):

∂ρt(x)

∂t= −Tr(∇x(A(x)ρt(x)))− Tr∇xρt(x)

∫RdB(x− y)ρt(y)dy.

Finite volume assumption?We are dealing with L1-solutions only!No results at all for bdd densities = infinite systems.


Setup General facts

Method is based on the use of the equation of motion and effectively means aversion of LLN

VE for Hamiltonian dynamics from BBGKY hierarchy (heuristic derivation):H.Spohn, 1980

V.P.Maslov, 1982: derivation of Vlasov equation from BBGKYfor ”random number of particles”(=zero density system).

Rigorous derivation for positive densities meets problems:the lack of detailed knowledge about the solution to BBGKY.

VE appeared originally in plasma physicsand in the stellar dynamical problems.


Setup General facts

Markov evolutions in continuum:

Diffusions (e.g., gradient diffusion)

Jumping particles Markov processes (e.g., Kawasaki dynamics)

Birth-and-death stochastic dynamics (e.g., Glauber, IBM in spatial ecology)

other stochastic IPS in Rd

Questions:

What is possible concept of related Vlasov equations?

May we work with infinite particle dynamics?

Is there a notion of a limiting IPS dynamics which creates Vlasov equation?

Other scalings and other kinetic equations?


Setup General facts

Lattice framework: kinetic equations from lattice stochastic dynamics

For lattice stochastic dynamics and their hydrodynamic scalings

we have several excellent results by

Lebowitz, Presutti, Varadhan, Yau, Kipnis, Landimand many other authors

showing how macroscopic kinetic equations appear from microscopic stochasticprocesses.

Particular examples of non-local RDE were derived from lattice stochasticdynamics by Durrett, 1995.

Front propagation in these mesoscopic models analyzed by Perthame/Souganidis,2005.

Scaling of lattice stochastic dynamics to super-processes etc.


Setup General facts

Our approach is based on the study of state evolutionfor the considered systems.

Markov dynamics of IPS ⇒ evolution of states (measures)

Particular scalings ⇒ different kinds of kinetic equations

General discussion for Hamiltonian dynamics:[Dobrushin/Sinai/Suhov], 1985.

Interacting diffusions:McKean-Vlasov limitwhich is a stochastic versionof the deterministic case.Classical results by McKean, Dawson, Gartner, Shiga et. al.


General scheme of Vlasov scaling

Markov evolutions

Let L be a Markov pre-generator defined on some set of functions F(Γ)given on the configuration space Γ.

(Backward) Kolmogorov equation:

∂Ft∂t

= LFt,

Ft|t=0 = F0;

Duality: < F, µ >:=∫

ΓFdµ

(Forward) Kolmogorov = Fokker- Planck equation:

∂µt∂t

= L∗µt,

µt|t=0 = µ0.



Vlasov scaling

Initial distribution: µ0 ∈M1fm(Γ) with correlation function k0 : Γ0 → R+,

k0 = k(n)0 , n ≥ 0.

µt ∈M1(Γ) the distribution at time t > 0 and kt its correlation function.dµtdt

= L∗µt

µt∣∣t=0

= µ0,

where L∗ is the adjoint to the generator on functionsdFtdt

= LFt

Ft∣∣t=0

= F0.dktdt

= L4kt

kt∣∣t=0

= k0

where L4 := L∗ is the generator of a semigroup T4t := T ∗t .Yuri Kondratiev (Bielefeld) Kinetic equations from stochastic dynamics in continuum 16 / 45


1st step in VS

Choose the initial state of the system:

∀ε > 0 correlation functions k(ε)0 as ε→ 0:

k(ε)0, ren(η) := ε|η|k

(ε)0 (η)→ r0(η), ε→ 0, η ∈ Γ0,

where correlation function r0 will be chosen properly.

In the case ofr0(η) = eλ(ρ0, η) =

∏x∈η

ρ0(x), η ∈ Γ0

ρ0 : Rd → (0,+∞)

the assumption about the initial conditions leads to a Poisson measure :

µ(ε)0, ren → πρ0 ,

where µ(ε)0, ren has correlation function ε|η|k

(ε)0 (η).



2nd step in VS

Scaling of the generator:L 7−→ Lε.

The concrete type of this scaling will depend on the model.

Suppose that there exist solution of the correlation functional evolutiondk

(ε)t

dt= L4ε k

(ε)t

k(ε)t

∣∣t=0

= k(ε)0

We expect (and this will be shown in concrete models) that order of the

singularity in ε for this solution will be the same as for initial function k(ε)0 .



