“Wald’s insight was simply to ask: where are the missing ...kurtz/Lectures/wald.pdfm(t) = number of blues on ﬁrst mlevels Then by exchangeability P(t) = lim m!1 1 m Z m(t) exists,

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Wikipedia

Abraham Wald 1902-1950

“Wald’s insight was simply to ask:where are the missing holes? Theones what would have been all overthe engine casing, if the damagehad been spread equally over theplane? Wald was pretty sure heknew. The missing holes were onthe missing planes.”

How Not to be Wrong

Jordan Ellenberg


2014 Wald Lectures

• Lookdown constructions for Moran models

• Exchangeable and conditionally Poisson constructions

• Particle representations for stochastic partial differential equa-tions

• References

• Abstracts


Lookdown constructions for Moran models

• A construction of a Moran model

• Infinite population limit

• Genealogies and the Kingman coalescent

• The infinite population type process

Donnelly and Kurtz (1996)


Wikipedia

Patrick Moran 1917-1988


Construction of continuous time Moran models

Number the population 1, . . . , n. For 1 ≤ i < j ≤ n, let Lij be inde-pendent unit Poisson processes, and set

Ln(t) =∑

1≤i


Ordered population model

The model (and picture) are the same as above, but now there are nocoins, and if Lij, i < j, jumps, we simply replace j by a copy of i (jlooks down to i and copies its color).

Assume that the initial colors are assigned uniformly at random tolevel i = 1, . . . , n.

Claim: Zn defined by

Zn(t) ≡ number of blue balls at time t

has the same distribution for both constructions.


Infinite population limit

Assume the initial color assignment is iid.

In the ordered construction, adding more levels has no effect on thelower levels.

Let n→∞.


Arbitrary types and mutation

The same construction works for arbitrary description of types andwith mutation

• Infinitely many alleles model (Kimura, Crow)

E = [0, 1]

Every mutant is new, for example, picked with the uniform dis-tribution on [0, 1].

• Infinite sites model (Kimura, Watterson)

E = [0, 1]∞

where mutation is given by

(x1, x2, . . .)→ (ξ, x1, x2, . . .)

ξ picked with the uniform distribution on [0, 1].


Reading off the genealogy

We’ve proved a limit theorem, but what have we proved?

For s < t, let Rk(s, t) be the level at time s of the ancestor of theparticle at level k at time t.

Rk(s−, t) = k −∑i


Exchangeability of the types at time t

Xk(t) the type of the particle at level k at time t.

{Xk(t)} is exchangeable: The empirical measure Zm(t) =∑m

k=1 δXk(t)gives no information about which type is at which level, that is

P{X1(t) = x1, . . . , Xm(t) = xm|Zm(t) =m∑i=1

δxi}

= P{X1(t) = xσ1, . . . , Xm(t) = xσm|Zm(t) =m∑i=1

δxi}


de Finetti’s theorem

X1, X2, . . . is exchangeable if

P{X1 ∈ Γ1, . . . , Xm ∈ Γm} = P{Xs1 ∈ Γ1, . . . , Xsm ∈ Γm}

(s1, . . . , sm) any permutation of (1, . . . ,m).

Theorem 1 (de Finetti) Let X1, X2, . . . be exchangeable. Then there existsa random probability measure Ξ such that for every bounded, measurable g,

limn→∞

g(X1) + · · ·+ g(Xn)n

=

∫g(x)Ξ(dx)

almost surely, and

E[m∏k=1

gk(Xk)|Ξ] =m∏k=1

∫gkdΞ


Infinite population type process

Back to two types:

Zm(t) = number of blues on first m levels

Then by exchangeability

P (t) = limm→∞

1

mZm(t)

exists, so P is a stochastic process giving the fraction of the infinitepopulation that is blue.

More generally, for arbitrary (not quite) type space

limm→∞

1

m

m∑k=1

g(Xk(t)) =

∫E

g(x)V (t, dx)

so V = {V (t), t ≥ 0} is a probability-measure-valued process givingthe distribution of types in the infinite population.


What is V ?

Come back tomorrow, same time, same place.


Second lecture

• Specification of Markov processes by martingale problems.

• Filtering methods–conditional distributions of Markov processes.

• Exchangeable and conditionally Poisson processes.

Third Lecture

• Specification by systems of stochastic differential equations.

• Limit theorems for exchangeable systems.

• Stochastic partial differential equations.


Exchangeable and conditionally Poisson constructions

• Generators for Markov processes

• Generator for the ordered construction

• Averaging out the extra information

• Identifying the infinite population limit

• Models with continuous levels

• Conditionally Poisson random measures

• Construction for general population models


Generators for Markov processesX is an E-valued Markov process if for t, s ≥ 0 and f ∈ B(E),

E[f(X(t+ s))|Ft] = E[f(X(t+ s))|X(t)].

Specify X by its “infinitesimal” behavior

E[f(X(t+ ∆t))|Ft] ≈ f(X(t)) + Af(X(t))∆t

for an appropriate collection of functions f ∈ D(A).

This intuitive description is made precise by the following definition.

Definition 1 X is a solution of the martingale problem for A if and onlythere exists a filtration {Ft} such that

Mf(t) = f(X(t))−∫ t

0

Af(X(s))ds

is an {Ft}-martingale for each f ∈ D(A).


Examples of generators

Standard Brownian motion (E = Rd)

Af =1

2∆f, D(A) = C2c (Rd)

Pure jump process (E arbitrary)

Af(x) = λ(x)

∫E

(f(y)− f(x))µ(x, dy)

ODE Ẋ(t) = F (X(t)) (E = Rd)

Af(x) = F (x) · ∇f(x)

Diffusion (E = Rd)

Af(x) =1

2

∑i,j

aij(x)∂2

∂xi∂xjf(x) +

∑i

bi(x)∂

∂xif(x), D(A) = C2c (Rd)


Sums of generators should be generators

For example,

Af(x) = λ1(x)

∫E

(f(y)− f(x))µ1(x, dy) + λ2(x)∫E

(f(y)− f(x))µ2(x, dy)

= (λ1(x) + λ2(x))( λ1(x)λ1(x) + λ2(x)

∫E

(f(y)− f(x))µ1(x, dy)

+λ2(x)

λ1(x) + λ2(x)

∫E

(f(y)− f(x))µ2(x, dy))

orAf(x) = λ(x)

∫E

(f(y)− f(x))µ(x, dy) + F (x)∇f(x),

piecewise deterministic in the sense of Davis (1993).∫ τk+1τk

λ(X(s))ds = ∆k,

where the ∆k are independent unit exponentials.


A martingale lemmaLet {Ft} and {Gt} be filtrations with Gt ⊂ Ft.

Lemma 2 Suppose U and V are {Ft}-adapted and

U(t)−∫ t

0

V (s)ds

is an {Ft}-martingale. Then

E[U(t)|Gt]−∫ t

0

E[V (s)|Gs]ds

is a {Gt}-martingale.

