Particle representations and limit theorems for stochastic
partial differential equations
(or, if the only tool you have is a hammer. . . )
• A stochastic McKean-Vlasov equation
• Exchangeability and de Finetti’s theorem
• Convergence of exchangeable systems
• From particle approximation to particle representation
• Uniqueness via Markov mapping
• Vanishing spatial noise correlations
• A stochastic Allen-Cahn equation
• Conditionally Poisson representations
• References
• Abstract
with Dan Crisan, Peter Kotelenez, Phil Protter, Eliane Rodrigues, Jie Xiong
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1
A stochastic McKean-Vlasov equation
Kotelenez (1995); Kurtz and Xiong (1999)
Consider the stochastic partial differential equation
Z t
hVε (t), ϕi = hVε (0), ϕi +
hVε (s), Lε ϕ(·, Vε (s))ids
0
Z
+
hVε (s), ∇ϕ(·)T Jε (·, u, Vε (s))iW (du × ds)
U ×[0,t]
R
where Vε is measure-valued, hµ, ϕi = Rd ϕ(x)µ(dx), W is space-time
Gaussian white noise on U × [0, ∞) with variance measure ν(dy) × ds,
1X
Dε,ij (x, µ)∂i ∂j ϕ(x) + F (x, µ) · ∇ϕ(x)
Lε ϕ(x, µ) =
2 ij
Z
Dε (x, µ) =
Jε (x, u, µ)JεT (x, u, µ)ν(du).
U
Note that hVε (t), 1i ≡ hVε (0), 1i; just assume hVε , 1i ≡ 1.
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2
Exchangeability and 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
exists a random probability measure Ξ such that for every bounded,
measurable g,
Z
g(X1 ) + · · · + g(Xn )
lim
= g(x)Ξ(dx)
n→∞
n
almost surely, and
m
m Z
Y
Y
E[ gk (Xk )|Ξ] =
gk dΞ
k=1
k=1
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3
Convergence of exchangeable systems
Kotelenez and Kurtz (2010)
n
} be exchangeable (allowing
Lemma 2 For n = 1, 2, . . ., let {ξ1n , . . . , ξN
n
n
Nn = ∞.) LetPΞ be the empirical measure (defined as a limit if Nn =
n
n
∞), Ξn = N1n N
i=1 δξi . Assume
• Nn → ∞
n
• For each m = 1, 2, . . ., (ξ1n , . . . , ξm
) ⇒ (ξ1 , . . . , ξm ) in S m .
Then
{ξi } is exchangeable and setting ξin = s0 ∈ S for i > Nn , {Ξn , ξ1n , ξ2n . . .} ⇒
{Ξ, ξ1 , ξ2 , . . .} in P(S) × S ∞ , where Ξ is the deFinetti measure for {ξi }.
n
If for each m, {ξ1n , . . . , ξm
} → {ξ1 , . . . , ξm } in probability in S m , then
Ξn → Ξ in probability in P(S).
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4
Lemma 3 Let X n = (X1n , . . . , XNn n ) be exchangeable families of DE [0, ∞)valued random variables such that Nn ⇒ ∞ and X n ⇒ X in DE [0, ∞)∞ .
Define
P n
n
Ξn = N1n N
i=1 δXi ∈ P(DE [0, ∞))
P
Ξ = limm→∞ m1 m
i= δXi
P n
n
Vn (t) = N1n N
i=1 δXi (t) ∈ P(E)
P
V (t) = limm→∞ m1 m
i=1 δXi (t)
Then
a) For t1 , . . . , tl ∈
/ {t : E[Ξ{x : x(t) 6= x(t−)}] > 0}
(Ξn , Vn (t1 ), . . . , Vn (tl )) ⇒ (Ξ, V (t1 ), . . . , V (tl )).
b) If X n ⇒ X in DE ∞ [0, ∞), then Vn ⇒ V in DP(E) [0, ∞). If X n →
X in probability in DE ∞ [0, ∞), then Vn → V in DP(E) [0, ∞) in
probability.