3rd step in VS

We considerk

(ε)t, ren(η) := ε|η|k

(ε)t (η), η ∈ Γ0,

and want to show that

k(ε)t, ren(η)→ rt(η), ε→ 0, η ∈ Γ0.

In fact, we consider renormalized version of the evolution equationdk

(ε)t, ren

dt= L4ε,renk

(ε)t, ren

k(ε)t, ren

∣∣t=0

= k(ε)0, ren

whereL4ε,ren = ε|η|L4ε ε

−|η|.



Vlasov hierarchy

We want to show that the solution k(ε)t, ren converges to rt which satisfied

new hierarchy =: Vlasov hierarchy

drtdt

= V 4rt

rt∣∣t=0

= r0

This equation describes an evolution of a virtual interacting particle systemappearing in the Vlasov limit.

In general, this evolution is not related with a new Markov generator.



Consider the case of an initial Poisson measure:

r0(η) = eλ(ρ0, η).

Under some general conditions,the scaling leads to solution rt of the same form:

rt(η) = eλ(ρt, η), η ∈ Γ0.

The Vlasov hierarchical equation in this case implies a non-linear equation for ρt:

∂

∂tρt(x) = υ(ρt)(x), x ∈ Rd,

which we will call Vlasov-type equation corresponding to the considered Markovevolution.


Derivation of Vlasov hierarchies

Birth and death evolutions

Birth-and-death generators

Lbad = L− + L+,

where

(L−F

)(γ) :=

∑x∈γ

d (x, γ \ x) [F (γ \ x)− F (γ)] ,

(L+F

)(γ) :=

∫Rdb (x, γ) [F (γ ∪ x)− F (γ)] dx


Derivation of Vlasov hierarchies

Denote L− = L−(d), L+ = L+(b).

We will use scaling of rates b, d, say, bε, dε,, correspondingly, ε > 0.

Scaling of Lbad

Lbad, ε = L−(dε) + ε−1L+(bε),

General conditions for the weak convergence of L4ε to the limiting Vlasovgenerator V 4 considered in

Finkelshtein/K/Kutoviy, 2009, to appear in JSP.Below we consider some examples from this paper.


Derivation of Vlasov hierarchies Particular models: B-A-D systems

Birth-and-death systems

Example (Contact model = branching with mortality)

(LF )(γ) = m∑x∈γ

[F (γ \ x)− F (γ)]

+ λ∑x∈γ

∫Rda(x− y)[F (γ ∪ y)− F (γ)]dy;

(VE) :∂

∂tρt(x) = −mρt(x) + λ(ρt ∗ a)(x).

Scaling:λ 7→ ε−1λ, a 7→ εa



Example (Migration model)

(LF )(γ) =∑x∈γ

∑y∈γ\x

a(x− y)[F (γ \ x)− F (γ)]

+ σ

∫Rd

[F (γ ∪ x)− F (γ)]dx;

(VE) :∂

∂tρt(x) = −ρt(x)(ρt ∗ a)(x) + σ.

Scaling:

a 7→ εa

σ 7→ ε−1σ



Example (Bolker-Pacala model in spatial ecology)

(LF )(γ) =∑x∈γ

(m+

∑y∈γ\x

a−(x− y))

[F (γ \ x)− F (γ)]

+∑x∈γ

∫Rda+(x− y)[F (γ ∪ y)− F (γ)]dy;

(VE) :∂

∂tρt(x) = −mρt(x)− ρt(x)(ρt ∗ a−)(x) + (ρt ∗ a+)(x).

Scaling:a− 7→ εa−

a+ 7→ ε−1εa+ = a+.

That is a non-local Fisher-Kolmogorov equation, see, e.g., Coville/Dupaigne, 2008.



Example (Ecological model with establishment)

(LF )(γ) = m∑x∈γ

[F (γ \ x)− F (γ)]

+ λ∑x∈γ

∫Rda(x− y)e−

∑u∈γ φ(y−u)[F (γ ∪ y)− F (γ)]dy;

(VE) :∂

∂tρt(x) = −mρt(x) + λ(a ∗ ρt)(x)e−(φ∗ρt)(x).

Scaling:a 7→ εa, φ 7→ εφ

λ 7→ ε−1λ

This equation is new one and shall be analyzed in detail.



