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Glauber Dynamics in Continuum
Yuri Kondratievjoint work with O.Kutovyi and R.Minlos
Bielefeld, Germany
October 28, 2008
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 1 / 24
Setup General facts and notations
The configuration space:
Γ :={γ ⊂ Rd
∣∣ |γ ∩ Λ| <∞ for all compact Λ ⊂ Rd}.
| · | - cardinality of the set.
Remark: Γ is a Polish space.
n-point configuration space:
Γ(n) :={η ⊂ Rd | |η| = n
}, n ∈ N0.
The space of finite configurations:
Γ0 :=⊔n∈N0
Γ(n).
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 2 / 24
Glauber dynamics with competition
Potential
φ : Rd 7→ R ∪ {+∞}:
(S) (Stability) There exists B > 0 such that, for any η ∈ Γ0, |η| ≥ 2.
E(η) :=∑
{x, y}⊂η
φ(x− y) ≥ −B|η|.
(SI) (Strong Integrability) For any β > 0,
Cst(β) :=∫
Rd
|1− exp [βφ(x)]|dx <∞.
Relative energy: γ ∈ Γ, x ∈ Rd \ γ
E(x, γ) :={ ∑
y∈γ φ(x− y), if∑y∈γ |φ(x− y)| <∞,
+∞, otherwise.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 3 / 24
Glauber dynamics with competition
Gibbs measures
∫Γ
∑x∈γ
F (γ, x)µ(dγ) =∫
Γ
∫Rd
e−βE(x, γ)F (γ ∪ x, x)zdxµ(dγ)
for any measurable function F : Γ× Rd → [0,+∞].
G(φ, z, β) - the set of all Gibbs measures
Remark: If φ is stable and
C(β) :=∫
Rd
|1− e−βφ(x)|dx < z−1e−1−2βB
then the class G(φ, z, β) is non-empty. Moreover,
kµ(η) ≤ constC(β)−|η|, η ∈ Γ0.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 4 / 24
Glauber dynamics with competition
Dirichlet forms
E+(F, F ) =∫
Γ
∫Rd
|F (γ ∪ x)− F (γ)|2 exp(−E(x, γ))zdxdµ(γ)
G+-dynamics
E−(F, F ) =∫
Γ
∫Rd
|F (γ ∪ x)− F (γ)|2zdxdµ(γ)
G−-dynamics
Corresponding equilibrium dynamics:K/Lytvynov, K/Lytvynov/Rockner
Spectral properties of equilibrium Markov generators:Bertini/Cancrini/Cesi, K/Lytvynov, Wu, K/Minlos/Zhizhina
We will study G− Glauber non-equilibrium stochastic dynamics
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 5 / 24
Glauber dynamics with competition
Pre-generator
Markov pre-generator on Γ:
(LF )(γ) :=∑x∈γ
eβE(x, γ)D−x F (γ) + z
∫Rd
D+x F (γ)dx,
D−x F (γ) = F (γ \ x)− F (γ) death of x ∈ γ;
D+x F (γ) = F (γ ∪ x)− F (γ) birth of x ∈ Rd;
Remark: Operator L is symmetric in L2(Γ, µ).Corresponding Dirichlet form is E−.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 6 / 24
Glauber dynamics with competition
Questions
1 Existence of non-equilibrium dynamics (via general analytic approach)
2 Spectral properties (symbol)
3 Asymptotic behavior of dynamics
4 Speed of convergence to equilibrium
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 7 / 24
Glauber dynamics with competition
Harmonic analysis on configuration space
L : dFt
dt = LFt Γ, F oo〈F,µ〉=
∫Γ F dµ //M1
fm(Γ)
K∗
��
SS
Lenard
L∗ : dµt
dt = L∗µt
L := K−1LK Γ0, G oo〈G, k〉=
∫Γ0Gk dλ
//
K
KK
��
K−1
K(Γ0) L∗ : dkt
dt = L∗kt
K-transform:KG(γ) :=
∑ξbγ
G(ξ), γ ∈ Γ.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 8 / 24
Glauber dynamics with competition
Construction of the process: main steps
The key ”function”: R(η) := C |η|, C > 0
Set of initial states
M1R(Γ) =M1
C(Γ) :={µ ∈M1(Γ) | kµ ≤ const · C|·|
}.
