行政院國家科學委員會專題研究計畫 成果報告
交互作用粒子系統的流力極限(7)
計畫類別： 個別型計畫 計畫編號： NSC92-2115-M-002-010- 執行期間： 92 年 08 月 01 日至 93 年 08 月 31 日 執行單位： 國立臺灣大學數學系暨研究所 計畫主持人： 張志中 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 93 年 12 月 13 日
行政院國家科學委員會專題研究計畫成果報告
交互作用粒子系統的流力極限 (7)
Hydrodynamic Limit of
Interacting Particle Systems (7)
計畫編號：NSC 92 – 2115 – M – 002 – 010
執行期限：92 年 8 月 1 日至 93 年 8 月 31 日
主持人：張志中 台灣大學數學系
Email:
[email protected]
一、中文摘要 本計劃中我們探討 2 維格子點空間 Z2 上對稱簡單互斥過程並導出了一個超指數估 計。此估計增進了對前作一個位置被粒子佔據的時間的大離差估計中速率函數的了解， 並有助於對數個位置被粒子佔據的時間差（總平均為零）的大離差估計的研究。 關鍵詞：對稱簡單互斥過程、粒子佔據的時間、大離差估計、超指數估計 AbstractIn this project we study the symmetric simple exclusion process (SEP) on two dimensional lattice space Z2 and derive a super-exponential estimate. This estimate gives more understanding of the rate function of the occupation time large deviations. Moreover, it helps investigating the large deviations of occupation time difference of several sites.
Keywords: symmetric simple exclusion process, occupation time, large deviations estimate,
super-exponential estimate
二、報告內容
Given T > 0, on the configuration space Ω = {0, 1}Z2
, consider the speeded-up symmetric simple exclusion process (SEP) generated by LT given by
(LTf )(η) = T 2 X x,y∈Z2 |x−y|=1 [f (σx,yη) − f (η)] ,
where the summation is carried over all nearest neighbor sites x, y, |x−y| = 1, of Z2. In this
formula, f is a local function and σx,yη is the configuration obtained from η by exchanging
the occupation variables η(x) and η(y): (σx,yη)(z) = η(z) if z 6= x, y, η(x) if z = y, η(y) if z = x .
For each 0 ≤ α ≤ 1, denote by να the Bernoulli product measure on Ω with marginals
given by
να{η, η(x) = 1} = α
for x ∈ Z2. Clearly, {ν
α, 0 ≤ α ≤ 1} is a one-parameter family of reversible invariant
measures. For 0 ≤ α ≤ 1, denote by Pα = PT,α the probability on the path space D(R+, Ω)
corresponding to SEP starting from να. From now on we fix an α ∈ (0, 1).
The large deviations principle of the occupation time of the origin:
AT =
Z 1
0
ηs(0) ds
under Pα = PT,α as T → ∞ has been established in [1]. It states that the decay rate is of
order T / log T , and the rate function Υα : [0, 1] → R+ is given by
Υα(β) =
π
2 n
sin−1(2β − 1) − sin−1(2α − 1)o2 .
To prove the above large deviations principle, we introduced in [1] the so-called polar
empirical measure. To define this we need some notation. Let Z2
∗ = Z2− {0} and for each
T > 1 define the projection σT : Z2 → [0, ∞) by
σT(x) = log |x| log T , |x| = q x2 1+ x22 , σT(0) = 0 .
Let µ1,T be the polar empirical measure on R
+ given by µ1,T(η) = 1 2π log T X x∈Z2 ∗ η(x) 1 |x|2δσT(x) , (1)
where δv is the Dirac measure concentrated on v. For H : R+ → R bounded and piecewise
continuous, denote ¿ H, µ1,T(η) À = 1 2πlogT X x∈Z2 ∗ H(σT(x)) 1 |x|2η(x) . (2)
Let ¯µT be the Radon measure on R
+ defined by ¯ µT = Z 1 0 µ1,T(ηs) ds . (3)
For c > 0, denote by Mc the set of positive Radon measures µ on R+ such that µ([a, b]) ≤ (b − a) + c for every 0 ≤ a ≤ b < ∞: Mc = n µ(dr), µ([a, b]) ≤ (b − a) + c for 0 ≤ a ≤ b < ∞ o .
The condition on the measure of intervals makes the set Mc, which is endowed with the
vague topology, a compact separable metric space. Let M0 be the subspace of M of all
measures which are absolutely continuous with respect to the Lebesgue measure and whose density is bounded by 1. The subspace M0 is closed (thus compact) and a sequence µn in
M0 converges to µ if and only if
lim n→∞ Z H(r) µn(dr) = Z H(r) µ(dr)
holds for all continuous functions H with compact support in (0, ∞).