Proof. The lemma follows by the definition and properties of condi-tional expectations. �


Martingale properties of conditional distributions

Corollary 3 If X is a solution of the martingale problem for A with respectto the filtration {Ft} and πt is the conditional distribution of X(t) givenGt ⊂ Ft, that is,

πtf ≡∫f(x)πt(dx) = E[f(X(t))|Gt],

then

πtf −∫ t

0

πsAfds (1)

is a {Gt}-martingale for each f ∈ D(A).


The generator for the ordered construction

For x = (x1, . . . , xn)

Anf(x) =∑

1≤i


The martingale problem for Zn

By Corollary 3,

πtf −∫ t

0

πsAnfds = αf(Zn(t))−∫ t

0

αAnf(Zn(s))ds

but

αAnf(k) =

(n

2

)[(1−k

n)k

n(αf(k+1)−αf(k))+(1−k

n)k

n(αf(k−1)−αf(k))

].

For f ∈ B({0, . . . , n}), define

Cnf(k) =

(n

2

)[(1− k

n)k

n(f(k+1)−f(k))+(1− k

n)k

n(f(k−1)−f(k))

]Then Zn is a solution of the martingale problem for Cn.


The infinite population Moran model

For x = (x1, x2, . . . , )

Af(x) =∑

1≤i


The conditional distribution given P

The conditional distribution of (X1(t), X2(t), . . .) given FPt is just thedistribution of a sequence of Bernoullis with success probability P (t).If f depends only on the first m variables, then

πtf =∑θ

m∏k=1

P (t)θk(1− P (t))1−θkf(θ1, . . . , θm) = αf(P (t))

where the sum is over all choices of θ = (θ1, . . . , θm) ∈ {0, 1}m.


Martingale problem for P

Let f(x) depend only on (x1, . . . , xm) and let |θ|m =∑m

k=1 θk. then

αf(p) =∑θ

p|θ|m(1− p)m−|θ|mf(θ1, . . . , θm)

As before, we need to compute αAf for

Af(x) =∑

1≤i


A converse to Corollary 3

Theorem 4 Kurtz and Nappo (2011) Let A ⊂ Cb(E)× Cb(E) satisfy cer-tain technical conditions, most significantly thatD(A) be closed under mul-tiplication and separate points. Suppose that {πt} is a P(E)-valued processadapted to a filtration {Gt} such that

πtf −∫ t

0

πsAfds

is a {Gt}-martingale for each f ∈ D(A). Then (essentially), there exists aprocess X that is a solution of the martingale problem for A satisfying

πtf = E[f(X(t))|Gt], f ∈ B(E).


A population model Kurtz and Rodrigues (2011)Consider a process with state space E = ∪n[0, r]n mod order.

0 ≤ g ≤ 1, g(r) = 1, and f(u, n) =∏n

i=1 g(ui) For a > 0, and −∞ <b ≤ ra, define

Arf(u, n) = f(u, n)n∑i=1

2a

∫ rui

(g(v)−1)dv+f(u, n)n∑i=1

(au2i−bui)g′(ui)

g(ui).

In other words, particle levels satisfy

U̇i(t) = aU2i (t)− bUi(t),

and a particle with level z gives birth at rate 2a(r − z) to a particlewhose initial level is uniformly distributed between z and r.

N(t) = #{i : Ui(t) < r}

αr(n, du) the joint distribution of n iid uniform [0, r] random vari-ables.


A calculation

f̂(n) =∫f(u, n)αr(n, du) = e

−λgn, e−λg = 1r∫ r

0 g(u)du

To calculate Cf̂(n) =∫Arfu, n)αr(n, du), observe that

r−12a

∫ r0

g(z)

∫ rz

(g(v)− 1)dv = are−2λg − 2ar−1∫ r

0

g(z)(r − z)dz

and

r−1∫ r

0

(az2 − bz)g′(z)dz = −r−1∫ r

0

(2az − b)(g(z)− 1)dz

= −2ar−1∫ r

0

zg(z)dz + ar + b(e−λg − 1).

Then

Cf̂(n) = ne−λg(n−1)(are−2λg − 2are−λg + ar + b(e−λg − 1)

)= arn(f̂(n+ 1)− f̂(n)) + (ar − b)n(f̂(n− 1)− f̂(n)).


Conclusion

Let N be a solution of the martingale problem for

Cf̂(n) = arn(f̂(n+ 1)− f̂(n)) + (ar − b)n(f̂(n− 1)− f̂(n)),

that is, N is a branching process with birth rate ar and death rate(ar − b). Then for f ∈ D(A) and πt(du) = αr(N(t), du),

πtf −∫ t

0

πsArfds = f̂(N(t))−∫ t

0

Cf̂(N(s))ds

is a {FNt }-martingale. Then there exists a solution

(U1(t), . . . , UN(t)(t), N(t))

of the martingale problem for A such that

E[

N(t)∏i=1

g(Ui(t))|FNt ] =∫ N(t)∏

i=1

g(ui)α(N(t), du)


Extinction for branching processLet U∗(0) be the minimum of U1(0), . . . , UN(0). Then for all t, all levelsare above

U∗(t) =U∗(0)e

−bt

1− abU∗(0)(1− e−bt)Let τ = inf{t : N(t) = 0}. If τ is finite, U∗(τ) = r. If N(0) = n, then

P{τ > t} = P{U∗(t) < r} = P{U∗(0) <r

e−bt − rab (e−bt − 1)}

= 1− (1− 1e−bt − rab (e−bt − 1)

)n.

Note that the assumption that b ≤ ra ensures that

e−bt − rab

(e−bt − 1) ≥ 1.


The limit as r →∞For f(u) =

∏i g(ui), 0 ≤ g ≤ 1, g(z) = 1, z ≥ ug, Arf converges to

Af(u) = f(u)∑i

2a

∫ ugui

(g(v)− 1)dv + f(u)∑i

(au2i − bui)g′(ui)

g(ui).

If nr−1 → y, then αr(n, du) → α(y, du) where α(y, du) is the distribu-tion of a Poisson process on [0,∞) with intensity y.

f̂(y) = αf(y) =

∫f(u)α(y, du) = e−y

∫∞0

(1−g(z))dz = e−yβg

and

αAf(y) = e−yβg(

2ay∞∫0g(z)

∞∫z

(g(v)− 1)dvdz + y∞∫0

(az2 − bz)g′(z)dz)

= e−yβg(ayβ2g − byβg)= ayf̂ ′′(y) + byf̂ ′(y)

≡ Cf̂(y)


Particle representation of Feller diffusion

If Y is a solution of the martingale problem for C and

πtf =

∫f(u)α(Y (t), du),

then

πtf −∫ t

0

πsAfds = f̂(Y (s))−∫ t

0

Cf̂(Y (s))ds

is a martingale with respect to {FYt }. Consequently, there exists asolution of the martingale problem for A such that conditioned onFYt , {Ui(t)} is a Poisson process with intensity Y (t). In particular,

Y (t) = limr→∞

1

r#{i : Ui(t) ≤ r}.

