Homogenization of diffusions on the lattice Z^{d} with periodic drift coefficients; Application of logarithmic Sobolev inequality

Sergio Albeverio ^{∗}, M. Simonetta Bernabei ^{†}, Michael Röckner ^{‡}and Minoru W. Yoshida ^{§}

Jan. 7th, 2005

0. Abstract

Suppose that on a complete probability space (Ω, F, P; F_{t}) with an increasing family of sub σfield {F_{t}}_{t∈R_{+}} we are given a family of independent 1-dimensional F_{t}-standard Brownian motion processes {B_{k}(t)}_{t∈R_{+}}, k ∈ Z^{d}.

For each ϵ ∈ (0, 1] and each x = {x_{k}}_{k∈Z^{d}} ∈ R^{Z^{d}}, consider the SDE such that

X_{k}^{ϵ}(t, x) = x_{k} + \sqrt{2}B_{k}(t)
+\frac{1}{ϵ} ∫_{0}^{t} b_{k}(\frac{X^{ϵ}(s, x)}{ϵ})ds, t ∈ R_{+}, k ∈ Z^{d},  (0. 1)

where we denote

X^{ϵ}(s, x) ≡ \{X_{k}^{ϵ}(s, x)\}_{k∈Z^{d}}, b_{k}(x) ≡ ∑_{Λ∈k}(−\frac{∂}{∂x_{k}} J_{Λ}(x)),

for a given family of potentials

J ≡ {J_{Λ} | Λ ⊂ Z^{d}, |Λ| < ∞}.

For J we assume the following: J-1) (Periodicity) For each Λ ⊂ Z^{d} such that |Λ| < ∞,

J_{Λ} ∈ C_{Λ}^{∞}(R^{Z^{d}}),

and it is a periodic function with respect to each variable with the period 2π; J-2) (Shift invariance)

J_{Λ+k} = J_{Λ} ∘ ϑ^{k}, ∀k ∈ Z^{d};

∗Universität Bonn, GERMANY.

†Università di Camerino, ITALY.

‡Universität Bielefeld, GERMANY.

§The Univ. ELECTRO-COMMUNICATIONS JAPAN.

J-3) (Finite range) there exists an L < ∞ and J_{Λ} = 0 holds for any Λ such that

Λ ∋ 0,  and  Λ ∉ [−L, +L]^{d}.

We give the following metric ρ on (C(R_{+} → R))^{Z^{d}}, and denote the Frechét space equipped with this metric by W:

ρ(x(·), x′(·)) ≡ ∑_{n∈N} \frac{1}{2^{n}} \{\{ ∑_{k∈Z^{d}} \frac{1}{2^{|k|}} sup_{0≤t≤n} |x_{k}(t) − x′_{k}(t)|^{2}\}^{\frac{1}{2}} ∧ 1\},  (0. 2)

for x(·) ≡ {x_{k}(·)}_{k∈Z^{d}}, x′(·) ≡ {x′_{k}(·)}_{k∈Z^{d}} ∈ (C(R_{+} → R))^{Z^{d}}.

For each t ≥ 0, let ξ_{t} be the measurable map such that

ξ_{t} : W ∋ x(·) ⟼ x(t) ∈ R^{Z^{d}},

and for each x ∈ R^{Z^{d}} and ϵ > 0, let P_{x}^{ϵ} be the probability measure on (W, B(W)) which is the probability law of the process {X^{ϵ}(t, x)}_{t∈R_{+}}:

P ({ω | X^{ϵ}(·, x) ∈ B}) = P_{x}^{ϵ}(ξ_{·} ∈ B), ∀B ∈ B(W).

Let T = {y ∈ R^{2} : |y| = 1} be the unit circle equipped with the Riemannianian metric. Let y = {y_{k}}_{k∈Z^{d}} denote the point in T^{Z^{d}}. In order to identify T with [0, 2π) we define

Θ : T^{Z^{d}} ∋ {y_{k}}_{k∈Z^{d}} ⟼ {θ_{k}}_{k∈Z^{d}} ∈ [0, 2π)^{Z^{d}}

where θ_{k} = θ(y_{k}) and the function θ : T ⟶ [0, 2π) is defined by

y = \binom{cos θ(y)}{sin θ(y)} ∈ T ⊂ R^{2}.

