McKean-Vlasov limit for interacting systems with simultaneous jumps - - PowerPoint PPT Presentation

Background Neuroscience models General class of models

McKean-Vlasov limit for interacting systems with simultaneous jumps

Luisa Andreis

• Prof. Paolo Dai Pra, Markus Fischer

Università degli studi di Padova

Berlin - Padova Young researchers Meeting in Probability

• Oct. 23-25, 2014

Background Neuroscience models General class of models

Outline of the talk

Brief overview on interacting particle systems, propagation of chaos and McKean-Vlasov limits Interacting particle systems for neuroscience A general class of models

Background Neuroscience models General class of models

Interacting particle systems

Mathematical models for different areas as Neuroscience, Genetics, Biology, Economics, etc., rely on interacting particle systems. These models start from a finite number N of particles interacting with each other. Particle system

Background Neuroscience models General class of models

Interacting particle systems

Mathematical models for different areas as Neuroscience, Genetics, Biology, Economics, etc., rely on interacting particle systems. These models start from a finite number N of particles interacting with each other. Particle system ւ ց microscopic macroscopic behaviour behaviour

Background Neuroscience models General class of models

Propagation of chaos

Definition (p-chaotic sequence) Let E be a separable metric space and pN a sequence of symmetric probabilities on

• EN. pN is p-chaotic, with p probability on E, if for any sequence φ1, . . . , φk of test

functions and for all k ≥ 1, lim

N→∞pN, φ1 ⊗ · · · ⊗ φk ⊗ 1 ⊗ · · · ⊗ 1 = k

• i=1

p, φi. (1)

Background Neuroscience models General class of models

Propagation of chaos

Definition (p-chaotic sequence) Let E be a separable metric space and pN a sequence of symmetric probabilities on

• EN. pN is p-chaotic, with p probability on E, if for any sequence φ1, . . . , φk of test

functions and for all k ≥ 1, lim

N→∞pN, φ1 ⊗ · · · ⊗ φk ⊗ 1 ⊗ · · · ⊗ 1 = k

• i=1

p, φi. (1) Definition (Propagation of chaos) Consider a Markovian system of particles with symmetric law PN on C(R+, E)N (or D(R+, E)N). Propagation of chaos means that when the initial conditions are p0-chaotic, for a certain p0 probability on E, then there exists a suitable p probability on C(R+, E) (or D(R+, E)), with initial condition p0, such that the sequence PN is P-chaotic.

Background Neuroscience models General class of models

Propagation of chaos as law of large numbers

Let us define the empirical measure ρN = 1 N

N

• i=1

δXi , (2) where (X1, . . . , XN) is a random variable distributed according pN and therefore ρN is a sequence of r.v. with values on M(E). Theorem (Tanaka, Sznitman) Let E be a separable metric space, p a probability measure on E and pN a sequence

• f symmetric probability measures on EN, the fact that pN is p-chaotic is equivalent to

law(ρN) − → δp, (3) where the convergence is in law.

Background Neuroscience models General class of models

Proving propagation of chaos

Classical approach to prove propagation of chaos: to prove tightness of the sequence; to identify the possible limits of the sequence; to prove uniqueness of this limit.

Background Neuroscience models General class of models

Proving propagation of chaos

Classical approach to prove propagation of chaos: to prove tightness of the sequence; to identify the possible limits of the sequence; to prove uniqueness of this limit. Example: McKean-Vlasov SDE of the particle system: dX i

t = b

• X i

t , ρN(t)

• dt + σ
• X i

t , ρN(t)

• dBi

t

for i = 1, . . . , N, (4) SDE of the limiting process: dXt = b (Xt, pt) dt + σ (Xt, pt) dBt. (5) Here X 1, . . . , X N, X are stochastic processes with values on Rd, b(·, ·) : Rd × M(Rd) → Rd, σ(·, ·) : Rd × M(Rd) → Rd×k, B1, . . . , BN, B are k-dimensional Brownian motion and pt is the law of the process Xt itself.

