Noise-Induced Stabilization of Planar Flows II

We continue the work started in Part I of this article, showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir's \cite{Peskir_07} generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.


Introduction
In Part I of this work [6], we investigated the following complexvalued dynamics (1.1) dz t = (az n+1 t + a n z n t + · · · + a 0 ) dt + σ dB t z 0 ∈ C where n ≥ 1 is an integer, a, a i ∈ C, σ ≥ 0, and B t = B (1) t + iB (2) t is a complex Brownian motion defined on a probability space (Ω, F , P). (We prove many results for the slight generalization of this equation given in (2.1).) There, we studied how the presence of noise (σ > 0 in (1.1) ) could stabilize the unstable underlying deterministic system (σ = 0 in (1.1)). To prove stability in the stochastic perturbation, we developed a framework for building Lyapunov functions and applied it to (1.1) assuming that the drift in equation (1.1) did not contain any "significant" lower-order terms; that is, we assumed that a j = 0 for n 2 − 1 ≤ j ≤ n. This was done in order to focus on the overarching elements of the construction of Lyapunov functions and to avoid any additional complexities presented by the presence of such lower-order terms. In this paper, we give an inductive asymptotic argument which shows how to remove this assumption, thereby proving the full version of Theorem 3.2 of Part I [6]. Here, we also provide a radially sharp lower bound on the decay rate of the invariant measure's density as stated in Theorem 5.5 of Part I [6]. This work extends and strengthens a stream of results on similar problems [1,2,4,5,8].
As first glance, it is surprising that the general case is substantially more complicated than those cases covered in Part I [6], as intuition suggests that the behavior of the process z t at infinity is determined by the leading-order term z n+1 and the noise. We will see here, however, that there is a range in which each of the intermediate lower-order terms becomes dominant at infinity as one moves towards to regions where noise dominates. The scaling analysis of Section 7.1 of Part I [6] hinted at this possibility when we employed our simplifying assumption, for it implied that the dominant balance of terms transferred directly from the leading order term z n+1 to the radial diffusion term without any interference from the remaining lower-order terms. In this paper, we will perform the analogous analysis for the general case in Section 3, showing how to correctly study the process at infinity in the presence of the intermediate lower-order terms. We will see, in particular, that the analysis used in Section 7.1 of Part I [6] breaks down in "small" regions containing the explosive trajectories of the deterministic system (σ = 0 in equation (1.1) ) and that the additional terms produce intermediate boundary layers which surround the inner most layer where noise dominates We begin in Section 2 by recalling the general setup of Part I [6]. There, we also state the main results we will prove in this paper. In Section 3, we perform the asymptotic analysis which guides and motivates the rest of the work. Specifically, we will use the asymptotically dominant operators yielded from it to define our Lyapunov functions in Section 4 by using a succession of associated PDEs based on these operators. In Section 5, we analyze boundary flux terms in order to show that the local Lyapunov functions can be patched together to produce a global Lyapunov function. Using these calculations, we verify the needed global Lyapunov structure in Section 6. In Section 7, we show that the family of Lyapunov functions we have constructed are radially optimal by establishing a matching lower bound at infinity of the invariant probability density function. In Section 8, we prove a version of Peskir's generalized Tanaka formula [7] which allows to avoid C 2 -smoothing along the boundaries of the local Lyapunov functions. Being able to avoid such gluing greatly simplifies former similar analyses [1,4,5]. In Section 9, we make some concluding remarks and suggestions for possible directions of future research.

Preliminaries
Throughout the remainder of this paper, we study more generally the complex-valued SDE where n ≥ 1, σ > 0, B t = B 1 t + iB 2 t is a complex Brownian motion and F (z,z) is a complex polynomial in the variables (z,z) satisfying F (z,z) = O(|z| n ) as |z| → ∞. Upon rescaling by a complex constant, all results obtained for (2.1) easily translate back to the more general case when the leading-order term in the drift has a non-zero complex coefficient. This is because any rotation of a complex Brownian motion is also a complex Brownian motion.