Proposition 1 ((Section V of [S2])) Let J be the potential that satisfies the conditions J-1), J-2) and J-3). Then there exists at least one Gibbs probability measure μ on (T^{Z^{d}}, B(T^{Z^{d}})) such that

< E^{Λ}φ, μ >=< φ, μ >, ∀Λ ⊂ Z^{d} s.t. |Λ| < ∞, ∀φ ∈ C_{0}(T^{Z^{d}}),  (0. 3)

where

[E^{Λ}φ](y) = \frac{1}{Z_{Λ}(y_{Λ^{c}})} ∫_{T^{Z^{d}}} φ(y′_{Λ} · y_{Λ^{c}}) exp(−U^{Λ}(Θ(y′_{Λ} · y_{Λ^{c}})))dy′,  (0. 4)
U^{Λ}(x) ≡ ∑_{Λ′∩Λ≠∅} J_{Λ′}(x) x ∈ R^{Z^{d}},  (0. 5)

Z_{Λ}(y_{Λ^{c}}) = ∫_{T^{Z^{d}}} exp[−U^{Λ}(Θ(y′_{Λ} · y_{Λ^{c}}))]dy′

Here we use the notation y′_{Λ} · y_{Λ^{c}} ≡ y′′ ∈ T^{Z^{d}}, so that y′′_{Λ} = y′_{Λ} and y′′_{Λ^{c}} = y_{Λ^{c}}.

Assumption 1

There exists a unique Gibbs measure μ_{0}, and there exists a c = c(J) > 0 and an α = α(J) > 1 which depend only on J, and for each Λ ∈ Z^{d} (|Λ| < ∞) there exists K(Λ) ∈ (0, ∞) such that the following holds:

∥ ∫_{T^{Z}} φ(y_{Λ})p(t, ·, dy)− < φ, μ > ∥_{L^{∞}}
≤ K(Λ)(c + t)^{−α}{∥∇φ∥_{L^{∞}} + ∥φ∥_{L^{∞}}},  ∀t > 0, ∀φ ∈ C_{Λ}^{∞}(T^{Z^{d}}).  (0. 6)

Lemma 1 Assume that J-1), J-2), J-3) and Assumption 1 are satisfied. For any k ∈ Z^{d} the following is well defined as a measurable function of y ∈ T^{Z^{d}} :

E^{Q_{y}}[∫_{0}^{∞}{b_{k}(Θ(η_{s}(·))}ds].

Denote this function by χ_{k}(y), y ∈ T^{Z^{d}}, then χ_{k}, (k ∈ Z^{d}), satisfies the following:

χ_{k}(·) ∈ C(T^{Z^{d}} → R) (in particular it is bounded) ,  (0. 7)
χ_{k}(·) ∈ D[E],  E(χ_{k}, χ_{k}) ≤ \frac{5}{4},  ∀k ∈ Z^{d};  (0. 8)

E(χ_{k}(·), v(·)) ≡ ∑_{j∈Z^{d}} ∫_{T^{Z^{d}}} (\frac{∂}{∂y_{j}} χ_{k}(y))(\frac{∂}{∂y_{j}} v(y))μ_{0}(dy)
= − ∫_{T^{Z^{d}}} b_{k}(Θ(y))v(y)μ_{0}(dy), ∀k ∈ Z^{d}, ∀v ∈ C_{0}^{∞}(T^{Z^{d}}).  (0. 9)

As the L^{2}(μ_{0}) sence, we define

χ′_{k,j}(y) = \{ \begin{matrix}\sqrt{2}\frac{∂}{∂y_{j}} χ_{k}(y) & j ≠ k \\ \sqrt{2}(1 − \frac{∂}{∂y_{k}} χ_{k}(y)) & j = k. \end{matrix} .  (0. 10)

Let χ′ be the matrix of which components are functions such that

χ′(y) ≡ (χ′_{k,j}(y))_{k,j∈Z^{d}},  (0. 11)

and define

Ā ≡ (ā_{k,j})_{k,j∈Z^{d}}, with ā_{k,l} ≡ ∑_{j∈Z^{d}} ∫_{T^{Z^{d}}} χ′_{k,j}(y) · χ′_{l,j}(y)μ_{0}(dy).  (0. 12)

We denote {Y_{t}}_{t∈R_{+}} with Y_{0} = 0 as the unique continuous Gaussian process taking values in R^{Z^{d}} (namely Y_{·} is a (C(R_{+} → R))^{Z^{d}} valued random variable) of which covariance matrix is t · Ā (t ∈ R_{+}) defined on a complete probability space.

