交互作用粒子系統的流力極限 (8)

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 (8)

中 華 民 國 94 年 12 月 31 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 (8)

Hydrodynamic Limit of

Interacting Particle Systems (8)

計畫編號：NSC 93 – 2115 – M – 002 – 009

執行期限：93 年 8 月 1 日至 94 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 develop a connection between an eigenvalue problem and the occupation time large deviations of two dimensional symmetric simple exclusion process established in [1].

Keywords: eigenvalue, symmetric simple exclusion process, occupation time, large

deviations

二、報告內容

In this report we briefly describe how to build a bridge between an eigenvalue problem and the occupation time large deviations established in [1]. We thank H.T. Yau for valuable discussions.

First we state the result of the occupation time large deviations of two dimensional symmetric simple exclusion process derived in [1]. 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,y_{η 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_{. It is well known that {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 generated by LT starting from να. From now on we fix an

α ∈ (0, 1).

The large deviations principle of the occupation time of the origin:

OT =

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 _.

The corresponding Laplace-Varadhan Theorem [2] under the present setting has the following form.

Theorem 1 Let H : [0, 1] → R be a bounded continuous function on [0, 1]. Then

lim T →∞ log T T log Eα · expn T log TH(OT) o¸ = sup β∈[0,1] n H(β) − Υα(β) o . (1)

For any configuration η, denote by Eη the conditional expectation of SEP Pα given that

the process starts from the specific configuration η0 = η. Let V : Ω = {0, 1}Z2

→ R be a local function on Ω satisfying να[ V ] 6= 0 (in fact, more

Though no T appears on right hand side, the subscript T of OT(V ) on left hand side

indicates the dependency on T via the generator LT of the process. By Feynman-Kac

formula, the function u(t, η) = uT(t, η; λ), t ∈ [0, 1] , given by

uT(t, η) = Eη · expn Z t 0 T log TλV (ηs) ds o¸ (2)

solves the differential equation    ∂tuT(t, η) = LTuT + T log T λV (η)uT, t ∈ [0, 1] , uT(0, η) = 1 . (3)

We remark that the large deviations estimate of OT(V ) is known when V = η(0) as

described above. Consequently, with the help of super-exponential estimate, since V is a local function, the large deviations estimate of OT(V ) is fully understood.

Since u is positive, E[u] ≤ (E[u2_])1/2_{= kuk}

L2_(να). Furthermore 1 2∂t Z u2dνα = Z u · ∂tu dνα = Z u · ½ LTu + T log TλV u ¾ dνα = kuk2L2_(να) ½Z ¯ uLTu +¯ T log TλV ¯u 2_dν α ¾ = kuk2 L2_(να) ½Z T log TλV ¯u 2_dν α− D(¯u) ¾ ≤ kuk2 L2_(να)× sup f, kf k_L2(να)=1 ½Z T log TλV f 2_dν α− D(f ) ¾ ,

where ¯u = u/kukL2_(να) and D(f ) =

R f (−LT)f dνα. Denoting Γ = ΓT(λ) = sup f, kf k_L2(να)=1 ½Z T log TλV f 2_dν α− D(f ) ¾ , we obtain ∂tkuk2L2_(να) ≤ 2Γkuk2_L2_(να).

By Gronwall’s inequality we get

ku(t)k2

L2_(να) ≤ ku(0)k2_L2_(να)e2Γt = e2Γt

since u(0, η) ≡ 1. Now we evaluate u(t) at t = 1, take logarithm, multiply log T /2T to get log T T ΓT(λ) ≥ log T T log ku(1)kL2(να)≥ log T T log µZ u(1, η) dνα ¶ .

For simplicity, from now on we only consider the case V (η) = η(0). Therefore Z u(1, η; λ) dνα = Eα · exp n Z 1 0 T log Tληs(0) ds o¸ = Eα · expn T log TλOT o¸ .

Theorem 2 For arbitrary λ ∈ R define ΓT(λ) = sup f, kf k_L2(να)=1 ½Z T log Tλη(0)f 2_dν α− D(f ) ¾ . Then sup β∈[0,1] n λβ − Υα(β) o ≤ lim T →∞ log T T ΓT(λ) . (4)

Proof: Since H(a) = λa is a bounded continuous function on [0, 1], we may apply Theorem 1 to conclude that sup β∈[0,1] n λβ − Υα(β) o = lim T →∞ log T T log Eα · expn T log TH(OT) o¸ ≤ lim T →∞ log T T ΓT(λ)

by the discussion above.

Naturally, the next step is to understand how close the two quantities appeared on the both sides of (4) are. This will be one of the subjects of the future project, and will appear in [3].

3

Self-evaluation

The eigenvalue problem is always an interesting subject in mathematics. In this project we develop an approach that connects the problem with the large deviation estimates. It is hoped that this method can give a different insight about the classical eigenvalue problem. Conceivably we believe that the progress made in this project is important.

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] Kipnis C., Landim C. (1999). Scaling Limits of Interacting Particle Systems, Grundl-heren der mathematischen Wissenschaften 320, Springer-Verlag, Berlin, New York. [3] Chang, C.-C.: In preparation.