Recalling the notion of Lyapunov pairs defined in Section 4 of Part I [6], we will first show the following result.
By the results of Section 4 of Part I [6], Theorem 2.2 implies almost all of the main results. In particular, all consequences of Theorem 3.2 and Theorem 3.3 of Part I [6] follow except C (1 + |z|) γ dµ(z) = ∞ if γ ≥ 2n (2.3) where µ denotes the unique invariant measure corresponding to the process z t defined by (1.1). To prove (2.3), we show the following stronger result.
Theorem 2.4. Let ρ denote the invariant probability density function of (2.1) with respect to Lebesgue measure i 2 dz ∧ dz on R 2 . Then there exist positive constants c, K such that |z| 2n+2 ρ(z,z) ≥ c for |z| ≥ K. (2.5) Much of the paper is devoted to the proof of Theorem 2.2. Throughout, we will assume that the reader is familiar with Section 6 of Part I [6] which gives the general outline of the construction procedure used to produce Lyapunov pairs. In Section 7, we change our focus and prove Theorem 2.4. Section 8 contains the proof of a version of Peskir's result [7].

The Asymptotic operators and their associated regions
As in Part I [6], we will identify the asymptotically dominant terms in equation (1.1) at infinity by analyzing the time-changed Markov generator as r = |z| → ∞. Recall from 6.1 of [6] that after the time change t → τ = t 0 |z s | n ds, the generator L of the resulting process has the following form when written in polar coordinates z = re iθ : for some bounded smooth real-valued functions f k , g k and some smooth cutoff function h defined by h(r) = c for r ≥ 2 0 for r ≤ 1.
As before, the inclusion of the term h(r)r −n−2 sin(θ) above is not required to cover general equations of the form (1.1), but is rather included to deal with a secondary calculation needed to prove the lower bound on the density as stated in Theorem 2.4. As in Section 6.2 and Section 6.3 of [6], we will build our Lyapunov function for all (r, θ) by restricting analysis to the principal wedge In [6], we recall that to do this construction, we divided R into four regions: a "priming" region S 0 to initialize the construction, a transport region S 1 , a transition region S 2 to blend between the transport region S 1 and a region S 3 where the noise still plays a role at infinity. Moreover, this division of the principal wedge R was implied by the asymptotic analysis of L carried out in Section 7.1 of [6]. Here, too, we will see that a division of R holds in the general case, but this time there are many more regions. Initially, the analysis in the general case will exactly coincide with that done previously. Specifically, the first two regions, S 0 and S 1 , will be of the same form as before.
Afterwards, however, the dynamics at infinity undergoes further incremental changes, and this results in a significant increase in the number of regions and, consequently, the number of asymptotic operators.
As done in Section 7.1 of Part I [6], we begin by studying L as r → ∞, (r, θ) ∈ R, in regions where |θ| is bounded away from zero; that is, we first consider L • S λ 0 as λ → ∞. Observing that to satisfy L ≈ T 1 for r ≫ 0 with (r, θ) restricted to a region S 1 of the form where θ * 0 , θ * 1 are any positive constants. To see what happens in the remainder of R, we now turn to analyzing L • S λ α as λ → ∞ for α > 0 fixed. By fixing the constant θ * 1 > 0 from the definition of S 1 above to be sufficiently small, it is reasonable to assume that the dominant behavior of L • S λ α in λ can be discovered in R \ S 1 by considering the power series expansion of the coefficients of L. Fixing a J > n 2 + 6, we write where α i , β ij , γ i , δ i , ǫ ij are constants and the remainder functions R P and R Q satisfy for some positive constants C P , C Q . We have switched from P and Q to P and Q because, in P and Q, we include higher order terms from the power series expansion of r cos(nθ) and sin(nθ), respectively.