Example (Ecological model with fecundity)

(LF )(γ) = m∑x∈γ

[F (γ \ x)− F (γ)]

+ λ∑x∈γ

∫Rda(x− y)e−

∑u∈γ\x φ(x−u)[F (γ ∪ y)− F (γ)]dy;

(VE) :∂

∂tρt(x) = −mρt(x) + λ(a ∗ (ρte

−(φ∗ρt)))(x).

Scaling:a 7→ εa, φ 7→ εφ

λ 7→ ε−1λ



Example (Dieckmann--Law model)

(LF )(γ) =∑x∈γ

(m+

∑y∈γ\x

a−(x− y))

[F (γ \ x)− F (γ)]

+∑x∈γ

∫Rda+(x− y)

(λ+

∑u∈γ\x

b(x− u))

[F (γ ∪ y)− F (γ)]dy;

(VE) :∂

∂tρt(x) = −mρt(x)− ρt(x)(ρt ∗ a−)(x)

+ λ(ρt ∗ a+)(x) + (((b ∗ ρt)ρt) ∗ a+)(x).

Scaling:a− 7→ εa−, a+ 7→ ε−1εa+ = a+.

b 7→ εb


Derivation of Vlasov hierarchies Particular models: conservative particle systems

Conservative particle systems

Example (Free Kawasaki)

(LF ) (γ) =∑x∈γ

∫Rda (x− y) [F (γ \ x ∪ y)− F (γ)] dy;

(VE) :∂

∂tρt (x) = (ρt ∗ a) (x)− ρt (x) 〈a〉 .



Example (Density dependent Kawasaki)

(LF ) (γ) =∑x∈γ

∫Rda (x− y)

∑u∈γ

b (x, y, u) [F (γ \ x ∪ y)− F (γ)] dy;

∂

∂tρt (x) =

∫Rdρt (y) a (x− y)

∫Rdρt (u) b (y, x, u) dudy

− ρt (x)

∫Rda (x− y)

∫Rdρt (u) b (x, y, u) dudy.

In particular, if b(x, y, u) = b(x− u) then

(VE) :∂

∂tρt (x) = ((ρt(ρt ∗ b)) ∗ a)(x)− 〈a〉ρt (x) (ρt ∗ b)(x).

If b(x, y, u) = b(y − u) then

∂

∂tρt (x) = (ρt ∗ b) (x) (ρt ∗ a) (x)− ρt (x) (ρt ∗ a ∗ b)(x).



Example (Gibbs--Kawasaki)

(LF ) (γ) =∑x∈γ

∫Rda (x− y) e−E

φ(x,γ) [F (γ \ x ∪ y)− F (γ)] dy;

(VE) :∂

∂tρt (x) = (ρt ∗ a) (x) exp − (ρt ∗ φ) (x)

− ρt (x) (a ∗ exp −ρt ∗ φ) (x) .

Scaling:φ 7→ εφ



Example (Gradient diffusion)

(LF )(γ) =∑x∈γ

∆xF (γ)−∑x∈γ

∑y∈γ\x

〈Oφ(x− y),OxF 〉

(VE)∂

∂tρt (x) = ∆ρt(x)−

∫φ(x− y) 〈∇ρt(x),∇ρt(y)〉dy

−ρt(x)

∫〈∇φ(x− y),∇ρt(y)〉 dy

Scaling:φ 7→ εφ

Gradient diffusion appeared in mathematical theory of electrolites,Bogoliubov-Streltsova, 1956.



Convergence problem

In all models above the weak convergence of the generators

L4ε → V 4, ε→ 0

is shown.

A difficult question: convergence of solutions of hierarchical equations to theVlasov hierarchy solution.

In each model even just the problem of the existence for the solution to hierarchyneed a special analysis and difficult technical work (non-equilibrium dynamics).

But we need essentially more:detailed control of the solution uniformly w.r.t. parameters on any finite timeinterval.

Two regimes:

zero density systems = L1 solutions to be considered

positive density systems = L∞ solutions



For the positive density case only knownresults (Finkelshtein/K/Kutoviy 2009-2010) concern

Contact Model

Migration Model

Glauber Dynamics

Bolker-Pacala model in spatial ecology

Zero density case should be possible to analyze also in other modelsand under more relaxing conditions on parameters (work in progress).