Overestimating η(x) by one, it is easy to show that the random measures ¯µT belong to
Mcfor T large enough. More precisely, there exists a finite universal constant C0 such that
¯
µT([a, b]) ≤ (b − a) + C0
log T
for all 0 ≤ a ≤ b < ∞, T > 1. In particular, for each c > 0, there exists a finite T (c) such that ¯µT belongs to M
c for all T > T (c). The same statement remains in force for µ1,T
in place of ¯µT. This property of the random measures ¯µT explains the introduction of the
spaces Mc. From now on we fix some c > 0.
For any α in (0, 1), let C2(R
+, α) be the space of twice continuously differentiable
functions γ : [0, ∞) → (0, 1) such that γ0 has a compact support in (0,1
2) and such that
γ(r) = α for r ≥ 1/2. There exists therefore 0 < β < 1 and 0 < ε < 1/4 such that γ(r) = β
for r ≤ ε, and γ(r) = α for r ≥ 1
2− ε. For each γ in C2(R+, α), let Γ = Γγ,αbe the function
given by
Γ(u) = 1 2log
γ(u) [1 − α]
[1 − γ(u)] α · (4) Notice that Γ is a twice continuously differentiable function with compact support in [0, 1/2) and whose derivative has compact support on (0, 1/2). Denote by Σ(R+) the space of
functions {Γ0
γ,α, γ ∈ C2(R+, α)}. Notice that Σ(R+) is a vector space (which is not the case
of C2(R
+, α) or {Γγ,α, γ ∈ C2(R+, α)}) and that every function h in Σ(R+) is continuously
differentiable and has compact support in (0, 1/2). Moreover, given h in Σ(R+) there exists
one and only one γ in C2(R
+, α) such that h = Γ0γ,α. This means that the map from
C2(R
+, α) to Σ(R+) is one to one.
For 0 < α < 1, let F (a) = a(1 − a) and Iα : Mc→ R+ be given by
Iα(µ) = π sup h∈Σ(R+)
n
− ¿ h0, m À − ¿ h2, F (m) À
o
if µ(dr) = m(r)dr is absolutely continuous with respect to the Lebesgue measure and has a density such that m(r) = α for r ≥ 1/2. In all other cases Iα(µ) = ∞. Note that
Proposition 2.2, which states that in the large deviations regime the measure ¯µT(dr) is
fixed on [1/2, ∞) and equal to αdr, justifies the concentration of Iα on such measures. In
[1] we show that the rate function Iα is lower semi-continuous, convex and such that
Iα(µ) = π 4 Z R+ m0(r)2 m(r)[1 − m(r)]dr .
Theorem 2.1 For every closed subset F of Mc and every open subset G of Mc, lim sup T →∞ log T T log Pα £ ¯ µT ∈ F¤ ≤ − inf µ∈FIα(µ) , lim inf T →∞ log T T log Pα £ ¯ µT ∈ G¤ ≥ − inf µ∈GIα(µ) .
Some observations about the law of large numbers type behavior of ¯µT are needed in
order to prove Theorem 2.1. The first one, i.e. Proposition 2.2, is needed for the upper bound, while the second one, i.e. Proposition 2.3, is needed for the lower bound.
Proposition 2.2 For r in R+ and ε > 0, let Φr,ε(r0) = ε−11{[r, r + ε]}(r0). For every
r ≥ 1/2, ε > 0 and δ > 0, lim sup T →∞ log T T log Pα h | ¿ Φr,ε, ¯µT À −α| > δ i = −∞ . Proposition 2.3 Fix γ in C2(R
+, α) and recall the definition of Γ = Γγ,α given by (4). As
T ↑ ∞, the measure ¯µT(η) converges in P
T,γ-probability to the measure γ(r) dr.
However, it is not clear from the above two observations what happens at (1/2)− for
¯
µT(dr). Therefore, to fill in the missing information, this year in this project we prove a
third Proposition, i.e. Proposition 2.5, which is based on the following Lemma.
Lemma 2.4 For r > 0 and ε > 0, let Φr,ε(r0) = ε−11{[r, r + ε]}(r0). For every ε > 0,
δ > 0, and sufficiently large R > 1/2,
lim sup T →∞ log T T log Pα h | ¿ Φ1 2−ε,ε, ¯µ T À − ¿ Φ R,ε, ¯µT À | > δ i = −∞ . Proposition 2.5 For every ε > 0 and δ > 0,
lim sup T →∞ log T T log Pα h | ¿ Φ1 2−ε,ε, ¯µ T À −α| > δi = −∞ .
The proofs can be found in [2] and are omitted here.
3
Self-evaluation
The study of the empirical polar measures made in this project helps a lot in the investi-gation of other scaling limit problems of two dimensional exclusion processes.
4
References
1. Chang, C.-C., Landim, C., Lee, T.Y. (2004). Occupation time large deviations of two dimensional symmetric simple exclusion process. Ann. Probab. 32 1B 661-691. 2. Chang, C.-C.: In preparation.