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5
Properties of cadlag processes
a) The set DΞ = {t : E[Ξ{x : x(t) 6= x(t−)}] > 0} is at most countable.
b) If for i 6= j, with probability one, Xi and Xj have no simultaneous
discontinuities, then DΞ = ∅ and convergence of X n to X in DE [0, ∞)∞
implies convergence in DE ∞ [0, ∞).
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6
From particle approximation to particle representation
N
Let XεN = {Xε,i
} satisfy
Z
N
N
Xε,i (t) = Xε,i (0) +
U ×[0,t]
N
Jε (Xε,i
(s), u, VεN (s))iW (du × ds)
Z
+
t
N
F (Xε,i
(s), VεN (s))ds
0
where VεN (t) =
able.
1
N
PN
N (t)
i=1 δXε,i
N
and {Xε,i
(0), 1 ≤ i ≤ N )} is exchange-
Assume
(x, µ) ∈ Rd × P(Rd ) → F (x, µ) ∈ Rd
and
(x, µ) ∈ Rd × P(Rd ) → Jε (x, ·, µ) ∈ L2 (ν)
are bounded and continuous. Then there exists an exchangeable (weak)
solution. (Construct an Euler approximation and pass to the limit.)
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7
Convergence to infinite system
Letting N → ∞ and applying Lemma 3, there exists an exchangeable
solution to the infinite system
Z
Jε (Xε,i (s), u, Vε (s))iW (du × ds)
Xε,i (t) = Xε,i (0) +
U ×[0,t]
t
Z
+
F (Xε,i (s), Vε (s))ds
0
where Vε (t) is the de Finetti measure of {Xε,i (t)}.
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8
Uniqueness for infinite system
Kurtz and Protter (1996); Kurtz and Xiong (1999)
Theorem 4 Let
Z
ρ(µ1 , µ2 ) =
sup
{f :|f (x)−f (y)|≤|x−y|}
|
Z
f dµ1 −
Rd
f dµ2 |
Rd
and assume
|F (x1 , µ1 ) − F (x2 , µ2 )| + kJε (x1 , ·µ1 ) − Jε (x2 , ·, µ2 )kL2 (ν)
≤ C(|x1 − x2 | + ρ(µ1 , µ2 )).
Then the solution of the infinite system is unique.
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9
The corresponding SPDE
Z
t
ϕ(Xε,i (t)) = ϕ(Xε,i (0)) +
Lε ϕ(Xε,i (s), Vε (s))ds
0
Z
∇ϕ(Xε,i (s))T Jε (Xε,i (s), u, Vε (s))W (du × ds)
+
U ×[0,t]
By the exchangeablity, averaging over i gives
Z t
hVε (t), ϕi = hVε (0), ϕi +
hVε (s), Lε ϕ(·, Vε (s))ids
0
Z
+
hVε (s), ∇ϕ(·)T Jε (·, u, Vε (s))iW (du × ds)
U ×[0,t]
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10
Uniqueness via Markov mapping
Define γ : (Rd )∞ → P(Rd ) by
n
1X
δx i
γ(x) = lim
n→∞ n
i=1
if the limit exists in P(Rd ) and γ(x) = µ0 otherwise.
Then Vε (t) = γ(Xε (t)) and a Markov mapping theorem implies that
every solution of the SPDE can be obtained in this way.
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11
Vanishing spatial noise correlations
Let d ≥ 2, U = Rd , ν be Lebesgue measure, and
Jε (x, u, µ) = ε−d/2 J (x, ε−1 (x − u), µ),
so that the stochastic intergral becomes
Z
Jε (Xε,i (s), u, Vε (s))iW (du × ds)
Rd ×[0,t]
Z
=
ε−d/2 J (Xε,i (s), ε−1 (Xε,i (s) − u), Vε (s))iW (du × ds)
d
ZR ×[0,t]
=
J (Xε,i (s), z, Vε (s))iWiε (dz × ds),
Rd ×[0,t]
where for each i, Wiε is a Gaussian white noise defined by
Z
Z
ε
ϕ(z, s)Wi (dz×ds) =
ε−d/2 ϕ(ε−1 (Xε,i (s)−u), s)iW (du×ds)
Rd ×[0,∞)
Rd ×[0,t]
(NOTE: The Wiε are not independent but are exchangeable.)