Banach space (”densities” for locally finite measures)
KC := {k : Γ0 → R | k · C−|·| ∈ L∞(Γ0, λ)},
where λ := λ1 is the Lebesgue-Poisson measure with intensity 1.
Banach space (quasi-observables)
LC := L1(
Γ0, C|η|λ(dη)
).
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 9 / 24
Glauber dynamics with competition
Construction of the process: main steps
To show
1 Symbol generates C0-semigroup in LC ;
2 Growth condition: kt(η) ≤ const · C|η| (satisfied !!!);
3 Positive definiteness: preservation of the correlation function property.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 10 / 24
Glauber dynamics with competition
The symbol of the operator (K- image) :
The formal K- image or symbol of the operator L:
L := K−1LK
Symbol of the Glauber dynamics with competition
L := L0 + L1 + L2,
whereL0G(η) := −A(η)G(η), A(η) =
∑x∈η
∏y∈η\x
eβφ(x−y);
L1G(η) := −∑
ξ⊂η, ξ 6=η
G(ξ)∑x∈ξ
∏y∈ξ\x
eβφ(x−y)∏y∈η\ξ
(eβφ(x−y) − 1);
L2G(η) := κ∫
Rd
G(η ∪ x)dx, κ > 0.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 11 / 24
Glauber dynamics with competition
Banach space and domain of the symbol
We considerL : D(L) ⊂ LC → LC
in the Banach space
LC := L1(
Γ0, C|η|λ(dη)
).
The domain of L
D(L) := {G ∈ LC : A ·G ∈ LC} .
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 12 / 24
Glauber dynamics with competition
Existence of a semigroup
Proposition
For any triple of positive constants C, κ and β which satisfies
2eCst(β)C + 2κe2BβC−1 < 3
the symbol L is a generator of a holomorphic semigroup Tt, t ≥ 0 in LC .
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 13 / 24
Glauber dynamics with competition
Idea of proof
1 The operator (L0G)(η) := −G(η)∑x∈η
∏y∈η\x e
βφ(x−y) with
D(L0) =
{G ∈ LC
∣∣∣∣∣∑x∈η
eβE(x, γ\x)G(η) ∈ LC
}.
is a generator of holomorphic semigroup;
2 The operator(L− L0, D(L)
)is relatively bounded w.r.t. (L0, D(L0));
3 The operator(L,D(L)
)is a generator of a holomorphic semigroup in LC
(Kato perturbation theory).
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 14 / 24
Glauber dynamics with competition
Existence of the process
Positive definiteness: approximation arguments
Result of the general construction approach:
Theorem (Kondratiev/K/Minlos)
For any µ ∈M1C(Γ), there exists a non-equilibrim Markov process with generator
L and initial distribution µ provided
2eCst(β)C + 2κe2BβC−1 < 3
holds.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 15 / 24
Glauber dynamics with competition
Symbol: spectral properties
1 The point z = 0 is an eigenvalue of the operator L with the eigenvector
ψ0(η) ={
1, η = ∅,0, η 6= ∅.
2 There exists z0 > 0 such that
I1 = {z ∈ C : Rez > −z0} \ {0}
and
I2 ={z ∈ C : |arg z| < 3π
4
}\ {0}
belong to the resolvent set of the operator L.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 16 / 24
Glauber dynamics with competition
Dual space
The dual rescaled space to the Banach space LC :
KC := {k : Γ0 → R | k · C−|·| ∈ L∞(Γ0, λ)}.