Background Neuroscience models General class of models

Interacting particle systems in neuroscience

Modelling membrane potential ւ ց neurons spikes neurons spikes

• ccur when their
• ccur randomly

potentials reach according some a treshold point processes

Background Neuroscience models General class of models

Toy model for interacting neurons

Presented by De Masi, Galves, Löcherbach, Presutti (2014) and deeper investigated by Fournier, Löcherbach (2014). UN(t) =

• UN

1 (t), . . . , UN N (t)

• ∈ RN

+,

configuration of membrane potentials for N interacting neurons. Chemical synapses: each neuron randomly spikes, it sets its energy at 0 and it gives all the other neurons a small and deterministic quantity of engergy, i.e. 1

N .

Electrical synapses: each neuron’s potential tends to reach the value of the center of mass. Infinitesimal generator of the N dimensional particle systems For all φ : RN

+ → R smooth test function,

LNφ(x) =

N

• i=1

f(xi)[φ(x + ∆i(x)) − φ(x)] − λ

N

• i=1
• [xi − ¯

x] ∂φ ∂xi (x)

• .

(6) Here f ∈ C(R+, R+) is strictly positive for x > 0 and non-decreasing, λ ≥ 0 and ∆i(x)j =

• 1

N

for j = i −xi for j = i

Background Neuroscience models General class of models

Approach to prove propagation of chaos

The peculiarity of this model are simultaneous jumps. In the framework of propagation of chaos, non-simultaneous jumps in particle systems and in non-linear limits have been treated before, Graham(1992), Méléard (1996). Authors approach for simultaneous jumps

• They build an ad hoc Markov process that is a simplification and a discretization
• f the model.
• They consider an elaborate coupling algorithm to prove that the discrete time

approximation of the initial Markov process and the simpler discrete process are closer with N → ∞.

• They obtain a limiting equation for the density of the simpler discrete process

when N → ∞ and the time interval goes to 0.

Background Neuroscience models General class of models

Toy model for interacting neurons

Infinitesimal generator of the limit process, for all ϕ : R+ → R smooth test function, Lϕ(x) = f(x)[ϕ(0) − ϕ(x)] + [−λ(x − ¯ ρ) + p] ∂ϕ ∂x (x). (7) Here f ∈ C(R+, R+) is strictly positive for x > 0 and non-decreasing, λ ≥ 0 and ¯ ρ = ∞ xρ(dx), (8) p = ∞ f(x)ρ(dx). (9)

Background Neuroscience models General class of models

Motivation

Do particle systems with simultaneous jumps need ad hoc treatment to prove propagation of chaos?

Background Neuroscience models General class of models

Motivation

Do particle systems with simultaneous jumps need ad hoc treatment to prove propagation of chaos? ↓ Our aim is to build a sufficiently general framework to include the previous model and to obtain limiting results with the classical tools of propagation of chaos.

Background Neuroscience models General class of models

Particle system

We start from a system of N interacting particles, each of them associated to a random process X N

i (t) with values on R, i.e.

X N(t) =

• X N

1 (t), . . . , X N N (t)

• ∈

Rd × · · · × Rd. Infinitesimal generator For every φ : Rd × · · · × Rd → R smooth test function, LNφ(x) = N

i=1

• F(xi, ρN) ▽i φ(x)

+λ(xi, ρN)

• [0,1]N [φ(x + ∆i(x, h)) − φ(x)] νN(dh)
• ,

(10) with ∆i(x, h)j =

• Θ(xi ,xj ,ρN,hi ,hj )

N

for j = i, ψ(xi, ρN, hi) for j = i. (11) Here λ(·, ·) is bounded and νN is the projection on N coordinates of a symmetric probability measure ν on [0, 1]N and gives the randomness of the amplitude of the jumps.

Background Neuroscience models General class of models

Candidate to be the limit process

Through heuristic computation we arrive at a possible limit process, X(t) ∈ Rd. Infinitesimal generator For every ϕ : Rd → R smooth test function, Lϕ(x) = F(x, ρ) ▽ ϕ(x) +

• Rd λ(y, ρ)
• [0,1]2 Θ(y, x, ρ, h1, h2)ν2(dh1, dh2)ρ(dy)
• ▽ ϕ(x)

+λ(x, ρ)

• [0,1]

[ϕ(x + ψ(x, ρ, h1)) − ϕ(x)] ν1(dh1)

• ,

(12) where ρ is the law of the process itself.

