{'fraction_non_alphanumeric': 0.05999928994923137, 'fraction_numerical': 0.021266020520467215, 'mean_word_length': 3.5988571428571428, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 28, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 7, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the long-time behavior of a perturbed conservative system with degeneracy

We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states, and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential equations related, for example, to turbulence problems.


Introduction.
Many Hamiltonian systems that arise in mechanics, mechanical engineering, as well as hydrodynamics are subject to group symmetry. As an example, in the study of the motion of an ideal incompressible fluid, V.I.Arnold had proposed (see [1], [2], [3]) a beautiful picture that describes the dynamics of ideal incompressible fluid as geodesic flows on the group of all diffeomorphisms of a certain domain (see also the author's related work [32] in this direction). The studies of random perturbations of Hamiltonian systems, or general dynamical systems with symmetry, in particular the long-time dynamics and problems about invariant measures of these systems are of interest (see also the author's related work [31], [18], [17]). Schematically, the general problem can be formulated as follows. We are given a dynamical systeṁ in an ambient space x ∈ M (M can be a Riemanian manifold). Usually we assume b(x) preserves the energy. Then we assume that for some group G the system (1) has some symmetry with respect to G. The last sentence about symmetry of the system (1) with respect to the group G is a bit vague and could be understood in many different ways. It can be understood in a strict way so that the group can act on the space M (in particular, it is such case when G = M ) and the dynamics of (1) is invariant with respect to G-action. It can also be understood as a more "rough" symmetry, in the sense for example that the stable attractors of (1) has equivalent dynamical properties under G-action (in [16] such dynamical property is in the sense of equivalence of logarithmic asymptotics of transition probabilities when we add a small noise to (1), this is the notion of "quasi-potential"). Our goal is to describe the effect of adding a small noise to (1). That is, we study systems of typė where ξ ε is a deterministic and/or stochastic perturbation depending on the small parameter(s) ε = (ε 1 , ..., ε k ). Recent progresses in this direction have shown that an effective description of the long-time behavior of (2) is the motion on the cone of invariant measures of the unperturbed system (1) (see [16]). Several examples of such description are recently demonstrated in [16], [23], [21], [24], [22]. The above paradigm is only a general scheme. In this work we are interested in studying a model problem that falls under the above general paradigm. Let us consider the following system (see [7], [4,Section 4.4]) corresponding to (1): dx t = −x t y t dt , dy t = x 2 t dt .
A phase picture of system (3) can be seen in Figure 1(a). We see that the whole line Oy A contains stable equilibriums and the whole line Oy B contains unstable equilibriums. This is different from the cases considered in [28], [25]. In this case we can understand the symmetry of (3) in a more rough way: the stable and unstable equilibriums are symmetric with respect to shifts in the directions of Oy A and Oy B , respectively. The unperturbed system (3) preserves the energy E(x, y) = x 2 + y 2 . The driving vector field b(x, y) = (−xy, x 2 ) is degenerate on x = 0. Let us add a perturbation to system (3) that consists of a deterministic friction and a random noise: Here W 1 t and W 2 t are two independent standard 1-dimensional Brownian motions; the small parameter ε > 0 is the intensity of the friction, and the small parameter √ ε > 0 represents the intensity of the noise. System (4) is a two-dimensional nonlinear stochastic equation involving a non-potential force. It is this non-potential force that has the essential effect of creating a line of stable fixed points (attracting line Oy A ) touching a line of unstable fixed points (repelling line Oy B ). In the subsequent text we sometimes refer to this model as the AB-model.
