Model Stati.Distr. Mean field Fluid limit Questions
Modelling flocks and prices: jumping particles
with an attractive interaction
Joint work in progress with Miklós Zoltán Rácz
Márton Balázs
Budapest University of Technology and Economics
Large Scale Stochastic Dynamics
Oberwolfach, November 11, 2010.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
xi
◮
◮
m − xi
m
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model (Bálint Tóth)
Goats jump on R.
m
◮
◮
R
n goats jump on R (state space is Rn ).
Given a configuration
x1 , x2 , . . . , xn of goats the center of
P
mass is m = n1 ni=1 xi .
◮
Goat i jumps with rate w(xi − m), where w is the jump rate
function: R → R+ , decreasing.
◮
Jumps are positive, random, independent of everything,
and are of density ϕ, mean one.
Model Stati.Distr. Mean field Fluid limit Questions
The model
Stationary distribution
Mean field equation
Exponential jumps
Extreme value statistics
Fourier methods
Fluid limit
Questions
Model Stati.Distr. Mean field Fluid limit Questions
The model
Can describe
◮ motion of flocks, herds (as you have seen...),
Model Stati.Distr. Mean field Fluid limit Questions
The model
Can describe
◮ motion of flocks, herds (as you have seen...),
◮ competing prices of goods (gyros / falafel / shawarma),
Model Stati.Distr. Mean field Fluid limit Questions
The model
Can describe
◮ motion of flocks, herds (as you have seen...),
◮ competing prices of goods (gyros / falafel / shawarma),
◮ prices of stocks,
Model Stati.Distr. Mean field Fluid limit Questions
The model
Can describe
◮ motion of flocks, herds (as you have seen...),
◮ competing prices of goods (gyros / falafel / shawarma),
◮ prices of stocks,
◮ etc.
Model Stati.Distr. Mean field Fluid limit Questions
The model
Can describe
◮ motion of flocks, herds (as you have seen...),
◮ competing prices of goods (gyros / falafel / shawarma),
◮ prices of stocks,
◮ etc.
Found results of the types:
◮ interacting diffusions with linear drift (A. Greven et. al.),
◮ rank dependent drift of Brownian motions (S. Chatterjee, S.
Pal 2007, S. Pal, J. Pitman 2007),
◮ relocation of random walking particles (A. Manita, V.
Shcherbakov 2005),
◮ reordering and steps by a joint Gaussian (A. Ruzmaikina,
M. Aizenman 2005, L.P. Arguin 2008, L.P. Arguin, M.
Aizenman 2009),
◮ multiplicative steps as well (I. Grigorescu, M. Kang 2009).
Model Stati.Distr. Mean field Fluid limit Questions
Stationary distribution
First question: what is the stationary distribution?
Model Stati.Distr. Mean field Fluid limit Questions
Stationary distribution
First question: what is the stationary distribution? As seen
from the center of mass m(t), of course.
Model Stati.Distr. Mean field Fluid limit Questions
Stationary distribution
First question: what is the stationary distribution? As seen
from the center of mass m(t), of course.
n = 2 particles: just an exercise.
Model Stati.Distr. Mean field Fluid limit Questions
Stationary distribution
First question: what is the stationary distribution? As seen
from the center of mass m(t), of course.
n = 2 particles: just an exercise. But I have never before seen
a density like cosh−2 (z) appearing (case ϕ ∼Exp(1) jumps,
w(x) = e−2x jump rates).
Model Stati.Distr. Mean field Fluid limit Questions
Stationary distribution
First question: what is the stationary distribution? As seen
from the center of mass m(t), of course.
n = 2 particles: just an exercise. But I have never before seen
a density like cosh−2 (z) appearing (case ϕ ∼Exp(1) jumps,
w(x) = e−2x jump rates).
n = 3 particles: already seems hopeless. The process is “very
irreversible”.
Model Stati.Distr. Mean field Fluid limit Questions
n = 3 particles, jump lengths are deterministically 1
(2,2,-4)
ω(0)
(0,3,-3)
ω(−2)
(-2,4,-2)
ω(−3)
ω(−3)
(-3,3,0)
ω(0)
(-2,1,1)
ω(−4)
(-4,2,2)
ω(0)
ω(−1)
ω(1)
ω(−3)
(-3,0,3)
ω(−2)
(2,-4,2)
ω(0)
(0,-3,3)
ω(−2)
(-2,-2,4)
ω(0)
ω(−4)
ω(1)
ω(−3)
ω(2)
(3,-3,0)
ω(1)
(1,-2,1)
ω(−1)
(-1,-1,2)
ω(−2)
ω(−3)
ω(−1)
ω(−2)
ω(0)
(4,-2,-2)
ω(2)
(2,-1,-1)
ω(0)
(0,0,0)
ω(−2)
ω(−2)
ω(−3)
ω(−1)
ω(−2)
ω(0)
ω(−1)
ω(1)
ω(1)
(1,1,-2)
ω(−1)
(-1,2,-1)
ω(2)
ω(−2)
ω(−4)
ω(1)
(3,0,-3)
ω(0)
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
̺(x, t)
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
̺(x, t)
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
density at x
̺(x, t)
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
density at x
̺(x, t)
jump rate at y
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
jump rate at y
density at x
̺(x, t)
density at y
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
jump rate at y
density at x
̺(x, t)
density at y
prob to jump to x
x
w(y − m(t)) ·
−∞
̺(y, t)
·
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
density at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
jump rate at y
̺(x, t)
density at y
prob to jump to x
x
w(y − m(t)) ·
̺(y, t)
·
−∞
and
m(t) =
Z
∞
x̺(x, t) dx.