Background Neuroscience models General class of models

Intermediate process

Exploiting an idea from Dai Pra, Fischer, Regoli (2013), we construct an intermediate process, starting from the hypothetical limit process. We consider a system of N particles, each of them is associated to a random process ˜ X N

i (t) ∈ Rd and the

comprehensive process is ˜ X N(t) = (˜ X N

1 (t), . . . , ˜

X N

N (t)) ∈ Rd × · · · × Rd.

Infinitesimal generator For every φ : Rd × · · · × Rd → R smooth test function, ˜ LNφ(x) = N

i=1

• F(xi, ˜

ρN) ▽i φ(x) +

• Rd λ(y, ˜

ρN)

• [0,1]2 Θ(y, x, ˜

ρN, h1, h2)ν2(dh1, dh2)˜ ρN(dy)

• ▽i φ(x)

+λ(xi, ˜ ρN)

• [0,1]N
• φ(x + ˜

∆i(x, h)) − φ(x)

• ν(dh)
• ,

(13) where ˜ ∆i(x, h)j = for j = i, ψ(xi, ρN, hi) for j = i. (14) The difference in this interacting particle system is that there are no more simultaneous jumps, since all the collateral jumps caused by one particle jump have been absorbed in an additional drift.

Background Neuroscience models General class of models

Propagation of chaos for the intermediate system

Propagation of chaos for the Markov process ˜ X N(t) = (˜ X N

1 (t), . . . , ˜

X N

N (t)) is a

consequence of classical tools, proved by Graham (1991) for diffusions with jumps. Theorem Let ˜ PN be the probability measure corresponding to the process ˜ X =

• ˜

X N(t)

• t∈[0,T]

with initial law p-chaotic and P be the probability measure corresponding to the process X = {X(t)}t∈[0,T] with initial law p. Under Lipschitz-type conditions on coefficient, the sequence of laws ˜ PN is P-chaotic.

Background Neuroscience models General class of models

Initial and intermediate particles systems

We prove that, under reasonable conditions, the initial Markov process X N and the intermediate Markov process ˜ X N are closer as N goes to infinity. To prove this, we do not exploit particular construction, by classical martigale inequalities and properties of diffusion with jumps. Theorem Under the same Lipschitz-type conditions on coefficients as before, we have ˆ dbL (law(ρN), law(˜ ρN)) → 0, (15) as N → ∞, where ˆ dbL(·, ·) is the bounded Lipschitz metric on M(M(D)). Recall that ρN = N

i=1 δXN

i (·) and ˜

ρN = N

i=1 δ˜ XN

i (·) are empirical measures

corresponding to process X N and ˜ X N; therefore they are two random variable on M(D([0, T], Rd)).

Background Neuroscience models General class of models

Conclusions and further developments

Conclusions We get a general procedure to prove propagation of chaos for a class of interacting particle systems with simultaneous jumps under Lipschitz-type conditions on coefficients and under the hypothesis that the jump rate λ is bounded.

Future possible developments:

• relax conditions on the coefficients of the model and prove propagation of chaos

through some estimates;

• consider particular models, belonging to the class we presented here, and

investigate solutions, stationary distributions and other features;

• develop possible fluctuation and large deviations results for this class of systems.

Background Neuroscience models General class of models

Some references

P . Dai Pra, M. Fischer, D. Regoli, A Curie-Weiss model with dissipation, 2013, J.

• Stat. Phys., vol 152,37-53
• A. De Masi, A. Galves,E. Löcherbach, E. Presutti, Hydrodynamic limit for

interacting neurons, 2014, arXiv preprint arXiv:1401.4264

• N. Fournier, E. Löcherbach, On a toy model of interacting neurons, 2014, arXiv

preprint arXiv:1410.3263

• C. Graham, McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with

discrete jump sets, 1992, Stoc. proc. appl., vol. 40, 69-82

• S. Méléard, Asymptotic behaviour of some interacting particle systems;

McKean-Vlasov and Boltzmann models, 1996, Probabilistic models for nonlinear partial differential equations, Springer

• A. S. Sznitman, Topics in propagation of chaos, 1991, Ecole d’Eté de Probabilités

de Saint-Flour XIX-1989, Springer

Background Neuroscience models General class of models

Thank you!

Luisa Andreis