Our goal in this paper is to study the long-time behavior of system (4) as ε ↓ 0. By further developing results in [16], [23], [21], [24], [22], we will characterize the limiting process as a diffusion process Y t on the positive-y semi-axis. The limiting diffusion process Y t behaves as a 2-dimensional radial Bessel process with linear damping, and henceforce we call it a damped 2-d radial Bessel process, abbreviated as damped-BES (2) (for Bessel process in arbitrary dimension see [38,Chapter XI,§1]). The origin O is an inaccessible point for damped-BES (2). Diffusion processes on singular 1-dimensional manifolds as the limit of averaging procedure has been considered in [26], [27], among many other literature. The major contribution in our work is that we consider the manifold of unstable equilibria touching the manifold of stable equilibria. This results in non-trivial analysis that leads to our limiting process Y t as well as the inaccessibility of the origin O. We will describe the limiting Markov diffusion process Y t via its infinitesimal generator, and we show the weak convergence by making use of tightness and the classical martingale problem method.
In a certain sense, our model problem here differs from the set-up in the classical Freidlin-Wentzell theory (see [25]) in that the point-like asymptotically stable attractor is replaced by a manifold. We can view our limiting process Y t on Oy A , the damped-BES(2) process, as a "process-level attractor" of our system. For ε > 0, the dynamics of the system as ε ↓ 0 corresponds to the "metastable" behavior (see [15]). We will show that under this scenario the "metastable" behavior of the system is characterized by jumps between points on Oy A and Oy B .
We are motivated by finite dimensional models for the inviscid stochastic 2-d Navier-Stokes equations of the form (see [33]) in which η(t, x) is a noise, and positive parameter ν → 0. The survey [33] considers an open problem (Open Problem 3 in the last section) that aims at studying the vanishing noise limit of stationary measures of the 2-d stochastic Navier-Stokes system. The difficulty there is that one has to put a rather restrictive hypothesis, namely the unperturbed dynamics has to be globally asymptotically stable. To remove this restriction, in the finite dimensional case this problem is rather well-understood, and one can establish the so-called Freidlin-Wentzell asymptotics for stationary measures (see Section 6.4 in [30]). As for stochastic PDEs, similar results can be proved, provided that the global attractor for the unperturbed dynamics has a regular structure. The latter means that the attractor consists of finitely many steady-states and the heteroclinic orbits joining them. A result in this direction has been proved in [35] for the case of a damped nonlinear wave equation. This motivates our consideration of system (3), which mimics the attractor for the 2-d Euler system, that has continuous sets of steady states (see [43,Lecture 68]). In fact, systems that arise in hydrodynamics, such as in the context of Euler's equation, typically possess equilibrium points that belong to an infinite dimensional "manifold" of other equilibria. These has been found in experiments (see [41], [42]), in numerical simulations (see [40]), explained using arguments based on statistical mechanics (see [8], [39], [36], [6]), as well as explained theoretically (see [5], [37]). When we add a damping to (3), we obtain for fixed ε > 0 the model system (4) without the stochastic noise, which admits only one single attractor O. This situation mimics the case of 2-d Navier-Stokes equations, that behaves similarly to a "generic" dissipative system. Actually the situstion will be much more complicated for the Navier-Stokes equations. For example, in low dimensions a good example is the famous Lorenz attractor (see [44]). In our case, when the dissipation term −ν∆u of (5) is small, the system (5) is close to an Euler's equation, and our model problem (4) suggests some geometric feature of the attractors. The paper is organized as follows. In Section 2 we will explain the heuristics of the limiting mechanism. In Section 3 we demonstrate the main convergence theorem as well as its proof. In Section 4 we prove auxiliary lemmas that are needed in Section 3. In Section 5 we describe the dynamics of our model system for small but nonzero ε > 0. Some remarks and generalizations are provided in Section 6.
2 Heuristic description of the limiting mechanism.
To describe the limiting motion as ε ↓ 0, we can first do a time rescaling In this way, we see the separation of a "fast" motion which is governed by the nonpotential force term, and a "slow" motion which is due to the random perturbation. Due to the effect of the fast motion, starting from anywhere (x 0 , y 0 ) that is not lying on the semi-axis Oy B , the process (X ε t , Y ε t ) will come close to the attracting line Oy A in a relatively short time. Let π denote this hitting operator, so that we have the following definition.
