交互作用粒子系統的流力極限 (6)

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 (6)

中 華 民 國 92 年 12 月 31 日

1

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 (6)

Hydrodynamic Limit of

Interacting Particle Systems (6)

計畫編號：NSC 91 – 2115 – M002 – 010

執行期限：91 年 8 月 1 日至 92 年 9 月 30 日

主持人：張志中 台灣大學數學系

Email:

[email protected]

In this project we study the large deviations estimate of occupation time difference of symmetric simple exclusion processes (SEP) on one and two dimensional lattice spaces Z1 and Z2.

Keywords: symmetric simple exclusion process, occupation time difference, large deviations estimate 二、緣由與目的 此處先說明期限延長與經費使用的情 形。原核定「出席國際會議」之部分，因 91 年 5 月時不確定能否獲得補助，故未能 在91 年 5 月限期截止前報名，以致未能如 預定計畫參加 ICM 2002 於 91 年 8 月 29 日 至 9 月 3 日 在 北 京 舉 行 的 satellite conference: stochastic analysis （主要行程） 及在京都於 9 月 4 日至 7 日舉行的 RIMS 2002 international project research: stochastic analysis and related topics （次要 行程）。92 年 6 月時因受國內 SARS 疫情 影響恐申請外國簽證不易、或出國後在外 國入境時遭隔離檢疫而影響行程，故申請 延期 2 個月並變更「出席國際會議」之補 助為「國外差旅費」。此申請案於6 月 13 日獲准（臺會綜二字第0920028200 號）， 故計畫執行期限延至 92 年 9 月 30 日。但 原計畫訪問的姚鴻澤教授於暑假中忙於自 紐 約 Courant Institute 搬 家 至 加 州 的 Stanford University ， 而 University of Maryland 的李宗祐教授則為胃癌侵襲所 苦，故國外訪問終未能成行。以下敘述計 畫之緣由與目的。

Consider the symmetric simple exclusion process (SEP) on d-dimensional lattice Zd, d=1 or 2. The configurations of this process are denoted byηso thatη(x) is equal to 1 or 0 if site x in Zd is occupied or not forη. For eachαin [0,1], denote by ν (α) the Bernoulli product measure on the configuration space Ω with marginals given by ν(α){η,η(x) =1 }=α, x in Zd. It is well-known that {ν(α), 0≦α≦1} is a one-parameter family of reversible invariant measures. In this project we study SEP accelerated by T starting from the reversible measureν(α) for a fixedαin (0,1). Given a local function b(η) on Ω satisfyingν(α) [b(η) |η--]=0, denote the occupation time difference B(T) associated with b by ( c(T)=square root of ln T )

B(T) = c(T) ∫

b(η

) ds.

We are interested in the large deviations of the occupation time difference B(T).

Given T > 0, on the configuration space Ω = {0, 1}Zd

, d=1,2, consider the accel-erated symmetric simple exclusion process (SEP) generated by LT given by

(LTf )(η) = T 2 X x,y∈Zd |x−y|=1 [f (σx,y_{η) − f (η)] ,}

where the summation is carried over all nearest neighbor sites x, y, |x − y| = 1, of Zd_{. In this formula, f is a local function} and σx,y_{η is the configuration obtained from} η by exchanging the occupation variables η(x) and η(y): (σx,y_{η)(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 ∈ Zd_{. Clearly, {ν}

α, 0 ≤ α ≤ 1}

is a one-parameter family of reversible in-variant measures. For 0 ≤ α ≤ 1, de-note 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).

Define the occupation time of the origin:

AT =

Z ₁

0

ηs(0) ds .

The large deviations principle of AT under Pα = PT,α as T → ∞, which is established in [1], states that the order is 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_.

Let b be a local function on Ω satisfying να[b(η) | ¯η ] = 0, where να[· | ¯η ] represents the να–expectation conditioned on the av-erage number of particles ¯η. Typical exam-ples are η(0) − η(e1) and η(e1) + η(−e1) +

η(e2) + η(−e2) − 4η(0).

Denote the occupation time difference BT associated with b by BT = p log T µ Z ₁ 0 b(ηs) ds ¶ ∈ R . There are at least two methods to study the large deviations principle of the joint distri-bution (AT, BT). The first is a probabilis-tic approach which basically is similar to the one used in [1] (see [3] also) with some natural modifications. As the object now is more complicated, one can expect that more detailed analysis is needed and the ar-guments would be much harder than those in [1]. The second is a PDE approach devel-oped by T.Y. Lee and has been applied suc-cessfully in several examples, see [2]. Here we outline the basic idea of PDE approach without proof. It is remarked that to ver-ify each step described below also requires lengthy and sophisticated arguments.