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12
Convergence
If
R
Rd
|J (x, z, µ)|2 dz < ∞ and x1 6= x2 , then
Z
ε−d/2 J (x1 , ε−1 (x1 − u), µ)ε−d/2 J (x2 , ε−1 (x2 − u), µ)T du
Rd
Z
=
J (x1 , ε−1 x1 − u), µ)J (x2 , ε−1 x2 − u), µ)T du
Rd
→0
Assume that the convergence is uniform on |x1 − x2 | ≥ δ > 0, for each
δ > 0, and on compact subsets of P(Rd ).
Assume the nondegeneracy condition
R
2
Rd (z · J (x, z, µ)) du
> 0.
inf inf
x,µ z
|z|2
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13
The zero correlation limit
Theorem 5 Assume Xε (0) = X(0), and an additional regularity condition for d = 2. As ε → 0, Xε converges in distribution to the solution
of
Z
Z t
Xi (t) = Xi (0)+
J (Xi (s), u, V (s))iWi (du×ds)+ F (Xi (s), V (s))ds
Rd ×[0,t]
0
where the Wi are independent and V (t) is the de Finetti measure for
{Xi }.
V is the unique solution of
Z
t
hV (t), ϕi = hV (0), ϕi +
hV (s), Lϕ(·, V (s))ids
0
P
where Lϕ(x, µ) = 12 ij Dij (x, µ)∂i ∂j ϕ(x) + F (x, µ) · ∇ϕ(x)
Z
D(x, µ) =
J (x, u, µ)J (x, u, µ)T du.
U
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14
Sketch of proof
Z
ε−d/2 J (Xε,i (s),
Xε,i (t) = Xi (0) +
U ×[0,t]
Z t
+
Z
Xε,i − u
, Vε (s))iW (du × ds)
ε
F (Xε,i (s), Vε (s))ds
0
= Xi (0) +
Rd ×[0,t]
J (Xε,i (s), u, Vε (s))iWiε (du
Z
× ds) +
t
F (Xε,i (s), Vε (s))ds
0
where
Miϕ,ε (t)
Z
=
Rd ×[0,t]
ϕ(u)Wiε (du×ds)
Z
=
Rd ×[0,t]
ε−d/2 ϕ(
Xε,i (s) − u
)W (du×ds)
ε
Relative compactness follows from boundedness of J and F and
Z
Xε,i (s) − u −d/2 Xε,j (s) − u
ψ,ε
ϕ,ε
ε−d/2 ϕ(
[Mi , Mj ]t =
)ε
)du → 0
ψ(
ε
ε
Rd ×[0,t]
for t < τij = inf{t : Xi (t) = Xj (t)}. If τij = ∞ a.s., then the limits Wi
and Wj are independent.
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15
A stochastic Allen-Cahn equation
Bertini, Brassesco, and Buttà (2009)
1
dm = ( mxx − (m2 − 1)m)dt + cẆ ,
2
m(t, b) = m+ , m(t, a) = m−
Find a signed measure-valued process M which will have a density with
respect to Lebesgue measure so we can write M (t, dx) = M (t, x)dx.
Let {Xi } be independent reflecting Browian motions on the interval
[a, b] with uniform initial distribution, and let Ai satisfy
t
Z
Ai (t) = Ai (0) +
Z
R×[0,t]
Z
+
0
where
ρ (Xi (s) − u)W (du × ds)
G(M (s, Xi (s))Ai (s)ds +
0
t
m+ − Ai (s−)
dΛ+ (s) +
|m+ − Ai (s−)| i
Z
0
t
m− − Ai (s−)
dΛ− (s),
|m− − Ai (s−)| i
n
1X
hM (t), ϕi = lim
ϕ(Xi (t))Ai (t).
n→∞ n
i=1
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16
Assumptions
• {Ai (0)} is an exchangeable sequence.