In the limit as ε ↓ 0, the process (X ε t , Y ε t ) is pushed by the flow onto Oy A , and will be close to π(x 0 , y 0 ) in short time. There, the Y -component Y ε t behaves as a 2-dimensional linearly damped radial Bessel process (damped-BES(2)) on Oy A : Indeed, when Y t is close to O, the large positive drift term 1 2Y t comes from the limit of the positive drift (X ε t ) 2 ε in the Y -equation of (6) as ε ↓ 0 (which is illustrated as Corollary 4.3). This makes the origin O an inaccessible point for Y t . However, for small ε > 0, the process (X ε t , Y ε t ) may still enter a thin strip around the half-line Oy B through O. Due to the strong Markov property of the process (X ε t , Y ε t ), once it enters the domain R 2 \Oy B , it will move along the fast flow to hit somewhere on Oy A . For any fixed ε > 0, the probability of hitting the level Y = −a for some a > 0 before moving along the fast flow and hit somewhere on Oy A decays to 0 as ε ↓ 0. As the process Y ε t is closer to the origin O, the positive drift term (X ε t ) 2 ε pushes the process Y ε t to bounce back to positive y-axis. Thus our limiting Y -process, the damped-BES(2), only lives on the positive Y -axis (see Figure 1(c)).
The above scenario can be roughly seen by considering the radial process In fact, by applying Itô's formula to (6) we see that where W r t is a standard Brownian motion on R. When the process X ε t is pushed by the flow to be close to the Y -axis, we have that Y ε t is close to r ε t , and thus (8) indicates the limiting Y -dynamics (7).
However, for fixed ε > 0, at a subexponential time scale, excursions of the process (X ε t , Y ε t ) moving from Oy A towards a level set y = −a will be observed. These excursions are directly crossing through a neighborhood of O. Due to the repelling nature of Oy B and the random perturbation, the process will not strictly lie on Oy B but it will move along the fast flow and come close to somewhere on Oy A . This induces jumps from points in Oy B to points in Oy A (see Figure 1(b)). At even larger time scale, such as an exponentially long time scale, large deviation effect makes the process (X ε t , Y ε t ) move from the attracting line Oy A to the repelling line Oy B . Such moves are not through O but are directed motions against the fast flow. Again the instability of Oy B and the random perturbation will make the process quickly jump back to Oy A . This induces back and forth jumps between points in Oy A and those in Oy B (see Figure 1(b)). As ε becomes smaller, motions of the process (X ε t , Y ε t ) to Oy B and jumping back become more and more rare, and in the limit no more such jumps appear, so that we come to the limiting process Y t which cannot penetrate through O. Thus as ε > 0 is close to 0, the description of the "metastable" behavior of system (4) involves both a diffusion part and a jump part. Figure 2 shows sample pathes of the X ε t and Y ε t processes, as well as the limiting Y -process (driven by the same Brownian motion as the driving Brownian motion for Y ε t ) starting from (X, Y ) = (0, 2) in 15000 steps for stepsize= 0.0001, with all steps rescaled to [0, 1]. In Figure 2(a), (b), the red curves are the sample pathes for Y t , and the blue curves are for sample pathes Y ε t when ε = 0.1 (Figure 2(a)) and ε = 0.01 (Figure 2(b)). In Figure 2(c), (d), the black curves are the sample pathes for X ε t when ε = 0.1 ( Figure  2(c)) and ε = 0.01 (Figure 2(d)). One can see that the process X ε t is mainly localized near 0, and the process Y ε t behaves similarly as the process Y t , especially when the parameter ε > 0 is small.
Let us also notice that, the cone formed by the set of extremal invariant measures of the unperturbed system (3) consists of both the lines Oy A and Oy B . And according to [16] the description of the limiting process shall be given by a Markov process on this cone. Our result is in a sense a specific example of this general paradigm. What we are demonstrating here is that the part Oy B of this cone is simply inaccessible, and the limiting process just lives on Oy A . This agrees with the heuristic that Oy A is the "stable" half-line of equilibriums and Oy B is the "unstable" half-line of equilibriums.