For simplicity consider d = 2 case only. To investigate the LDP of the joint distri-bution of AT and BT, by Laplace-Varadhan theorem, it suffices to study

lim T →∞ log T T log Eα · expn T log T ³ σAT + λBT ´o¸ = lim T →∞ log T T log Eα[VT(1, η; σ, λ)] , where VT is defined for t ∈ [0, 1], (σ, λ) ∈ R2 as VT(t, η) = VT(t, η; σ, λ) = Eη · exp n Z t 0 ³ σT log Tηs(0) + λT √ log Tb(ηs) ´ ds o¸ . By Feynman-Kac formula, VT solves the

dif-ferential equation            ∂tVT(t, η) =LTVT + µ σT log Tη(0) +√λT log Tb(η) ¶ VT, t ∈ [0, 1] , VT(0, η) =1 .

For convenience let

log VT = vT ⇔ VT = exp(vT) , φT(η; σ, λ) = σT log Tη(0) + λT √ log Tb(η) .

It follows that vT(t, η) = vT(t, η; σ, λ) satis-fies, t ∈ [0, 1],    ∂tvT(t, η) = e−vT(LTevT) + φT(η) = LTvT + φT(η) + RT(vT), vT(0, η) = 0 . (1) Here RT(f ) = e−f(LTef) − LTf = T 2 X x,y∈Z2 |x−y|=1 h exp n f (σx,y_{η) − f (η)}o −1 − ³ f (σx,y_{η) − f (η)}´i = QT(f ) + eT(f ) , and QT(f ) = T 4 X x,y∈Z2 |x−y|=1 h f (σx,y_{η) − f (η)}i2_, eT(f ) = RT(f ) − QT(f ) .

The first claim is that eT is negligi-ble in the large deviations limit T → ∞. Denote by v(1)_{(t, η) = v}(1)

T (t, η; σ, λ), t ∈ [0, 1], the solution of differential equation (1) in which RT is replaced solely by QT. The claim implies that it suffices to study v(1)_{(t, η).}

To study v_T(1)(t, η) we need to introduce an auxiliary function. Note that by as-sumption b has να–mean 0 on each hyper-plane ¯η =constant. Therefore there exists a gT(η) = gT(η; λ) such that

LTgT(η) + λT √

log Tb(η) = 0 . For simplicity we take a typical example: gT(η) = 2λ √ log Tη(0), b = 4η(0)− X j=1,2 χ=1,−1 η(χej) . Let v(2)_T (t, η; σ, λ) = v(1)_T (t, η; σ, λ) − gT(η; λ). When we write down the differen-tial equation for v_T(2), and replace the term QT(v_T(2)+ gT) by QT(v(2)_T ) + QT(gT), we ac-tually write down a new equation. Denote

by v_T(3) the solution of this new differential equation. The second claim is that the contribution of gT within vT(j), j = 1, 2, is negligible and one can substitute QT(vT(2)+ gT) with QT(v(2)T ) + QT(gT). In summary we have lim T →∞ log T T log Eα h exp{vT(1, η)} i = lim T →∞ log T T log Eα h exp{v_T(3)(1, η)} i . Observe that QT(f + h) = QT(f ) + QT(h) +T 2 X x,y∈Z2 |x−y|=1 h f (σx,yη) − f (η) ih h(σx,yη) − h(η) i , QT(gT) = λ2_T log T X l=1,2 χ=−1,1 h η(χej) − η(0) i₂ = λ 2_T log T ½ 4η(0) + h 1 − 2η(0)i³ X l=1,2 χ=−1,1 η(χej) ´¾

which satisfies να[ QT(gT) | ¯η ] 6= 0. Let UT(t, η) = Eη · exp n Z t 0 T log TU(ηs) ds o¸ , uT = log UT, and u(1)_T (t, η) the solution of the following differential equation :

       ∂tu(1)_T (t, η) =LTu(1)_T + QT(u(1)_T ) + T log TU(η), t ∈ [0, 1] , u(1)_T (0, η) =0 .

Now it is clear that by choosing U(η) = ση(0) + 4λ2_η(0) +λ2h_{1 − 2η(0)}i³ X l=1,2 χ=−1,1 η(χej) ´ , and applying superexponential estimate and large deviations estimate of the occu-pation time established in [1] one can derive the rate function for the large deviations of the joint distribution (AT, BT), T → ∞, as

Iα(c, d) =

d2

32c(1 − c)+ Υα(c) . 3

四、計畫成果自評

The large deviations results obtained in this project are contained in [4].

五、參考文獻

[1] Chang, C.C., Landim, C., Lee, T.Y.: Occupation time large deviations of two dimensional symmetric simple exclusion process. Ann. Probab. To appear

[2] Lee, T.Y.: Asymptotic results for super Brownian motions and semilinear differential equations. Ann. Probab. (2002)

[3] Landim, C.: Occupation time large deviations for the symmetric simple exclusion process. Ann. Probab. 20, 206--231 (1992)

[4] Chang, C.C., Landim, C., Lee, T.Y.: In preparation.