• W is space-time
Gaussian white noise governed by Lebesgue meaR
sure so R×[0,t] ρ (Xi (s) − u)W (du × ds) is a Brownian motion with
R
variance t R ρ (u)2 du.
−
+
• Λ+
i and Λi are right continuous, nondecreasing processes, Λi increases only when Xi = b and Λ−
i increases only when Xi = a, and
+
+
if Xi (s) = b, Λi (s) − Λi (s−) = |m+ − Ai (s−)| and if Xi (s) = b,
−
Λ−
i (s) − Λi (s−) = |m− − Ai (s−)|.
• Note that if Xi (s) = b, then Ai (s) = m+ and if Xi (s) = a, Ai (s) =
m− .
• We require the solution {(Xi , Ai )} to be exchangeable.
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17
Derivation of SPDE
M is the signed measure given by
n
1X
ϕ(Xi (t))Ai (t)
hϕ, M (t)i = lim
n→∞ n
i=1
and M is the density of M .
Note that the limit will exist by the exchangeability requirement and
M is absolutely continuous with respect to Lebesgue measure by the
uniformity of the {Xi (t)}.
Let Xi be given by the Skorohod equation
Xi (t) = Xi (0) + σBi (t) + λai (t) − λbi (t),
where the Bi are independent standard Browian motions independent
of W and {Xi (0)}, the Xi (0) are independent and uniformly distributed
over [a, b], and λai and λbi are the local times at a and b.
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18
For ϕ ∈ Cc2 (a, b),
Z
t
ϕ(Xi (t))Ai (t) = ϕ(Xi (0))Ai (0) +
ϕ(Xi (s))dAi (s)
0
Z t
Z t
1 00
0
ϕ (Xi (s))Ai (s)dBi (s) +
+
ϕ (Xi (s))Ai (s)ds
0 2
0
Z t
= ϕ(Xi (0))Ai (0) +
ϕ(Xi (s))G(M (s, Xi (s)))Ai (s)ds
0
Z
+
ϕ(Xi (s))ρ (Xi (s) − u)W (du × ds)
R×[0,t]
t
0
Z
+
Z
ϕ (Xi (s))Ai (s)dBi (s) +
0
0
t
1 00
ϕ (Xi (s))Ai (s)ds
2
Averaging both sides of this identity,
Z
t
hϕ, M (t)i = hϕ, M (0)i +
hϕG(M (s, ·)), M (s)ids
0
Z
Z
Z t
1
+
ϕ(x)ρ (x − u)dxW (du × ds) +
h ϕ00 , M (s)ids
R×[0,t] R
0 2
subject to the boundary conditions M (t, b) = m+ and M (t, a) = m− .
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19
What happens when goes to zero?
Let be the diameter of the support of ρ .
Z
Z
ϕ(x)ρ (x − u)dxW (du × ds)
R×[0,t]
R
converges to
Z
Z
ϕ(u)W (du × ds)
c
R×[0,t]
if
R
R ρ (x)dx
R
→ c.
Wi (t)
Z
ρ (Xi (s) − u)W (du × ds)
=
R×[0,t]
converge to independent Brownian motions with parameter σ 2 if
Z
ρ2 (x)dx → σ 2 .
R
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20
Conditionally Poisson representations
Kurtz and Rodrigues (2010)
Consider
Z
t
Z
t
σ(Xi (s))dBi (s) +
c(Xi (s))ds
Xi (t) = Xi (0) +
0
0
Z
σ
b(Xi (s), u)W (du × ds)
+
Γ×[0,t]
and
Z
t
Z
t
Ui (t) = Ui (0) +
Ui (s)γ0 (Xi (s))dBi (s) −
Ui (s)d(Xi (s))ds
0
0
Z
+
Ui (s)γ1 (Xi (s), u)W (du × ds)
Γ×[0,t]
Bi independent Brownian motions,
{(Xi (0), Ui (0))} a conditionally Poisson point process with mean measure V (0, dx) × du.
