Dichotomy in a Scaling Limit under Wiener Measure with Density

In general, if the large deviation principle holds for a sequence of probability measures and its rate functional admits a unique minimizer, then the measures asymptotically concentrate in its neighborhood so that the law of large numbers follows. This paper discusses the situation that the rate functional has two distinct minimizers, for a simple model described by the pinned Wiener measures with certain densities involving a scaling. We study their asymptotic behavior and determine to which minimizers they converge based on a more precise investigation than the large deviation's level.


Introduction and results
This paper deals with a sequence of probability measures {µ N } N =1,2,... on the space C = C(I, R), I = [0, 1] defined from the pinned Wiener measures involving a proper scaling with densities determined by a class of potentials W .The large deviation principle (LDP) is easily established for {µ N } and the unnormalized rate functional is given by Σ W ; see (1.3) below.The aim of the present paper is to prove the law of large numbers (LLN) for {µ N } under the situation that Σ W admits two minimizers h and ĥ.We will specify the conditions for the potentials W , under which the limit points under µ N are either h or ĥ as N → ∞.

Model
Let ν 0,0 be the law on the space C of the Brownian bridge such that x(0) = x(1) = 0.The canonical coordinate of x ∈ C is described by x = {x(t); t ∈ I}.For a, b ∈ R, x ∈ C and N = 1, 2, . .., we set where h = ha,b is the straight line connecting a and b, i.e. h(t) = (1 − t)a + tb, t ∈ I; see Figure 1 below.The law on C of h N with x distributed under ν 0,0 is denoted by ν N = ν N,a,b .
In other words, ν N is the law of the Brownian bridge connecting a and b with covariance Let W = W (r) be a (measurable) function on R satisfying the condition: We consider the distribution µ N = µ N,a,b on C defined by where Z N is the normalizing constant.Under µ N,a,b , negative h has an advantage since the density becomes larger if it takes negative values.This causes a competition, especially when a, b > 0, between the effect of the potential W pushing h to the negative side and the boundary conditions a, b keeping h at the positive side.
The model introduced here can be regarded as a continuous analog of the so-called ∇ϕ interface model in one dimension under a macroscopic scaling; see Section 3.

LDP and LLN
The LDP holds for µ N on C as N → ∞ under the uniform topology.The speed is N and its unnormalized rate functional is given by for h ∈ H 1 a,b (I), i.e., for absolutely continuous h with derivatives ḣ(t) = dh/dt ∈ L 2 (I) satisfying h(0) = a and h(1) = b, where |{• • • }| stands for the Lebesgue measure.For more precise formulation, see Theorem 6.4 in [2] for a discrete model.Under our continuous setting, the proof is essentially the same or even easier than that.Indeed, when W = 0, the LDP follows from Schilder's theorem, while, when W = 0, W (N h(t)) in (1.2) behaves as −A1 {h(t)≤0} from the condition (W.1) and can be regarded as a weak perturbation.We omit the details.The LDP immediately implies the concentration property for µ N : lim for every δ > 0, where H W = {h * ; minimizers of Σ W } and dist ∞ denotes the distance in C under the uniform norm • ∞ .In particular, if Σ W has a unique minimizer h * , then the LLN holds under µ N : lim for every δ > 0.

Structure of H W
It is easy to see that H W = { h} when a, b ≤ 0, and H W = { ȟ} when a > 0, b < 0 (or a < 0, b > 0), where ȟ is a certain line connecting a and b with a single corner at the level 0; see Section 6.3, Case 2 in [2] for details.The interesting situation arises when a > 0, b ≥ 0 (or a ≥ 0, b > 0).We now assume that a, b > 0. The straight line h is always a possible minimizer of Σ W .If a+b < √ 2A, there is another possible minimizer ĥ of Σ W . Indeed, let ĥ be the curve composed of three straight line segments connecting four points (0, a), P 1 (t 1 , 0), P 2 (1 − t 2 , 0) and (1, b) in this order; see Figure 2. The angles at two corners P 1 and P 2 are both equal to θ ∈ [0, π/2], which is determined by the Young's relation (free boundary condition): tan θ = √ 2A.More precisely saying, we have Then, { h, ĥ} is the set of all critical points of Σ W (see Section 6.3, Case 1 in [2]), and this implies that H W ⊂ { h, ĥ}.

Results
This paper is concerned with the critical case where both h and ĥ are minimizers of Σ W , i.e.
In fact, in the following, we always assume the conditions (W.1) and which is actually equivalent to the condition: a, b > 0 and is fulfilled for some K ∈ R, then (1.5) holds with h * = h.
The rate functional Σ W of the LDP is determined only from the limit values W (±∞), but for Theorems 1.1 and 1.2 we need more delicate information on the asymptotic properties of W as r → ±∞ to control the next order to the LDP.The roles of the above conditions might be explained in a rather intuitive way as follows: The condition (W.3) (with K = 0) means that W is large at least for r ≥ 0 so that the force pushing the Brownian path downward is weak and not enough to push it down to the level of ĥ.On the other hand in Theorem 1.2, since the values of N h(t) in (1.2) are very large for t close to 0 or 1, compared with (W.3), the Brownian path is pushed downward because of the condition (W.4) and, once it reaches near the level 0, the condition (W.5) forces it to stay there.This makes the Brownian path reach the level of ĥ.In the special case where a = b = A/8 (t 1 = t 2 = 1/4), the second condition in (W.6) is fulfilled if 1/2 < α 1 < 1, and such α 1 , which simultaneously satisfies the first condition in (W.6), exists if α 2 > 1.
Section 2 gives the proofs of Theorems 1.1 and 1.2.Section 3 explains the relation between the (continuous) model discussed in this paper and the so-called ∇ϕ interface model (discrete model) in one dimension in a rather informal manner.The analysis is, in general, simpler for continuous models than discrete models.The same kind of problem is discussed for weakly pinned Gaussian random walks, which may involve hard walls, by [1] in which the coexistence of h and ĥ in the limit is established under a certain situation; see also [3].In our setting, the pinning effect can be generated from potentials having compact supports and taking negative values near r = 0. Our condition (W.1) on W excludes the potentials of pinning type and of hard wall type.

Proofs
From (1.4) followed by LDP together with our basic assumption H W = { h, ĥ}, for the proofs of Theorems 1.1 and 1.2, it is sufficient to show that the ratio of probabilities converges either to 0 or to +∞, respectively, as N → ∞ for small enough δ > 0. This will be established by (2.2) and (2.3) for Theorem 1.1 and by (2.5)-(2.7)for Theorem 1.2, below.

Proof of Theorem 1.1
In view of the scaling, we may assume K = 0 in the condition (W.3) without loss of generality.
Lemma 2.1.The joint distribution of (τ 1 , τ 2 ) under ν N is given by Proof.Let Q N be the law on C of y(t) = √ N h N (t) under ν N , let P a be the Wiener measure starting at a and where p(s, a, b) is the heat kernel and Pb where τ is the hitting time to 0. Therefore the conclusion of the lemma follows from 2s ds, a > 0, see, e.g., (6.3) in [5], p.80.