Exercise: Let τ = inf{t : Y (t) = 0}. Compute P{τ


Branching Markov processesf(x, u, n) =

∏ni=1 g(xi, ui), where g : E × [0,∞)→ (0, 1]

As a function of x, g is in the domain D(B) of the generator of aMarkov process inE, g is continuously differentiable in u, and g(x, u) =1 for u ≥ r.

Af(x, u, n) = f(x, u, n)n∑i=1

Bg(xi, ui)

g(xi, ui)

+f(x, u, n)n∑i=1

2a(xi)

∫ rui

(g(xi, v)− 1)dv

+f(x, u, n)n∑i=1

(a(xi)u2i − b(xi)ui)

∂uig(xi, ui)

g(xi, ui)

Each particle has a location Xi(t) in E and a level Ui(t) in [0, r].


Behavior of the process

The locations evolve independently as Markov processes with gen-erator B, the levels satisfy

U̇i(t) = a(Xi(t))U2i (t)− b(Xi(t))Ui(t)

and particles that reach level r die.

Particles give birth at rates 2a(Xi(t))(r − Ui(t)); the initial location ofa new particle is the location of the parent at the time of birth; andthe initial level is uniformly distributed on [Ui(t), r].


Generator for X(t) = (X1(t), . . . , XN(t))

Setting e−λg(xi) = r−1∫ r

0 g(xi, z)dz and f̂(x, n) = e−∑n

i=1 λg(xi), and cal-culating as in the previous example, we have

Cf̂(x, n) =n∑i=1

Bxif̂(x, n) +n∑i=1

a(xi)r(f̂((x, xi), n+ 1)− f̂(x, n))

+n∑i=1

(a(xi)r − b(xi))(f̂(d(x|xi), n− 1)− f̂(x, n)),

whereBxi is the generatorB applied to f̂(x, n) as a function of xi andd(x|xi) is the vector obtained from x be eliminating the ith compo-nent.


Infinite population limit

Letting r →∞, Af becomes

Af(x, u) = f(x, u)∑i

Bg(xi, ui)

g(xi, ui)+ f(x, u)

∑i

2a(xi)

∫ ugui

(g(xi, v)− 1)dv

+f(x, u)∑i

(a(xi)u2i − b(xi)ui)

∂uig(xi, ui)

g(xi, ui)

Particle locations evolve as independent Markov processes with gen-erator B, and levels satisfy

U̇i(t) = a(Xi(t))U2i (t)− b(Xi(t))Ui(t)

A particle with level Ui(t) gives birth to new particles at its loca-tion Xi(t) and initial level in the interval [Ui(t) + c, Ui(t) + d] at rate2a(Xi(t))(d− c).

A particle dies when its level hits∞.


The measure-valued limit

For µ ∈ Mf(E), let α(µ, dx × du) be the distribution of a Poissonrandom measure on E × [0,∞) with mean measure µ × m. Thensetting h(y) =

∫∞0 (1− g(y, v))dv

αf(µ) =

∫f(x, u)α(µ, dx× du) = exp{− ∫

Eh(y)µ(dy)},

and

αAf(µ) = exp{− ∫Eh(y)µ(dy)}[

∫E

∫ ∞0

Bg(y, v)dvµ(dy)

+

∫E

∫ ∞0

2a(y)g(y, z)

∫ ∞z

(g(y, v)− 1)dvdzµ(dy)

+

∫E

∫ ∞0

(a(y)v2 − b(y)v)∂vg(y, v)dvµ(dy)]

= exp{− ∫Eh(y)µ(dy)}

∫E

(−Bh(y) + a(y)h(y)2 − b(y)h(y)

)µ(dy)


Conditionally Poisson systems

Let ξ be a random counting measure on S and Ξ be a locally finiterandom measure on S.

ξ is conditionally Poisson with Cox measure Ξ if, conditioned on Ξ, ξis a Poisson point process with mean measure Ξ.

E[e−∫Sfdξ] = E[e−

∫S

(1−e−f )dΞ]

for all nonnegative f ∈M(S).


Particle representations of random measuresIf ξ is conditionally Poisson system on S × [0,∞) with Cox measureΞ×m where m is Lebesgue measure, then for f ∈M(S)

E[e−∫S×[0,K] fdξ] = E[e−K

∫S

(1−e−f )dΞ]

and for f ≥ 0,

Ξ(f) = limK→∞

1

K

∫S×[0,K]

fdξ = lim�→0

�

∫S×[0,∞)

e−�uf(x)ξ(dx× du) a.s.


Representation of the Dawson-Watanabe process

It follows that the Cox measure (or more precisely, the E marginal ofthe Cox measure) corresponding to the particle process at time t, callit Z(t), is a solution of the martingale problem for

A = {(αf, αAf) : f ∈ D}.


You can do this at home: Justification of representation

A the generator of model with assigned levels (typically integer-valued, [0, r]-valued, or [0,∞)-valued).

α(y, du) the distribution of level assignments given population modelstate y.

C the generator of the desired population model (must satisfy)

Cαf = αAf

Then if Y is a solution of the martingale problem forC, πt = α(Y (t), du)is the conditional distribution of the levels given observations of Y .

cf. Rogers and Pitman (1981)


You can do this at home: Constructing AEtheridge and Kurtz (2014)

Independent thinning

A collection of particles with types in E, x1, x2, . . .

Each particle xi has a level ui uniformly distributed on [0, r].

A function ρ : E → [1,∞).

Update levels: u′i = ρ(xi)ui and eliminate all particles with u′i ≥ r.

P{u′i ≥ r} = P{ui ≥r

ρ(xi)} = 1− 1

ρ(xi),

so take ρ(x) = 11−p(x) , if you want the probability of death to be p(x).

Note that conditioned on u′i < r, u′i is uniformly distributed on [0, r].


A birth event with k − 1 offspring

Let σ(x) ≥ 0 control the chances of an individual of type x becominga parent.

Select k points independently and uniformly in [0, r].

Let v∗ denote the minimum of the k new levels.

If σ(xi) > 0, let τui be determined by

e−σ(xi)τui =r − uir − v∗

if ui > v∗, and e−σ(xi)τui =uiv∗

if ui < v∗.

Note that in both cases, τui is exponentially distributed with param-eter σ(xi).

Taking (x∗, u∗) to be the point satisfying τu∗ = min(xi,ui) τui, we have

P{(x∗, u∗) = (xi, ui)} =σ(xi)∑j σ(xj)

.


Revision of the levels

After the event, the new levels for ui 6= u∗ are

u′i = r − (r − ui)eσ(xi)τu∗ if ui > v∗

andu′i = uie

σ(xi)τu∗ .

Conditioned on ui 6= u∗, u′i is uniformly distributed on [0, r].