3 The limiting process and weak convergence theorem.
Let Y t be defined as the diffusion process on R with infinitesimal generator given by the operator A and domain of definition D(A) (see [11]). For any continuous function f : R → R that is twice continuously differentiable in y ≥ 0 we have and For y < 0 we further define The domain of definition of the operator A is given by the set of continuous functions f : R → R such that f (y) are twice continuously differentiable in y ≥ 0, with the limit By (10) and (12) we infer further that exists.
The existence of such a process Y t is guaranteed by the Hille-Yosida theorem (see [14], [34]). The closure A| D(A) of the operator A in the space of continuous functions on R exists and it actually defines a Markov process on {y ≥ 0}, which is a 2-dimensional radial Bessel process with linear damping on R + , that is inaccessible to the origin O, and it contains isolated points on {y < 0}. Our main theorem can be stated as follows.
(b) The measures on C [0,T ] (R) induced by the process Y ε t converge weakly as ε ↓ 0 to the measure induced by Y t with Y 0 = y π (x 0 , y 0 ).
Proof. Let δ = δ(ε) = ε α > 0 with δ → 0 as ε ↓ 0. We pick α = 1 10 . Set σ 0 = 0 and Our proof intuitively goes as follows: Step 1. We show that if Y ε t ≥ δ, then as ε ↓ 0 the process X ε t is very close to the Y -axis. This is proved in Lemma 4.1. We then show in Lemma 4.2 and Corollary 4.3 that as X ε t is small, the quantity In particular, this makes the process Y ε t behaves close to a 2-dimensional radial Bessel process with linear damping when Y ε t ≥ δ.
Step 2. We show that during the time τ k ≤ t ≤ σ k we have |X ε t | ≤ 3δ with high probability. This is because whenever |X ε t | ≥ 2δ the flow (6) with small ε > 0 will quickly bring the particle back to the region Y ≥ δ, and during this process the |X|-value is less or equal than 3δ. This is done in Lemma 4.4.
Step 3. We show that P(Y ε σ k = 2δ) → 1 as ε ↓ 0 and therefore δ(ε) → 0. This is because if Y ε t ≤ −1.5δ, then the flow of (6) with small ε > 0 will quickly bring the particle back to Y ≥ δ, and during this process the Y -coordinate is ≥ −1.99δ with probability → 1 as ε ↓ 0. This is done in Lemma 4.5.
Steps 1 and 2 together help us to settle (14) so part (a) of this Theorem. To prove part (b) of this Theorem, we shall make use of a modification of Lemma 3.1 in [28,Chapter 8]. This has been used in the works [23], [22], [26], [9], [10], [20], [19]. First, in Lemma 4.9 we show that the family of processes Y ε t is tight in C [0,T ] (R). Secondly, we show that for every continuous function f : R + → R such that f ∈ D(A) and every T > 0, having bounded derivatives up to the third order, uniformly in the initial condition (x 0 , y 0 ) ∈ R 2 we have as ε ↓ 0. The desired convergence in (b) then follows from (15) by the argument using martingale problem formulation of Markov processes (see [13,Chapter 4]). We are left with proving (15). To this end, we decompose Let us first estimate (I). In fact, we can estimate, by Lemma 4.2, that This helps us to conclude, by further making use of Lemma 4.8 that . To estimate (II), we notice that Y ε σ k = 2δ and Y ε τ k = δ. Thus by using the fact that f ′ (0) = 0 we obtain This combined with the fact that N (ε) · E(σ k − τ k ) → 0 and N · δ 2 → 0 as ε ↓ 0 from Lemmas 4.8 and 4.6, help us to conclude that |(II)| → 0 as ε ↓ 0. Finally it is easy to see that |(III)| → 0 as ε ↓ 0. Thus (15) is proved.