We begin by considering the region just next to S 1 ; that is, we analyze L • S λ α as λ → ∞ when α > 0 is fixed and small. Looking at L • S λ α for λ > 0 large, the following four terms are candidates for any dominate balance of all terms in L: where ⌊x⌋ is the greatest integer less than or equal to x. Note that we have neglected the δ i θ i ∂ θ , i ≥ 2, terms since for |θ| small they are dominated by nθ∂ θ . Similarly, we have neglected all of the ǫ ij r −i θ j ∂ θ terms since for θ small the corresponding (III i ) = γ i r −i ∂ θ term always dominates it. We have also neglected all of the α i rθ i ∂ r and β ij r −i θ j ∂ r terms since they are always dominated by the r∂ r term for r large and θ small. The terms (III i ) must be included since there is always a region, dictated by the value of α > 0, where θ is small enough so the (II) is dominated by some collection of the (III i ) as r → ∞. It is also important to realize why we have truncated the sum (III i ) at i = ⌊ n 2 ⌋ + 1. Comparing the diffusion term (IV ) with the terms (III i ), observe that we need only consider indices i of (III i ) satisfying α − i ≥ 2α − (n + 2). Rearranging this conditions produces the restriction i ≤ n + 2 − α. To obtain the claimed condition i ≤ ⌊ n 2 ⌋ + 1, we must first understand the relevant range of α. When 2α − (n + 2) = 0 the term (II) balances the term (IV ). Solving this condition to find α = n 2 + 1 and substituting this value of α into i ≤ n + 2 − α, we obtain the claimed bound i ≤ ⌊ n 2 ⌋ + 1. Assumption 5.7 from [6] translated to this context implies that γ i = 0 for i ∈ {1, . . . ⌊ n 2 ⌋+1}; and hence in the case considered previously, none of the (III i ) terms we have retained in (3.7) are present. To further illustrate the differences encountered here, we now analyze L • S λ α as λ → ∞ for α > 0 fixed in the relevant range (0, ⌊ n 2 ⌋ + 1].
For 0 < α < 1, observe that When α = 1, however, as λ → ∞ we see that Hence we expect to satisfy L ≈ T 2 as r → ∞ when all paths to infinity are restricted to a region of the form for some large constant c > 0. Note that c > 0 is chosen to be large to assure that the term (III 1 ) is not also dominant in the region defined above. We also leave open the choice of a specific curve b because what will happen in the remaining part of R: {(r, θ) ∈ R : r ≥ r * , |θ| ≤ b(r)} will suggest its definition.
When α = 1, (III 1 ) also becomes dominant suggesting that (3.11) r∂ r + nθ∂ θ + γ 1 r −1 ∂ θ should be the asymptotic operator in the next region. However if γ 1 = 0, on the curve nθ = −γ 1 r −1 and all of the ∂ θ terms in (3.11) vanish. Hence we must turn to the terms neglected above and do a further analysis of L • S λ α as λ → ∞ to find the dominate ∂ θ term. In fact it is likely that dominate balance expressed in (3.11) will fail to hold before the terms above exactly cancel.
To help see which terms need to be included in a neighborhood of the curve defined by nθ = −γ 1 r −1 , we make a convenient change of coordinates. The basic idea is to remove the term (III 1 ) by means of a coordinate transformation, returning us to a setting like that considered above when α ∈ (0, 1). Introducing the mapping (r, θ) → (s 3 , φ 3 ) defined by (3.12) s 3 = r and φ 3 = rθ + γ 1 n + 1 .
we see that the operator L transforms to where (3.14) ij are constants and, because |θ| ≤ θ * 1 , the remainders R P 3 , . We have chosen not to write out the term σ 2 2r ∂ 2 r in (3.13) in the variables (s 3 , φ 3 ) because it is too long of an expression and since it is always dominated, by considering the appropriate scaling transformation, by the other terms in L (s 3 ,φ 3 ) as s 3 → ∞.