Exercise: What happens as r →∞ and kr → z?


Related constructions: Integer levels

Donnelly and Kurtz (1999a) General neutral population models (death and off-spring production does not depend on type).

Donnelly and Kurtz (1999b) Models with selection and recombination (Krone andNeuhauser ancestor selection graph, Griffiths recombination graph).

Birkner, Blath, Möhle, Steinrücken, and Tams (2009) Models with multiple parentsproducing offspring simultaneously.

Temple (2010) Models with dependent motion.

Gupta (2012) Model of cell polarity.

Hénard (2013) Change of measure.

Berestycki, Berestycki, and Limic (2014) Relationship of Λ-coalescents and contin-uous state branching processes.


Related constructions: Continuous levels

Greven, Limic, and Winter (2005) Spatial Moran model in which reproductionevents occur in separate colonies with migration among the colonies.

Veber and Wakolbinger (2013) Lookdown construction for the Barton-Etheridgemodel of evolution driven by local catastrophes.


Particle representations for stochastic partialdifferential equations

• Exchangeability and de Finetti’s theorem

• Convergence of exchangeable systems

• From particle approximation to particle representation

• Derivation of SPDE

• Vanishing spatial noise correlations

• Stochastic Allen-Cahn equation

• Particle representation

• Boundary conditions

• Tightness in D of approximations

• Uniqueness

• References

Dan Crisan, Chris Janjigian, Peter Donnelly, Phil Protter, Jie Xiong, Yoonjung Lee, Peter Kotelenez


Exchangeability and de Finetti’s theoremX1, X2, . . . is exchangeable if

P{X1 ∈ Γ1, . . . , Xm ∈ Γm} = P{Xs1 ∈ Γ1, . . . , Xsm ∈ Γm}

(s1, . . . , sm) any permutation of (1, . . . ,m).

Theorem 1 (de Finetti) Let X1, X2, . . . be exchangeable. Then there existsa random probability measure Ξ such that for every bounded, measurable g,

limn→∞

g(X1) + · · ·+ g(Xn)n

=

∫g(x)Ξ(dx)

almost surely, and

E[m∏k=1

gk(Xk)|Ξ] =m∏k=1

∫gkdΞ


Convergence of exchangeable systemsKotelenez and Kurtz (2010)

Lemma 2 For n = 1, 2, . . ., let {ξn1 , . . . , ξnNn} be exchangeable (allowingNn = ∞.) Let Ξn be the empirical measure (defined as a limit if Nn = ∞),Ξn = 1Nn

∑Nni=1 δξni . Assume

• Nn →∞• For each m = 1, 2, . . ., (ξn1 , . . . , ξnm)⇒ (ξ1, . . . , ξm) in Sm.

Then

{ξi} is exchangeable and setting ξni = s0 ∈ S for i > Nn, {Ξn, ξn1 , ξn2 . . .} ⇒{Ξ, ξ1, ξ2, . . .} in P(S)× S∞, where Ξ is the de Finetti measure for {ξi}.

If for eachm, {ξn1 , . . . , ξnm} → {ξ1, . . . , ξm} in probability in Sm, then Ξn →Ξ in probability in P(S).


Lemma 3 Let Xn = (Xn1 , . . . , XnNn) be exchangeable families of DE[0,∞)-valued random variables such that Nn ⇒∞ and Xn ⇒ X in DE[0,∞)∞.Define

Ξn = 1Nn∑Nn

i=1 δXni ∈ P(DE[0,∞))

Ξ = limm→∞1m

∑mi= δXi

V n(t) = 1Nn∑Nn

i=1 δXni (t) ∈ P(E)

V (t) = limm→∞1m

∑mi=1 δXi(t)

Then

a) For t1, . . . , tl /∈ {t : E[Ξ{x : x(t) 6= x(t−)}] > 0}(Ξn, V

n(t1), . . . , Vn(tl))⇒ (Ξ, V (t1), . . . , V (tl)).

b) If Xn ⇒ X in DE∞[0,∞), then V n ⇒ V in DP(E)[0,∞). If Xn → Xin probability in DE∞[0,∞), then V n → V in DP(E)[0,∞) in proba-bility.


From particle approximation to particle representation

Xni (t) = Xni (0) +Bi(t) +W (t) +

1

n

n∑j=1

∫ t0

b(Xni (s)−Xnj (s))ds

= Xni (0) +Bi(t) +W (t) +

∫ t0

∫Rb(Xni (s)− z)V n(s, dz)ds

V n(t) =1

n

n∑i=1

δXni (t)

If b is bounded and continuous and {Xni (0)} ⇒ {Xi(0)}, then relativecompactness is immediate and any limit point satisfies

Xi(t) = Xi(0) +Bi(t) +W (t) +

∫ t0

∫Rb(Xi(s)− z)V (s, dz)ds

Assuming uniqueness for the infinite system, V n ⇒ V where V (t) isthe de Finetti measure for {Xi(t)}.


Derivation of SPDE

Applying Itô’s formula

ϕ(Xi(t)) = ϕ(Xi(0)) +

∫ t0

ϕ′(Xi(s))dBi(s) +

∫ t0

ϕ′(Xi(s))dW (s)

+

∫ t0

L(V (s))ϕ(Xi(s))ds

whereL(v)ϕ(x) = ϕ′′(x) +

∫b(x− z)v(dz)ϕ′(x).

Averaging gives

〈V (t), ϕ〉 = 〈V (0), ϕ〉+∫ t

0

〈V (s), ϕ′〉dW (s) +∫〈V (s), L(V (s))ϕ(·)〉ds


Gaussian white noise

W (du×ds) will denote Gaussian white noise on U×[0,∞) with meanzero and variance measure µ(du)ds.

For example, W (C× [0, t]), t ≥ 0, is Brownian motion with mean zeroand variance µ(C).

For appropriately adapted and integrable Z,

MZ(t) =

∫U×[0,t]

Z(u, s)W (du× ds)

is a square integrable martingale with quadratic variation

[MZ ]t =

∫U×[0,t]

Z(u, s)2µ(du)ds.


Coupling through the center of mass

Let Xn(t) = 1n

∑ni=1Xi(t) =

∫Rd xV

n(t, dx) for

Xni (t) = Xi(0) +

∫ t0

σ(Xni (s), Xn(s))dBi(s) +

∫ t0

b(Xi(s), Xn(s))ds

+

∫U×[0,t]

α(Xni (s), Xn(s), u)W (du× ds)

Passing to the limit

Xi(t) = Xi(0) +

∫ t0

σ(Xi(s), X(s))dBi(s) +

∫ t0

b(Xi(s), X(s))ds

+

∫U×[0,t]

α(Xi(s), X(s), u)W (du× ds)

where Xn ⇒ X ,

X(t) = limm→∞

1

m

m∑i=1

Xi(t) =

∫RdxV (t, dx).