−∞
ϕ(x − y)
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
Take n → ∞, do not rescale space, and first let us guess for a
limiting PDE for the density of particles.
jump rate at x
density at x
∂̺(x, t)
= − w(x − m(t)) ·
∂t
+
Z
jump rate at y
̺(x, t)
density at y
prob to jump to x
x
w(y − m(t)) ·
̺(y, t)
·
ϕ(x − y)
−∞
and
m(t) =
Z
∞
x̺(x, t) dx.
−∞
R
TheseRequations conserve 1 = ̺(x, t) dx and give
ṁ(t) = w(x − m(t)) · ̺(x, t) dx.
dy,
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
We look for stationary solution of this equation as seen from the
center of mass.
Idea: as n → ∞, in a stationary distribution m(t) would
stabilize. So assume
m(t) = ct
and
̺(x, t) = ̺(x − ct).
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
We look for stationary solution of this equation as seen from the
center of mass.
Idea: as n → ∞, in a stationary distribution m(t) would
stabilize. So assume
m(t) = ct
and
̺(x, t) = ̺(x − ct).
Plug this in to get
′
−c̺ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Cases we can solve:
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮ When the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮ When the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx ,
̺(x) = G 1 (const · x),
β
G 1 is the generalized Gumbel density.
β
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮ When the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx ,
̺(x) = G 1 (const · x),
β
◮
G 1 is the generalized Gumbel density.
β
When w is a (down-)step function, ̺ is the Laplace density.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮ When the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx ,
̺(x) = G 1 (const · x),
β
◮
◮
G 1 is the generalized Gumbel density.
β
When w is a (down-)step function, ̺ is the Laplace density.
When w is a (down-)step function, but with a linear
decrease around 0, ̺ is Laplace with a normal segment in
the middle.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
−c̺′ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮ When the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx ,
̺(x) = G 1 (const · x),
β
◮
◮
G 1 is the generalized Gumbel density.
β
When w is a (down-)step function, ̺ is the Laplace density.
When w is a (down-)step function, but with a linear
decrease around 0, ̺ is Laplace with a normal segment in
the middle.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Extreme value statistics (Attila Rákos)
When the jumps are Exp(1): ϕ(x) = e−x ,
jump rate is exponential: w(x) = e−x ,
̺(x) = G(const · x), standard Gumbel density. Why?
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Extreme value statistics (Attila Rákos)
When the jumps are Exp(1): ϕ(x) = e−x ,
jump rate is exponential: w(x) = e−x ,
̺(x) = G(const · x), standard Gumbel density. Why?
Fix a particle X (t). Probability it jumps between t and t + dt is
approx. ect−X (t) dt. And when it jumps, it jumps Exp(1).
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Extreme value statistics (Attila Rákos)
When the jumps are Exp(1): ϕ(x) = e−x ,
jump rate is exponential: w(x) = e−x ,
̺(x) = G(const · x), standard Gumbel density. Why?
Fix a particle X (t). Probability it jumps between t and t + dt is
approx. ect−X (t) dt. And when it jumps, it jumps Exp(1).
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Extreme value statistics (Attila Rákos)
When the jumps are Exp(1): ϕ(x) = e−x ,
jump rate is exponential: w(x) = e−x ,
̺(x) = G(const · x), standard Gumbel density. Why?
Fix a particle X (t). Probability it jumps between t and t + dt is
approx. ect−X (t) dt. And when it jumps, it jumps Exp(1).
Take now more and more iid. Exp(1) variables. At time t, let we
have N(t) = ect /c of them. Define Y (t) as the maximum.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Extreme value statistics (Attila Rákos)
When the jumps are Exp(1): ϕ(x) = e−x ,
jump rate is exponential: w(x) = e−x ,
̺(x) = G(const · x), standard Gumbel density. Why?
Fix a particle X (t). Probability it jumps between t and t + dt is
approx. ect−X (t) dt. And when it jumps, it jumps Exp(1).
Take now more and more iid. Exp(1) variables. At time t, let we
have N(t) = ect /c of them. Define Y (t) as the maximum.
Between t and t + dt, dN(t) = ect dt many new Exp(1) particles
try to break the record. So the probability that Y (t) jumps is
1 − 1 − e−Y (t)
ect dt
≃ ect−Y (t) dt
(for large Y (t)).