Proof. Let Y ε s ≥ δ for 0 ≤ s ≤ t. Let s ∈ [0, t] and we consider applying Itô's formula to (X ε s ) 2 . In this way, we obtain from (6) that Therefore taking expectation in (17) we obtain As we have Y ε s ≥ δ and (X ε s ) 2 ≥ 0, we can estimate Integrating the above differential inequality in the argument s from 0 to t we see that we have So finally we obtain the estimate As we have δ = ε α , the above estimate (19) implies that we have From here we infer that as t ≥ t 0 (ε) and ε > 0 sufficiently small we have E(X ε t ) 2 ≤ Cε 1−α for some C > 0. In particular, we can pick t 0 (ε) = ε (1−α)/2 .
The above estimate (16) cannot provide a precise estimate for (X ε t ) 2 ε , which enters as the first term in the right-hand side of the equation for Y ε t . In fact, this estimate can be obtained by first noticing the following Lemma.
Here the constant C > 0 depends on the bounds for the derivatives of f .
Proof. Let us assume that there exist uniform constants As we have Y ε t ≥ δ and r ε t = ( From (21) and (22), taking into account (4.1), we know that, as ε > 0 is small, for some constant M > 0 we have (23) As we have, by martingale formulation of Markov processes, that we see that the claim (20) follows from (23).
Proof. Let us consider a function f ∈ D(A) having bounded derivatives up to the third order. We can apply Itô's formula to the Y -dynamics of (6) and we obtain, for any From the proof of Lemma 4.2 we see that the estimate (20) is valid also for the integral from t 1 to t 2 . In fact, a finer estimate can be obtained by improving (23) via the estimate .
Thus by (25) we see that We can pick a function f ∈ D(A) with bounded derivatives up to third order, such that f ′ (y) ≥ δ 2 for y ≥ δ. From here we derive (24).
Proof. Let us introduce the angular variable θ ε t = arctan Here we take the principal branch of the function tan θ as θ ∈ − π 2 , π 2 . Due to symmetry of the system (6) with respect to the Y -axis, if the point θ ∈ π 2 , 3π 2 , then we can equivalently consider θ = π − θ as a replacement of θ. In this way, if θ ε t = π 2 , then the diffusion particle is on the Oy A axis, and if θ ε t = − π 2 , then the diffusion particle is on the Oy B axis. Let us apply Itô's formula from (6) to θ ε t and we obtain The processes W θ t and W r t are two driving standard Brownian motions on R. Set the slow time clock t = (δ/ε)t and let us consider a time-rescaled pair of processes Θ ε t = θ ε (ε/δ)t and R ε t = r ε (ε/δ)t . Then the stochastic differential equations satisfied by (Θ ε t , R ε t ) are given by Without loss of generality, let us start the process from some (X ε 0 , Y ε 0 ) such that X ε 0 ≥ 2δ and Y ε 0 ≤ δ. In this case we have R ε 0 ≥ 2δ. Consider the stopping time and let T ε X = (δ/ε)T ε X . We see that for t ∈ [0, T ε X ] we have X ε t ≥ δ and thus We pick δ = ε α with α = 1 10 . Since ε δ = ε 9/20 and ε δ 2 = ε 4/5 = ε 16/20 , it is seen from the R-equation in (28) that for finite t, In this case, ). Therefore, the Θequation in (28) can be viewed as a perturbation of the dynamical equation such that in finite t. From (29), (30), (31) and (32) we know that T ε X is finite, and thus T ε X ∼ O(ε 9/10 ) and Y ε T ε X ≥ δ. From here we know that whenever X ε 0 ≥ 2δ, the flow will quickly bring the particle to the region Y ≥ δ, and during this process X ε t ≤ 3δ. Thus we see that with high probability, we have X ε t ≤ 3δ. The other-side estimate X ε t ≥ −3δ is obtained in a same fashion.
Proof. This is proved in the same way as the proof for Lemma 4.4.