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21
Measure-valued process
X
1 X
δXi (t) = lim e−Ui (t) δXi (t)
r→∞ r
→0
i
Ui (t)≤r
X
1 3 X −Ui (t) 2
−Ui (t)
2
e
Ui (t)δXi (t)
Ui (t)e
δXi (t) = = lim →∞
2
i
i
V (t) = lim
Define
1
L1 ϕ = aϕ00 + cϕ0 ,
2
Z
L2 ϕ = dϕ − (γ0 σ +
β(x) = γ0 (x)2 +
Γ
Z
γ1 σ
bdµ)ϕ0
γ12 (x, u)µ(du)
Γ
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22
Applying Itô’s formula
e−Ui (t) ϕ(Xi (t))
=e
−Ui (0)
Z
ϕ(Xi (0)) − t
Ui (s)e−Ui (s) ϕ(Xi (s))γ0 (Xi (s))dBi (s)
0
t
Z
e−Ui (s) σ(Xi (s))ϕ0 (Xi (s))dBi (s)
+
Z0
−
Z
+
Ui (s)e−Ui (s) ϕ(Xi (s))γ1 (Xi (s), u)W (du × ds)
Γ×[0,t]
e−Ui (s) σ
b(Xi (s), u)ϕ0 (Xi (s))W (du × ds)
Γ×[0,t]
Z
1 2 t −Ui (s)
e
Ui2 (s)ϕ(Xi (s))β(Xi (s))ds
+ 2
Z t 0
Z t
+
e−Ui (s) L1 ϕ(Xi (s))ds + Ui (s)e−Ui (s) L2 ϕ(Xi (s))ds
0
0
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23
Corresponding SPDE
Z
hV (t), ϕi = hV (0), ϕi −
Γ×[0,t]
Z
hV (t), (γ1 (·, u)ϕ + σ
b(·, u)ϕ0 )iW (du × ds)
t
hV (t), Lϕids
+
0
1
Lϕ = aϕ00 + (c − γ0 σ −
2
Z
Γ
γ1 σ
bdµ)ϕ0 + (d + β)ϕ
For the Zakai equation, take
σ
b = γ0 = 0,
d = −β
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24
Markov mapping theorem
Theorem 6 A ⊂ C(E)×C(E) a pre-generator with bp-separable graph.
D(A) closed under multiplication and separating.
γ : E → E0 , Borel measurable.
α a transition function from E0 into E satisfying
α(y, γ −1 (y)) = 1
R
Let µ0 ∈ P(E0 ), ν0 = α(y, ·)µ0 (dy), and define
Z
Z
C = {( f (z)α(·, dz), Af (z)α(·, dz)) : f ∈ D(A)} .
E
E
If Ye is a solution of the MGP for (C, µ0 ), then there exists a solution
Z of the MGP for (A, ν0 ) such that Y = γ ◦ Z and Ye have the same
distribution on ME0 [0, ∞).
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25
References
Lorenzo Bertini, Stella Brassesco, and Paolo Buttà. Boundary effects on the interface dynamics for the
stochastic AllenCahn equation. In Vladas Sidoravičius, editor, New Trends in Mathematical Physics,
pages 87–93. Springer, 2009.
Peter M. Kotelenez. A class of quasilinear stochastic partial differential equations of McKean-Vlasov type
with mass conservation. Probab. Theory Related Fields, 102(2):159–188, 1995. ISSN 0178-8051.
Peter M. Kotelenez and Thomas G. Kurtz. Macroscopic limits for stochastic partial differential equations
of McKean-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 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.
Thomas G. Kurtz and Eliane Rodrigues. Poisson representations of branching Markov and measure-valued
branching processes. preprint, 2010.
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.
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26
Abstract
Particle representations and limit theorems for stochastic partial differential equations
Solutions of the a large class of stochastic partial differential equations can be represented in terms of the de
Finetti measure of an infinite exchangeable system of stochastic ordinary differential equations. These representations provide a tool for proving uniqueness, obtaining convergence results, and describing properties of
solutions of the SPDEs. The basic tools for working with the representations will be described. Applications
include the convergence of an SPDE as the spatial correlation length of the noise vanishes, uniqueness for a
class of SPDEs, and consistency of approximation methods for the classical filtering equations.
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