Wasserstein Lipschitz conditionDonnelly and Kurtz (1999a); Kurtz and Protter (1996); Kurtz andXiong (1999)

Xni (t) = Xni (0) +

∫ t0

σ(Xni (s), Vn(s))dBi(s) +

∫ t0

b(Xni (s), Vn(s))ds

+

∫U×[0,t]

α(Xni (s), Vn(s), u)W (du× ds)

Wasserstein metric on {ν ∈ P(Rd) :∫|x|ν(dx)


Convergence

Lemma 4 If supnE[|Xni (0)|2]


Derivation of SPDEApplying Itô’s formula

ϕ(Xi(t)) = ϕ(Xi(0)) +

∫ t0

∇ϕ(Xi(s))Tσ(Xi(s), V (s))dBi(s)

+

∫U×[0,t]

∇ϕ(Xi(s)) · α(Xi(s), V (s), u)W (du× ds)

+

∫ t0

L(V (s))ϕ(Xi(s))ds

where for

a(x, ν) = σ(x, ν)σ(x, ν)T +

∫α(x, ν, u)α(x, ν, u)Tµ(du)

L(ν)ϕ(x) =1

2

∑i,j

aij(x, ν)∂i∂jϕ(x) + b(x, ν) · ∇ϕ(x).


The SPDE

Averaging gives

〈V (t), ϕ〉 = 〈V (0), ϕ〉+∫U×[0,t]

〈V (s), α(·, V (s), u) · ∇ϕ〉W (du× ds)

+

∫ t0

〈V (s), L(V (s))ϕ(·)〉ds

where V (t) = limk→∞ 1k∑k

i=1 δXi(t) and X(t) =∫zV (t, dz).

The equation is the weak form of

dv(t, x) = v(t, x)

∫Uα(x, V (t), u)W (du× dt) + L∗(V (t))v(t, x)


Vanishing spatial noise correlations

Let d ≥ 2, U = Rd, µ be Lebesgue measure, and σ = 0, and

αε(x, ν, u) = ε−d/2α(x, ε−1(x− u), ν).

The SPDE

〈Vε(t), ϕ〉 = 〈V (0), ϕ〉+∫U×[0,t]

〈Vε(s), αε(·, Vε(s), u) · ∇ϕ〉W (du× ds)

+

∫ t0

〈Vε(s), Lε(Vε(s))ϕ(·)〉ds

is represented by

Xε,i(t) = Xi(0) +

∫ t0

b(Xε,i(s), Vε(s))ds

+

∫U×[0,t]

αε(Xε,i(s), Vε(s), u)W (du× ds)


Change of variableThe stochastic integral can be written∫

Rd×[0,t]αε(Xε,i(s), Vε(s)), u〉W (du× ds)

=

∫Rd×[0,t]

ε−d/2α(Xε,i(s), ε−1(Xε,i(s)− u), Vε(s))W (du× ds)

=

∫Rd×[0,t]

α(Xε,i(s), z, Vε(s))Wεi (dz × ds),

where for each i, W εi is a Gaussian white noise defined by∫Rd×[0,∞)

ϕ(z, s)W εi (dz×ds) =∫Rd×[0,t]

ε−d/2ϕ(ε−1(Xε,i(s)−u), s)〉W (du×ds)

(NOTE: The W εi are not independent but are exchangeable.)


Convergence

If∫Rd |α(x, z, ν)|

2dz 0, foreach δ > 0, and on compact subsets of P(Rd).

Assume the nondegeneracy condition

infx,ν

infz

∫Rd(z · α(x, z, ν))

2du

|z|2> 0.


The zero correlation limit Kotelenez and Kurtz (2010)

Theorem 5 Assume α and b are bounded, V (0) has no atoms, and an ad-ditional regularity condition for d = 2. As ε → 0, Xε converges in distri-bution to the solution of

Xi(t) = Xi(0)+

∫Rd×[0,t]

α(Xi(s), u, V (s))Wi(du×ds)+∫ t

0

b(Xi(s), V (s))ds

where theWi are independent and V (t) is the de Finetti measure for {Xi(t)}.

V is the unique solution of

〈V (t), ϕ〉 = 〈V (0), ϕ〉+∫ t

0

〈V (s), Lϕ(·, V (s))〉ds

where Lϕ(x, ν) = 12∑

ij aij(x, ν)∂i∂jϕ(x) + b(x, ν) · ∇ϕ(x)

a(x, ν) =

∫Uα(x, u, ν)α(x, u, ν)Tdu.


Sketch of proof

Xε,i(t) = Xi(0) +

∫U×[0,t]

ε−d/2α(Xε,i(s),Xε,i − u

ε, Vε(s))〉W (du× ds)

+

∫ t0


= Xi(0) +

∫Rd×[0,t]

α(Xε,i(s), u, Vε(s))〉W εi (du× ds) +∫ t

0


where

Mϕ,εi (t) =

∫Rd×[0,t]

ϕ(u)W εi (du×ds) =∫Rd×[0,t]

ε−d/2ϕ(Xε,i(s)− u

ε)W (du×ds)

Relative compactness follows from boundedness of α and b and

[Mϕ,εi ,Mψ,εj ]t =

∫Rd×[0,t]

ε−d/2ϕ(Xε,i(s)− u

ε)ε−d/2ψ(

Xε,j(s)− uε

)du→ 0

for t < τij = inf{t : Xi(t) = Xj(t)}. If τij = ∞ a.s., then the limits Wiand Wj are independent.


Stochastic Allen-Cahn equation (Crisan, Janjigian, in progress)

Consider a family of SPDEs of the form

dv = ∆vdt+ F (v)dt+ noise,v(0, x) = h(x), x ∈ D,v(t, x) = g(x), x ∈ ∂D, t > 0,

where F (v) = G(v)v and G is bounded above. For example,

F (v) = v − v3 = (1− v2)v.To be specific, in weak form the equation is

〈V (t), ϕ〉 = 〈V (0), ϕ〉+∫ t

0

〈V (s),∆ϕ〉ds+∫ t

0

〈V (s), ϕG(v(s, ·))〉ds

+

∫U×[0,t]

∫D

ϕ(x)ρ(x, u)dxW (du× ds),

for ϕ ∈ C2c (D). cf. Bertini, Brassesco, and Buttà (2009)


Is it a nail?{Xi} independent, stationary, reflecting Brownian motions in D.

dAi(t) = G(v(t,Xi(t))Ai(t)dt+

∫Uρ(Xi(t), u)W (du× dt)

Ai(0) = h(Xi(0))

If Xi hits the boundary at time t, Ai(t) is reset to g(Xi(t)).

V (t) = limk→∞1k

∑ki=1Ai(t)δXi(t)

〈V (t), ϕ〉 =∫D ϕ(x)v(t, x)π(dx) where π is the stationary distribution

for Xi (normalized Lebesgue measure on D).