For each x ∈ R^{Z^{d}}, let P_{x} be the probability measure on (W, B(W)) that is the probability law of the process {x + Y_{t}}_{t∈R_{+}}. Heuristically, {Y_{t}}_{t∈R_{+}} can be expressed by

Y_{t} = ∫_{0}^{t} Ā^{\frac{1}{2}} dB_{t}, t ∈ R_{+},

where {B_{t}}_{t∈R_{+}} ≡ {{B_{k,t}}_{t∈R_{+}}}_{k∈Z^{d}} and {B_{k,t}}_{t∈R_{+}} (k ∈ Z^{d}) are some idependent sequence of one-dimensional standard Brownian motion processes.

Theorem 1 Assume that J-1), J-2), J-3) and Assumption 1 are satisfied. For each ϵ > 0 and x ∈ R^{Z^{d}}, on a probability space (W̄ , B(W̄), P̄_{x}^{ϵ}; B_{t}(W̄)), which is a standard extension of (W, B(W), P_{x}^{ϵ}; B_{t}(W)), there exists a B_{t}(W)-adapted R^{Z^{d}}-valued continuous process {ζ_{t}^{ϵ}}_{t∈R_{+}} that satisfies the following: For any φ ∈ C_{b}(W → R),

lim_{ϵ↓0} EP̄_{x}^{ϵ}[φ(ζ_{·}^{ϵ}(·))] = E^{P_{x}}[φ(ξ_{·}(·))],  ∀x ∈ R^{Z^{d}}.  (0. 13)

as a modification of above

lim_{ϵ↓0} EP̄_{ϵx}^{ϵ}[φ(ζ_{·}^{ϵ}(·))] = E^{P_{0}}[φ(ξ_{·}(·))],  ∀x ∈ [0, 2π)^{Z^{d}},  (0. 14)

{{ζ_{t}^{ϵ}}_{t∈R_{+}}}_{ϵ>0} is the approximation sequence in the sense that

lim_{ϵ↓0} ∫_{T^{Z^{d}}} EP̄_{ϵỹ}^{ϵ}[ρ(ξ_{·}(·), ζ_{·}^{ϵ}(·))]μ_{0}(dy) = 0,  (0. 15)

where ỹ = Θ(y) for Θ : T^{Z^{d}} → [0, 2π)^{Z^{d}}, and P_{x} is the probability law of the process {x+Y_{t}}_{t∈R_{+}}.

Theorem 2 Assume that J-1), J-2), J-3) and Assumption 1 are satisfied. Let P_{0} be the probability law of the process {Y_{t}}_{t∈R_{+}}.

Then

lim_{ϵ↓0} ∫_{T^{Z^{d}}} |E^{P_{ϵỹ}^{ϵ}}[φ(ξ_{·}(·))] − E^{P_{0}}[φ(ξ_{·}(·))]| μ_{0}(dy) = 0, ∀φ ∈ C_{b}(W → R),  (0. 16)

where ỹ = Θ(y) for Θ : T^{Z^{d}} → [0, 2π)^{Z^{d}}.

Also, there exists an N ∈ B(T^{Z^{d}}) such that μ_{0}(N) = 0, and a subsequence

{ϵ_{n}}_{n∈N} ⊂ {ϵ | ϵ ∈ (0, 1]},

and the following holds:

lim_{ϵ_{n}↓0} E^{P_{ϵnx}^{ϵn}}[φ(ξ_{·}(·))] = E^{P_{0}}[φ(ξ_{·}(·))], ∀φ ∈ C_{b}(W → R),

∀x ≡ {x_{k}}_{k∈Z^{d}} ∈ R^{Z^{d}} such that Θ^{−1}Φ(x) ∈ T^{Z^{d}} \ N , sup_{k∈Z^{d}} |x_{k}| < ∞,  (0. 17)

where

Θ^{−1} ∘ Φ : R^{Z^{d}} ∋ x = {x_{k}}_{k∈Z^{d}} ⟼ {θ^{−1} ∘ Φ(x_{k})}_{k∈Z^{d}} ∈ T^{Z^{d}}  (0. 18)

with

Φ(x_{k}) = θ_{k} ∈ [0, 2π) if x_{k} = θ_{k} mod 2π.