We consider alsoK≥1 := {k ∈ KC : k(∅) = 0}.
L∗ - the corresponding adjoint operator to the operator L on KCDuality:
� G, k �:=∫
Γ0
G · k dλ1 =∫
Γ0
G · C |·| · k
C |·|dλ1, G ∈ LC
Evolution on KC :� TtG, k �=:� G, T ∗t k �
kt := T ∗t k, t ≥ 0.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 17 / 24
Glauber dynamics with competition
Evolution of correlation functions
T ∗t - the semigroup which corresponds to L∗ (in the weak sense)
K≥1 - invariant subspace
T ∗t = T ∗t |K≥1 restriction to the invariant subspace K≥1.
Theorem (Kondratiev/K/Minlos)
The semigroup T ∗t is such that for t ≥ 1
||T ∗t || ≤ const e−z02 t.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 18 / 24
Glauber dynamics with competition
Idea of Proof
Space decomposition
L10 := {G ∈ LC |G = const ψ0}, ψ0(η) =
{1, η = ∅,0, η 6= ∅
and L≥1 := {G ∈ LC : G(∅) = 0}
LC = L10 + L≥1.
L10 is invariant with respect to the semigroup Tt;
Tt/L10
- factor semigroup on the factor space LC/L10
which can be identifiedwith the space L≥1
factor generator (L/L1
0
)[G] = L[G] := [LG],
where [G] ∈ LC/L10
is an equivalent class of the element G ∈ LC .
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 19 / 24
Glauber dynamics with competition
Idea of Proof
Similarity: L corresponds to the operator L11 := L|L≥1 in the sense ofsimilarity, i.e. there exists isomorphism J
J−1L11J = L/L10.
DefineTt := J Tt/L1
0J−1,
which generator coincides with L11.
T ∗t = T ∗t |K≥1 - adjoint semigroup to the semigroup Tt.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 20 / 24
Glauber dynamics with competition
Idea of Proof
The inverse formula for a semigroup via resolvent
− 12πi
∫Υ
(L|L≥1 − z11)−2eztdz = tTt,
where integral is taken over the contour Υ of the form
Υ = {z = −z0/2 + iw : w ∈ R}
The resolvent bound∣∣∣∣∣∣(L|L≥1 − (−z0
2+ iw)11)−1
∣∣∣∣∣∣ ≤ 3
(e−4βB + w2)12
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 21 / 24
Glauber dynamics with competition
Ergodicity
Theorem (Kondratiev/K/Minlos)
Let initial measure µ0 ∈M1(Γ) has the correlation function k0 ∈ KC for someC > 0. Suppose that the parameters of our system satisfy the following conditions
1
exp {e2βBCst(β)C}+ 2κe2βBC−1 <32
2
κe2βBC−1 exp {e2βBCCst(β)} < 1
Denote kµ the correlation function corresponding to µ ∈ G(κ, β). Then,
||kt − kµ||KC≤ const e−
z02 t||k0 − kµ||KC
.
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 22 / 24
Glauber dynamics with competition
Proof
Condition 1: exp {e2βBCst(β)C}+ 2κe2βBC−1 < 32
1 Existence of the semigroup T ∗t in the Banach space KC2 Bound for the semigroup T ∗t
||T ∗t || ≤ const e−z02 t.
Condition 2: κe2βBC−1 exp {e2βBCCst(β)} < 1
1 Existence of µ ∈ G(κ, β)2 Bound for the corresponding correlation function
kµ(η) ≤ C|η|, η ∈ Γ0
rt := kt − kµ = T ∗t (k0 − kµ) ∈ K≥1 and as result
||kt − kµ||KC= ||T ∗t r0||KC
≤ const e−z02 t||k0 − kµ||KC
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 23 / 24
Glauber dynamics with competition
Thank you for attention! :)
Yuri Kondratiev (Bielefeld) Glauber Dynamics in Continuum October 28, 2008 24 / 24