Particle representationMore generally, let B be the generator of a reflecting diffusion X inD and assume that X is ergodic with stationary distribution π. Let{Xi, i ≥ 1} be independent, stationary diffusions with generator B.

Assume that the boundary of D is regular for both Xi and the timereversal of Xi. Let τi(t) = 0 ∨ sup{s < t : Xi(s) ∈ ∂D, and

Ai(t) = g(Xi(τi(t)))1{τi(t)>0} + h(Xi(0))1{τi(t)=0} (1)

+

∫ tτi(t)

G(v(s,Xi(s)), Xi(s))Ai(s)ds+

∫ tτi(t)

b(Xi(s))ds

+

∫U×(τi(t),t]

ρ(Xi(s), u)W (du× ds),

where

〈V (t), ϕ〉 = limn→∞

1

n

n∑i=1

ϕ(Xi(t))Ai(t) =

∫ϕ(x)v(t, x)π(dx).


Corresponding SPDE

Define Mϕ,i(t) = ϕ(Xi(t))−∫ t

0 Bϕ(Xi(s))ds.

Then

ϕ(Xi(t))Ai(t) = ϕ(Xi(0))Ai(0) +

∫ t0

ϕ(Xi(s))dAi(s)

+

∫ t0

Ai(s)dMϕ,i(s) +

∫ t0

Bϕ(Xi(s))Ai(s)ds

= ϕ(Xi(0))Ai(0) +

∫ t0

ϕ(Xi(s))G(v(s,Xi(s)), Xi(s))Ai(s)ds

+

∫ t0

ϕ(Xi(s))b(Xi(s))ds

+

∫U×[0,t]

ϕ(Xi(s))ρ(Xi(s), u)W (du× ds)

+

∫ t0

Ai(s)dMϕ,i(s) +

∫ t0

Bϕ(Xi(s))Ai(s)ds


Averaging

〈V (t), ϕ〉 = 〈V (0), ϕ〉+∫ t

0

〈V (s), ϕG(v(s, ·), ·)〉ds+∫ t

0

∫bϕdπds

+

∫U×[0,t]

∫D

ϕ(x)ρ(x, u)π(dx)W (du× ds) +∫ t

0

〈V (s), Bϕ〉ds

which is the weak form of

v(t, x) = v(0, x) +

∫ t0

(G(v(s, x), x)v(s, x) + b(x))ds

+

∫U×[0,t]

ρ(x, u)W (du× ds) +∫ t

0

B∗v(x, s)ds,

where B∗ is the adjoint determined by∫gBfdπ =

∫fB∗gdπ.


Approximating systemsLet ψ be an L1(π)-valued stochastic process that is compatible withW , and assume (W,ψ) is independent of {Xi} Define Aψi to be thesolution of

Aψi (t) = g(Xi(τi(t)))1{τi(t)>0} + h(Xi(0))1{τi(t)=0}

+

∫ tτi(t)

G(ψ(s,Xi(s)), Xi(s))Aψi (s)ds+

∫ tτi(t)

b(Xi(s))ds

+

∫U×(τi(t),t]

ρ(Xi(s), u)W (du× ds).

The {Aψi } will be exchangeable, so we can define Φψ(t, x) to be thedensity of the signed measure determined by

〈ΦΨ(t), ϕ〉 ≡∫D

ϕ(x)Φψ(t, x)π(dx) = limN→∞

1

N

N∑i=1

Aψi (t)ϕ(Xi(t)).


A priori boundsAssume

K1 ≡ supx,D|b(x)|


Lemma 7 Suppose that (W,ψ) is independent of {Xi}. Then Φψ is {FW,ψt }-adapted and for each i,

E[Aψi (t)|W,ψ,Xi(t)] = Φψ(t,Xi(t))

soΦψ(t,Xi(t)) ≤ E[Γi(t)|W,ψ,Xi(t)]

Remark 8 Let GXit = σ(Xi(r) : r ≥ t). Then the Markov property and theindependence of (W,ψ) and Xi imply

Φψ(t,Xi(t)) = E[Aψi (t)|W,ψ,Xi(t)] = E[A

ψi (t)|σ(W,ψ) ∨ G

Xit ].

The properties of reverse martingales and Doob’s inequality give

E[ sup0≤t≤T

|Φψ(t,Xi(t))|2] ≤ 4E[ sup0≤t≤T

|Aψi (t)|2] ≤ 4E[ sup

0≤t≤TΓi(t)

2]


Proof. By exchangeability,

E[Aψi (t)ϕ(Xi(t))F (W,ψ)] = E[

∫ϕ(x)ΦΨ(t, dx)F (W,ψ)]

= E[

∫ϕ(x)Φψ(t, x)π(dx)F (W,ψ)]

= E[ϕ(Xi(t))Φψ(t,Xi(t))F (W,ψ)].

The last equality follows by the independence of Xi(t) and (W,ψ),and the lemma follows by the definition of conditional expectation.�


Boundary conditionsRecall Φψ(t, x) is the density of the signed measure determined by

〈ΦΨ(t), ϕ〉 = limN→∞

1

N

N∑i=1

Aψi (t)ϕ(Xi(t)).

If Xi(t) is close to ∂D, then by the regularity assumption, with highprobability t − τi(t) is small and Aψi (t) ≈ g(Xi(τi(t))). Consequently,for y ∈ ∂D

Φψ(t, y) = lim�→0

ΦΨ(B�(y))

π(B�(y))= g(y).


Tightness in D of approximations

Aψi (t) = g(Xi(τi(t)))1{τi(t)>0} + h(Xi(0))1{τi(t)=0}

+

∫ tτi(t)

G(ψ(s,Xi(s)), Xi(s))Aψi (s)ds+

∫ tτi(t)

b(Xi(s))ds

+

∫U×(τi(t),t]


Let

Zψi (t) = −g(Xi(t)) +∫ t

0

G(ψ(s,Xi(s)), Xi(s))Aψi (s)ds

+

∫ t0

b(Xi(s))ds+

∫U×[0,t]


Then

Aψi (t) = g(Xi(t)) + Zψi (t)− Z

ψi (τi(t)) + (h(Xi(0))− g(Xi(0)))1{τi(t)=0}.


Estimates on modulus of continuity

The Skorohod modulus of continuity of Aψi can be bounded in termsof the ordinary modulus of continuity of Zψi .

Lemma 9 Define γ0i = inf{t : Xi(t) ∈ ∂D}. Then with probability one,γ0i > 0, and for δ < γ0i ,

w′(Aψi , δ, T ) ≤ w(g ◦Xi, 4δ, T ) + 2w(Zψi , 4δ, T ). (2)

The relative compactness for {Aψi } for fixed i then follows from the relativecompactness of {Zψi } in CRd[0,∞).