And when it jumps, it jumps Exp(1). But we know that
Y (t) − ct + log c converges to standard Gumbel.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
′
−c̺ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
′
−c̺ (x) = −w(x)̺(x) +
Cases we can solve:
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
′
−c̺ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮
Seen: when the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
′
−c̺ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮
Seen: when the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx is exponential: take Fourier transform
to get
ciτ ̺b(τ ) = ϕ(τ
b ) − 1 · ̺b(τ + iβ).
Hope to solve the recurrence relation on the Im line, then
analytic continuation gives a hint on the form of ̺b, to be
verified.
Model Stati.Distr. Mean field Fluid limit Questions
Exponential jumps Extreme val. stat. Fourier methods
Fluid limit: a mean field equation
′
−c̺ (x) = −w(x)̺(x) +
Z
x
w(y)̺(y)ϕ(x − y) dy.
−∞
Cases we can solve:
◮
Seen: when the jumps are Exp(1): ϕ(x) = e−x , the above
becomes a linear second order ODE, easy to solve.
◮
When w(x) = e−βx is exponential: take Fourier transform
to get
ciτ ̺b(τ ) = ϕ(τ
b ) − 1 · ̺b(τ + iβ).
Hope to solve the recurrence relation on the Im line, then
analytic continuation gives a hint on the form of ̺b, to be
verified.
◮
Method tested when ϕ(x) = e−x (also seen before), hope to
work with other ϕ’s too.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
Recall the original fluid equation:
∂̺(x, t)
= −w(x − m(t)) · ̺(x, t)
∂t
Z x
w(y − m(t)) · ̺(y, t) · ϕ(x − y) dy,
+
−∞
or, for all f testfunctions:
hf , µ(t)i − hf , µ(0)i
Z t
=
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
0
m(s) = hx, µ(s)i.
Here E refers to expectation of Z w.r.t. density ϕ.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
Define the n-particle empirical measure µn (t) =
Goal:
1
n
Pn
i=1 δxi (t) .
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
2. Weak limits convergence to a solution µ(·) of the above
equation.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
2. Weak limits convergence to a solution µ(·) of the above
equation.
3. Uniqueness of solutions of the above equation.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i.
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
2. Weak limits convergence to a solution µ(·) of the above
equation.
3. Uniqueness of solutions of the above equation.
Assumptions: the rate function w is bounded; third moment of
the jump distribution ϕ.
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i
!!!
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
2. Weak limits convergence to a solution µ(·) of the above
equation.
3. Uniqueness of solutions of the above equation.
Assumptions: the rate function w is bounded; third moment of
the jump distribution ϕ.
Problem: bounded functions and “just measures” are not
enough!
Model Stati.Distr. Mean field Fluid limit Questions
Taking the fluid limit
The fluid equation:
hf , µ(t)i − hf , µ(0)i
Z t
E[f (x + Z )] − f (x) w(x − m(s)), µ(s) ds,
=
0
m(s) = hx, µ(s)i
!!!
P
Define the n-particle empirical measure µn (t) = n1 ni=1 δxi (t) .
Goal:
1. Tightness of {µn (·)}n≥1 in some path space of measures.
2. Weak limits convergence to a solution µ(·) of the above
equation.
3. Uniqueness of solutions of the above equation.
Assumptions: the rate function w is bounded; third moment of
the jump distribution ϕ.
Dealing with the space of probability measures having first
moments, and the Wasserstein 1 metric seems to work.
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
Complete the fluid limit.
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Stepfunction with linear segment jump rates, exponential
jumps: γ ≃ 1, 1/2 ≤ α ≤ 1.
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Stepfunction with linear segment jump rates, exponential
jumps: γ ≃ 1, 1/2 ≤ α ≤ 1.
In general, limit distribution theorems?
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
◮
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Stepfunction with linear segment jump rates, exponential
jumps: γ ≃ 1, 1/2 ≤ α ≤ 1.
In general, limit distribution theorems?
Can we really not find the stationary distribution for three
goats?
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
◮
◮
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Stepfunction with linear segment jump rates, exponential
jumps: γ ≃ 1, 1/2 ≤ α ≤ 1.
In general, limit distribution theorems?
Can we really not find the stationary distribution for three
goats?
And for the fluid limit, general rate functions / jump
distributions?
Model Stati.Distr. Mean field Fluid limit Questions
Questions
◮
◮
Complete the fluid limit.
Variance of the center of mass should scale:
tγ
Var(mn (t)) ∼ α .
n
Miklós did some (small) simulations. It seems that:
◮
◮
◮
◮
◮
◮
Exponential jump rates, exponential jumps: γ ≃ α ≃ 1.
Stepfunction jump rates, exponential jumps:
γ ≃ 1, 1/2 ≤ α ≤ 1.
Stepfunction with linear segment jump rates, exponential
jumps: γ ≃ 1, 1/2 ≤ α ≤ 1.
In general, limit distribution theorems?
Can we really not find the stationary distribution for three
goats?
And for the fluid limit, general rate functions / jump
distributions?
Thank you.