Proof. Let us introduce the auxiliary OU-process By Lemma 4.5, we know that as ε is small, with probability close to 1 we have Y ε τ k = 2δ. Taking this into account, as we have (X ε t ) 2 ε ≥ 0, we can estimate by comparison that Here σ is the first time that the OU-process Y t starting from Y 0 = δ hits Y = ±2δ.

As we have
we can further estimate We denote u(δ) = Eσ. By the standard theory of stochastic differential equations we know that u(y), y ∈ [−2δ, 2δ] is the solution to the ODE    −yu ′ (y) + 1 2 u ′′ (y) = −1 , Solving the above ODE system, we obtain that It is easy to see that as y ∈ [−2δ, 2δ] we have 0 ≤ y −2δ In particular, this implies that u(δ) ≤ Cδ 2 for some C > 0. Taking into account (34), we obtain the statement of this Lemma.
Lemma 4.9. The process Y ε t is weakly compact in C [0,T ] (R).
Proof. Let (Ω, F, P) be the probability space for Y ε t , 0 ≤ t ≤ T , such that for any ω ∈ Ω the sample path Y ε t (ω), 0 ≤ t ≤ T is a trajectory in C [0,T ] (R). We would like to show that from any sequence ε k ↓ 0, k = 1, 2, ... as k → ∞ one can extract a further subsequence ε k j ↓ 0, j = 1, 2, ... as j → ∞ such that for any bounded continuous for some j → ∞ and some random element Here E is the expectation with respect to P.
Unlike any of the previous Lemmas, here we will pick some fixed δ > 0. It is easy to see that if we replace δ = ε α by a fixed δ, then Lemmas 4.5, 4.6, 4.7 remain valid (The stopping times σ k and τ k can also be defined in a same way as for δ = ε α ), while the estimate (16) in Lemma 4.1 shall be modified into Henceforce we will make use of Lemmas 4.5, 4.6, 4.7 in below by directly adapting it to a fixed δ > 0.
Let for any small ε > 0 the family of sample pathes By Lemma 4.5 we know that P(Ω ε,δ bad ) → 0 as ε ↓ 0. Let us introduce a new probability measure P on (Ω, F, P) as follows. For any event A ∈ F we define P(A) = P(A\Ω ε,δ bad ) P(Ω\Ω ε,δ bad ) .
Let the corresponding expectation be defined by E. As we have P(Ω ε,δ bad ) → 0 as ε ↓ 0, we have that EX → EX for any random variable X as ε ↓ 0. From here we see that to show (35) it suffices to show that for some j → ∞ and some random element Y 0 t in C [0,T ] (R). We then understand (39) is just saying that Y ε t is weakly-compact under P. We will then make use of Lemma 5.1 in [29]. In fact, Lemma 5.1 in [29] indicates that in order to show weak-compactness of the family of sample paths in Y ε t in C [0,t] (R) under the measure P, it suffices to show, for each δ > 0, weak-compactness of the family of sample paths Y ε,δ The limiting measure µ 0 (Γ) has been studied in [7] via invariant measure and Kolmogorov (Fokker-Plank) equation, and has been shown to concentrate on Oy A . In the classical theory regarding random perturbations of dynamical systems (see [30, Section 6.6]), one is interested in considering the above two limits in a coordinated way. Namely we consider the case when t = t(ε) → ∞ as ε ↓ 0, and the asymptotic distribution of (X ε t(ε) , Y ε t(ε) ). In the classical case such as those demonstrated in [28], [30], the ω-limit sets of the unperturbed system consists of isolated compactum. In this case, if t(ε) increases sufficiently slowly, then over time t(ε) the trajectory of (X ε t(ε) , Y ε t(ε) ) cannot move far from that stable compactum in whose domain of attraction the initial point is. Over larger time intervals there are passages from the neighborhood of this compactum to neighborhoods of others; first to the "closest" compactum (in the sense of the action functional) and then to more and more "far away" ones. Such a phenomenon has been quantitatively characterized as the "metastable" behavior of the system.