UniquenessL1 ≡ supv,x∈D

|G(v,x)|1+|v|2 C}Γi(s)L3(1 + E[Γi(s)|W,Xi(s)]2)ds


Uniqueness for nonlinear SPDE

Theorem 10 Uniqueness for the linear infinite system and the nonlinearinfinite system and uniqueness for the linear SPDE

〈V ψ(t), ϕ〉 = 〈V (0), ϕ〉+∫ t

0

〈V ψ(s), ϕG(ψ(s, ·), ·)〉ds+∫ t

0

∫bϕdπds

+

∫U×[0,t]

∫D

ϕ(x)ρ(x, u)π(dx)W (du× ds) +∫ t

0

〈V ψ(s), Bϕ〉ds

implies uniqueness for the nonlinear SPDE.

Proof. Suppose ψ is a solution of the nonlinear SPDE. Use ψ as theinput into the linear infinite system. Uniqueness of the linear infinitesystem implies Φψ is a solution of the linear SPDE, but ψ is also asolution of the linear SPDE, so ψ = Φψ and uniqueness of the non-linear infinite system implies there is only one such ψ. (See Section 3of Kurtz and Xiong (1999).) �


Additional references

Bush, Hambly, Haworth, Jin, and Reisinger (2011) A model of credit risk.

Crisan, Kurtz, and Lee (2011) Absolute continuity and positivity of measure-valuedsolutions.

Kwon and Kang (2005) Equations with jumps.

Kurtz and Xiong (2004) Central limit theorem for finite particle approximations.


ReferencesJulien Berestycki, Nathanaël Berestycki, and Vlada Limic. A small-time coupling between Λ-coalescents and

branching processes. Ann. Appl. Probab., 24(2):449–475, 2014. ISSN 1050-5164. doi: 10.1214/12-AAP911.URL http://dx.doi.org.ezproxy.library.wisc.edu/10.1214/12-AAP911.

Lorenzo Bertini, Stella Brassesco, and Paolo Buttà. Boundary effects on the interface dynamics for the stochas-tic AllenCahn equation. In Vladas Sidoravičius, editor, New Trends in Mathematical Physics, pages 87–93.Springer, 2009.

Matthias Birkner, Jochen Blath, Martin Möhle, Matthias Steinrücken, and Johanna Tams. A modified look-down construction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottle-necks. ALEA Lat. Am. J. Probab. Math. Stat., 6:25–61, 2009. ISSN 1980-0436.

N. Bush, B. M. Hambly, H. Haworth, L. Jin, and C. Reisinger. Stochastic evolution equations in portfolio creditmodelling. SIAM J. Financial Math., 2(1):627–664, 2011. ISSN 1945-497X. doi: 10.1137/100796777. URLhttp://dx.doi.org.ezproxy.library.wisc.edu/10.1137/100796777.

Dan Crisan, Thomas G. Kurtz, and Yoonjung Lee. Conditional distributions, exchangeable particle sys-tems, and stochastic partial differential equations. Submitted, 2011. URL http://www.math.wisc.edu/˜kurtz/papers/crikurlee.pdf.

M. H. A. Davis. Markov models and optimization, volume 49 of Monographs on Statistics and Applied Probability.Chapman & Hall, London, 1993. ISBN 0-412-31410-X.

Peter Donnelly and Thomas G. Kurtz. A countable representation of the Fleming-Viot measure-valued diffu-sion. Ann. Probab., 24(2):698–742, 1996. ISSN 0091-1798.

Peter Donnelly and Thomas G. Kurtz. Particle representations for measure-valued population models. Ann.Probab., 27(1):166–205, 1999a. ISSN 0091-1798.

http://dx.doi.org.ezproxy.library.wisc.edu/10.1214/12-AAP911http://dx.doi.org.ezproxy.library.wisc.edu/10.1137/100796777http://www.math.wisc.edu/~kurtz/papers/crikurlee.pdfhttp://www.math.wisc.edu/~kurtz/papers/crikurlee.pdf


Peter Donnelly and Thomas G. Kurtz. Genealogical processes for Fleming-Viot models with selection andrecombination. Ann. Appl. Probab., 9(4):1091–1148, 1999b. ISSN 1050-5164.

Alison Etheridge and Thomas G. Kurtz. Genealogical constructions of population models. 2014. URL http://arxiv.org/abs/1402.6724.

Andreas Greven, Vlada Limic, and Anita Winter. Representation theorems for interacting Moran models,interacting Fisher-Wright diffusions and applications. Electron. J. Probab., 10:no. 39, 1286–1356 (electronic),2005. ISSN 1083-6489. URL http://www.math.washington.edu/˜ejpecp/EjpVol10/paper39.abs.html.

Ankit Gupta. Stochastic model for cell polarity. Ann. Appl. Probab., 22(2):827–859, 2012. ISSN 1050-5164. doi: 10.1214/11-AAP788. URL http://dx.doi.org.ezproxy.library.wisc.edu/10.1214/11-AAP788.

Olivier Hénard. Change of measure in the lookdown particle system. Stochastic Process. Appl., 123(6):2054–2083, 2013. ISSN 0304-4149. doi: 10.1016/j.spa.2013.01.015. URL http://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2013.01.015.

Peter M. Kotelenez and Thomas G. Kurtz. Macroscopic limits for stochastic partial differential equations ofMcKean-Vlasov type. Probab. Theory Related Fields, 146(1-2):189–222, 2010. ISSN 0178-8051. doi: 10.1007/s00440-008-0188-0. URL http://dx.doi.org/10.1007/s00440-008-0188-0.

Thomas G. Kurtz and Giovanna Nappo. The filtered martingale problem. In Dan Crisan and Boris Rozovskii,editors, Handbook on Nonlinear Filtering, chapter 5, pages 129–165. Oxford University Press, 2011.

Thomas G. Kurtz and Philip E. Protter. Weak convergence of stochastic integrals and differential equations. II.Infinite-dimensional case. In Probabilistic models for nonlinear partial differential equations (Montecatini Terme,1995), volume 1627 of Lecture Notes in Math., pages 197–285. Springer, Berlin, 1996.

http://arxiv.org/abs/1402.6724http://arxiv.org/abs/1402.6724http://www.math.washington.edu/~ejpecp/EjpVol10/paper39.abs.htmlhttp://www.math.washington.edu/~ejpecp/EjpVol10/paper39.abs.htmlhttp://dx.doi.org.ezproxy.library.wisc.edu/10.1214/11-AAP788http://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2013.01.015http://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2013.01.015http://dx.doi.org/10.1007/s00440-008-0188-0


Thomas G. Kurtz and Eliane R. Rodrigues. Poisson representations of branching markov and measure-valuedbranching processes. Ann. Probab., 39(3):939–984, 2011. doi: 10.1214/10-AOP574. URL http://dx.doi.org/10.1214/10-AOP574.

Thomas G. Kurtz and Jie Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Process.Appl., 83(1):103–126, 1999. ISSN 0304-4149.