Inter-spices interactions in spatial ecology

Two component density dependent ecological model

(LF )(γ+, γ−) :=∑x∈γ+

m+ +∑

x′∈γ+x

a−1 (x− x′) +∑y∈γ−

b−1 (x− y)

×[F (γ+ \ x, γ−)− F (γ+, γ−)

]

+∑y∈γ−

m− +∑

y′∈γ−y

a−2 (y − y′) +∑x∈γ+

b−2 (x− y)

×[F (γ+, γ− \ y)− F (γ+, γ−)

]



+∑x′∈γ+

∫Rd

a+1 (x− x′) +

∑y∈γ−

b+1 (y, x, x′)

[F (γ+ ∪ x, γ−)− F (γ+, γ−)

]dx

+∑y′∈γ−

∫Rd

a+2 (y − y′) +

∑x∈γ+

b+2 (x, y, y′)

[F (γ+, γ− ∪ y)− F (γ+, γ−)

]dy.



Scaling:

a±1,ε = εa±1 , b−1,2,ε = εb−1,2

b+1,2,ε = ε2b+1,2

Birth intensity:

1 7→ ε−1



Vlasov equation in ecology

∂

∂tρ+t (x) = −m+ρ+

t (x)− ρ+t (x)

(a−1 ∗ ρ

+t

)(x)

−ρ+t (x)

(ρ−t ∗ b−1

)(x) +

(ρ+t ∗ a+

1

)(x)

+

∫Rd

∫Rdρ+t (x′) ρ−t (y) b+1 (y, x, x′) dx′dy

and∂

∂tρ−t (y) = −m−ρ−t (y)− ρ−t (y)

(a+

1 ∗ ρ−t

)(y)

−ρ−t (y)(ρ+t ∗ b+1

)(y) +

(ρ−t ∗ a−1

)(y)

+

∫Rd

∫Rdρ−t (x) ρ+

t (y′) b+1 (x, y, y′) dxdy′


Hydrodynamic equations

Hydrodynamic scaling

Initial state µ(ε)0 s.t.

k(ε)0, ren(η) := εd|η|k

(ε)0 (η)→ r0(η), ε→ 0, η ∈ Γ0,

Scaled potentials:U 7→ U(ε−1·)

In the case of BAD-processes we need scaled (accelerated)intensity of birth:

z 7→ ε−dz

Corresponding generator for corr. functions

Lhdε



Rescaled hierarchy dk

(ε)t, ren

dt= Lhd

ε,renk(ε)t, ren

k(ε)t, ren

∣∣t=0

= k(ε)0, ren

whereLhdε,ren = εd|η|Lhd

ε ε−d|η|.

We want to show that the solution k(ε)t, ren converges to rt which satisfied

HD hierarchy

drtdt

= Lhdrt

rt∣∣t=0

= r0

This equation describes an evolution of a virtual interacting particle systemappearing in the HD limit.



Consider the case of an initial Poisson measure:

r0(η) = eλ(ρ0, η).

Under general conditions, the scaling leads to the solution rt of the same form:

rt(η) = eλ(ρt, η), η ∈ Γ0.

(chaos preservation property)

The HD hierarchical equation in this case implies a non-linear equation for ρt:

∂

∂tρt(x) = υ(ρt)(x), x ∈ Rd,

which we will call hydrodynamic equation (HDE) corresponding to the consideredMarkov evolution.



Example (Plankton model)

(LF )(γ) =∑x∈γ

∆xF (γ)

+∑x∈γ

(m+

∑y∈γ\x

a−(x− y))

[F (γ \ x)− F (γ)]

+∑x∈γ

∫Rda+(x− y)[F (γ ∪ y)− F (γ)]dy;

(HDE) :∂

∂tρt(x) = ∆ρt(x)−mρt(x)− κ−ρ2

t (x) + κ+ρt(x)

Fisher-KPP equation.Scaling:

a− 7→ a−(ε−1·)

a+ 7→ ε−da+(ε−1·)


Mixed scaling for BAD dynamics

Mixed scaling in Bolker-Pacala model

(LF )(γ) =∑x∈γ

(m+

∑y∈γ\x

a−(x− y))

[F (γ \ x)− F (γ)]

+∑x∈γ

∫Rda+(x− y)[F (γ ∪ y)− F (γ)]dy;

Vlasov scaling for dispersion (birth) partand HD scaling for competition (death) part of generator


Mixed scaling for BAD dynamics

Scaling:a− 7→ a−(ε−1·)

a+ 7→ ε−dεda+ = a+.

(HDE) :∂

∂tρt(x) = (ρt ∗ a+)(x)−mρt(x)− κ−ρ2

t (x) =∫a+(x− y)(ρt(y)− ρt(x))dy + κ+ρt(x)−mρt(x)− κ−ρ2

t (x).

That is a non-local Fisher-Kolmogorov equation.


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