The particular feature of the system (4) that we consider here has been in that the unperturbed system admits a continuum of stable attractors. At the level of timerescaled process (6), this leads to possible "jumps" of (X ε t(ε) , Y ε t(ε) ) between Oy A and Oy B . To illustrate this, let us imagine that we start our process ( As ε is small, in very short time ∼ O(ε), the process (X ε t , Y ε t ) first comes close to the Y -axis along the deterministic flow, and it hits a neighborhood of (0, y π (X ε 0 , Y ε 0 )) 1 . For any a > 0, let the stopping time (X ε t , Y ε t ) will then be quickly (at time scale ∼ O(ε)) brought back to a neighborhood of (0, a).
Under the above mechanism, as ε > 0 is small, what we actually see is that the process (X ε t , Y ε t ), although mostly stays within the half-plane of positive Y -value, being close to the Y -axis, makes rare excursions to (0, −a) along Y -axis, and after that quickly jumps back to (0, a). As ε > 0 becomes smaller and smaller, the excursion to (0, −a) becomes rarer and rarer, so that in the limit ε ↓ 0, the process (X ε t , Y ε t ) will not enter Oy B any more, and we arrive at the "process level stable attractor" (0, Y t ). This characterizes the metastable behavior of the system (6), and when changed back to the slow time, the perturbed system (4).

Let us introduce the elliptic operator
The above elliptic operator can be written as and In this way, the operator L 0 degenerates on x = 0. One can consider a corresponding Cauchy problem ∂u ε ∂t = L ε u ε , u ε (0, x, y) = f (x, y) , where f (x, y) is a bounded continuous function in (x, y) ∈ R 2 . The solution is represented by u ε (t, x, y) = E (x,y) f (X ε t , Y ε t ) .
By our Theorem 3.1, we infer that lim ε↓0 E (x,y) f (X ε t , Y ε t ) = lim ε↓0 E (x,y) f (0, Y ε t ) = E (0,y π (x,y)) f (0, Y t ). This gives the following Corollary 6.1. Let the initial condition f (x, y) be a bounded continuous function of (x, y). Then as ε → 0 we have u ε (t, x, y) → u(t, y π (x, y)) where u(t, y) is the solution of the equation 2. One can consider a more general system such as the one shown in Figure 4. Here the 3 axes Oy A 1 , Oy A 2 and Oy A 3 consist of stable equilibriums and the other 3 axes Oy B 1 , Oy B 2 , Oy B 3 consist of unstable equilibriums. One can analyze this system in a similar fashion as we did in this work, so that we expect to see the limiting process as a diffusion process on a tree Γ (see [26]). The tree Γ = Oy A 1 ∪ Oy A 2 ∪ Oy A 3 consists of 3 edges that are the semi-axes Oy A 1 , Oy A 2 , Oy A 3 . On each edge the limiting process is a Bessel-like process and the interior vertex O is inaccessible. The proof of these facts follows from the method we adopted in this paper as well as the techniques used in [28,Chapter 8], [27], [26]. One can first obtain "localization" type of results as we showed in Lemmas 4.4, 4.5. With such localization results at hand, we then show that the process localized onto Γ converges weakly to a diffusion process on the graph Γ, similarly as we did in the current work.
3. If the system (6) do not have the dissipative terms, so that it looks like Then the argument of the Lemmas 4.1-4.9 and the proof of Theorem 3.1 still go through, with minor changes in the estimates. The limiting Y -process will be a process of the form dY t = 1 Y t dt + dW 2 t , Y 0 = y π (x 0 , y 0 ) .
In particular, this implies that the Y t process keeps growing in the direction Oy A . That is to say, the energy grows in the direction of the stable manifold Oy A . Geometrically, this phenomenon comes from the fact that the energy constraint given by the conservative flow b(x, y) = (−xy, x 2 ) provides a positive force around the stable line Oy A . Thus the energy can keep growing at Oy A due to the random noise. Such a geometric phenomenon might be related to some problems in 2-d turbulence (see [12]).