Thomas G. Kurtz and Jie Xiong. A stochastic evolution equation arising from the fluctuations of a classof interacting particle systems. Commun. Math. Sci., 2(3):325–358, 2004. ISSN 1539-6746. URL http://projecteuclid.org.ezproxy.library.wisc.edu/euclid.cms/1109868725.

Youngmee Kwon and Hye-Jeong Kang. A class of nonlinear stochastic differential equations (SDES) withjumps derived by particle representations. J. Korean Math. Soc., 42(2):269–289, 2005. ISSN 0304-9914. doi:10.4134/JKMS.2005.42.2.269. URL http://dx.doi.org.ezproxy.library.wisc.edu/10.4134/JKMS.2005.42.2.269.

L. C. G. Rogers and J. W. Pitman. Markov functions. Ann. Probab., 9(4):573–582, 1981. ISSN 0091-1798. URL http://links.jstor.org/sici?sici=0091-1798(198108)9:42.0.CO;2-G&origin=MSN.

Kathryn E. Temple. Particle representations of superprocesses with dependent motions. Stochastic Process.Appl., 120(11):2174–2189, 2010. ISSN 0304-4149. doi: 10.1016/j.spa.2010.06.005. URL http://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2010.06.005.

Amandine Veber and Anton Wakolbinger. The spatial Lambda-Fleming-Viot process; an event-based con-struction and a lookdown representation. To Appear in Ann. Inst. H. Poincaré Probab. Stat., 2013.

http://dx.doi.org/10.1214/10-AOP574http://dx.doi.org/10.1214/10-AOP574http://projecteuclid.org.ezproxy.library.wisc.edu/euclid.cms/1109868725http://projecteuclid.org.ezproxy.library.wisc.edu/euclid.cms/1109868725http://dx.doi.org.ezproxy.library.wisc.edu/10.4134/JKMS.2005.42.2.269http://dx.doi.org.ezproxy.library.wisc.edu/10.4134/JKMS.2005.42.2.269http://links.jstor.org/sici?sici=0091-1798(198108)9:42.0.CO;2-G&origin=MSNhttp://links.jstor.org/sici?sici=0091-1798(198108)9:42.0.CO;2-G&origin=MSNhttp://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2010.06.005http://dx.doi.org.ezproxy.library.wisc.edu/10.1016/j.spa.2010.06.005


AbstractLecture 1: Lookdown constructions for Moran models

Moran models of population genetics are just one example of models involving finite collections of similarparticles whose types and/or locations evolve in time. Similar models arise in finance, queueing, and manyother areas. The problem of approximating these models when the number of particles is large leads naturallyto diffusions, measure-valued processes, and stochastic partial differential equations. The approximations aretypically derived by normalizing the empirical measure of the set of type/location values and passing to thelimit as the number of particles in the model goes to infinity. Here, we keep the particles discrete and obtaina limiting model given by a countable collection of discrete particles. A simple lookdown construction for aMoran model makes this convergence obvious, although identifying exactly what the limit is may not be soobvious.

In many other settings, limit arguments can be simplified and/or additional insight obtained by first obtaininga limiting model with a countably infinite collection of particles. The more familiar limits are then defined interms of this countable collection. The second and third lectures will discuss some of the technical tools neededand provide additional examples.

References

Peter Donnelly and Thomas G. Kurtz. A countable representation of the Fleming-Viot measure-valued diffu-sion. Ann. Probab., 24(2):698-742, 1996.

Peter Donnelly and Thomas G. Kurtz. Particle representations for measure-valued population models. Ann.Probab., 27(1):166-205, 1999.

Matthias Birkner, Jochen Blath, Martin Mhle, Matthias Steinrcken, Johanna Tams. A modified lookdown con-struction for the Xi-Fleming-Viot process with mutation and populations with recurrent bottlenecks. ALEA


Lat. Am. J. Probab. Math. Stat. 6 (2009), 2561.

Lecture 2: Exchangeable and conditionally Poisson constructions

From the example of the lookdown construction of the Moran model, we see the central role that exchangeabil-ity plays. Each particle has an integer-valued level, and the types and locations associated with the particlesform an exchangeable sequence indexed by the levels. The de Finetti measure associated with the sequencethen gives a measure-valued approximation to the empirical measure determined by the finite populationmodel.

Similar constructions for a larger class of models can be given if we assign real-valued rather than integer-valued levels. For constructions of this type, the analog of exchangeability in the finite population case is thatconditioned on the collection of types and locations, the levels are independent uniform random variables,and in the infinite population limit, the point process given by the set of type-location-level triples is a con-ditionally Poisson process with a Cox measure given as the product of a random measure on type/locationspace times Lebesgue measure on the nonnegative half-line. The random measure on type location space thengives the state of the measure-valued process that approximates the empirical measure determined by thefinite population model.

References

Andreas Greven, Vlada Limic, and Anita Winter. Representation theorems for interacting Moran models,interacting Fisher-Wright diffusions and applications. Electron. J. Probab., 10:no. 39, 1286-1356 (electronic),2005.

Thomas G. Kurtz and Eliane R. Rodrigues. Poisson representations of branching Markov and measure-valuedbranching processes. Ann. Probab., 39(3):939-984, 2011.

Amandine Veber and Anton Wakolbinger. The spatial lambda-Fleming-Viot process; an event-based construc-tion and a lookdown representation. To Appear in Ann. Inst. H. Poincare Probab. Stat., 2013.


Alison M. Etheridge and Thomas G. Kurtz. Genealogical constructions of population models. Preprinthttp://arxiv.org/abs/1402.6724

Lecture 3: Particle representations for stochastic partial differential equations

Many stochastic partial differential equations arise as limits of finite particle models, and particle representa-tions can be constructed for many of these. In a sense these constructions are lookdown constructions withoutthe lookdowns. Natural examples include stochastic versions of McKean-Vlasov models. Not so naturalexamples include stochastic partial differential equations with boundary conditions. For many of these con-structions, the empirical measures that give the solutions are weighted. Examples with weights include theclassical filtering equations, and applications of the constructions include derivation of consistent numericalschemes.

References

Thomas G. Kurtz and Philip E. Protter. Weak convergence of stochastic integrals and differential equations.II. Infinite-dimensional case. In Probabilistic Models for Nonlinear Partial Differential Equations (MontecatiniTerme, 1995), volume 1627 of Lecture Notes in Math., pages 197285. Springer, Berlin, 1996.

Thomas G.Kurtz and Jie Xiong. Particle representations for a class of nonlinear SPDEs. Stochastic Process.Appl. 83 (1999), no. 1, 103126.

Peter M. Kotelenez and Thomas G. Kurtz. Macroscopic limits for stochastic partial differential equations ofMcKean-Vlasov type. Probab. Theory Related Fields, 146(1-2): 189222, 2010.

Documents

“Wald’s insight was simply to ask: where are the missing ...kurtz/Lectures/wald.pdfm(t) = number of blues on ﬁrst mlevels Then by exchangeability P(t) = lim m!1 1 m Z m(t) exists,