After this change of variables, note that (II) + (III 1 ) has transformed into (n + 1)φ 3 ∂ φ 3 , hence we have "removed" (III 1 ). However, note that a new γ term is generated, playing the same role as (III 1 ) did in the previous coordinate system. While this may not seem like progress, notice that angular diffusion term (analogous to (IV )) now has a coefficient s −n 3 where it was r −n−2 in the old coordinates. Hence this term is more powerful. By a similar line of reasoning to the above, only the terms analogous to (III i ) with i ∈ {1, . . . , ⌊ n 2 ⌋} are not dominated by the noise. This is one less than previously. Therefore by performing such substitutions iteratively, we will be able to remove enough terms so that in the final coordinate system, the angular diffusion term would dominate all analogous terms to the (III i )'s . An important point which makes this iteration possible is that P 3 and Q 3 have that same forms as P and Q, respectively, in that the lower limits of the sums do not change. Even though the upper limits of the sums will increase, these added contributions are of lower order so they do not change the analysis.
To finish the analysis in the variables (s 3 , φ 3 ), we need to complete our understanding of the boundary |θ| = b(r), extract the dominant operator which replaces T 2 after we cross this boundary, and determine the lower limit of the region where this new operator remains dominant.
In a subset of the region |φ 3 | < φ * , the approximation L (s 3 ,φ 3 ) ≈ T 3 holds as s 3 → ∞. To discover the boundary of this region, we now study L (s 3 ,φ 3 ) • S λ α for α > 0. As before in the previous coordinate system, there are four terms which are potentially involved in any dominant balance of the terms in L (s 3 ,φ 3 ) • S λ α as λ → ∞: Notice that the terms (I) 3 , (II) 3 , (III) 3 , (IV ) 3 are completely analogous the terms (I), (II), (III 1 ), (IV ) from the preceding discussion. As already noted, after the change to the (s 3 , φ 3 ) coordinates, only terms γ . . , ⌊ n 2 ⌋} could possibly dominate or balance the angular diffusion term (IV ) 3 , and the term (III) 3 is the leading order term of this form. Hence when n = 1 or n = 2, there is only one such dominant term of this form, namely (III) 3 , and it is of either the same (case n = 2) or lesser (case n = 1) order as (IV ) 3 . In general (n ≥ 3), we will have to perform additional transformations to remove all of the possibly dominant terms. Before considering general n ≥ 3, we pause to finish the analysis in the cases when n = 1 and n = 2.
3.1. Remaining regions when n = 1. For every α ≥ 0, we see that When α = 1/2, the term (IV ) 3 also becomes dominant in λ. In particular, the region where we expect L (s 3 ,φ 3 ) ≈ T 3 as s 3 → ∞ is precisely for some η * > 0. Additionally, the operator Summing this up, we have seen that when n = 1, the approximating operators are T 1 , T 2 , T 3 , A with corresponding regions S 1 , S 2 , S 3 , S 4 where we expect the approximation to be valid for r > 0 large. Remark 3.20. We have already introduced a number of parameters (e.g. θ * 1 , φ * , η * , r * ) thus far that will have to be chosen to satisfy a number of properties. Instead of writing these properties explicitly, we simply need to make sure that we vary the parameters in a consistent way to obtain them. That is, we will always choose θ * 1 > 0 small enough, then pick φ * = φ * (θ * 1 ) > 0 large enough, then choose η * = η * (θ * 1 , φ * ) > 0 large enough, and then finally pick r * = r * (θ * 1 , φ * , η * ) > 0 large enough. For example, to assure that S 3 and S 4 defined above are of the required form outlined in Section 6.2 of [6], we can choose the parameters θ * 1 , φ * , η * , and r * in this way to see that in fact

3.2.
Remaining regions when n = 2. Notice that for 0 ≤ α < 1 When α = 1, then (III) 3 + (IV ) 3 also becomes dominant in λ. Therefore, this suggests that the region where for some η * > 0. Here again, we have picked the parameters in the way discussed in Remark 3.20. Notice also that the operator where we have again picked φ * and r * according to Remark 3.20. Summing this up, we have seen that when n = 2, the asymptotic operators are T 1 , T 2 , T 3 , A with corresponding regions S 1 , S 2 , S 3 , S 4 where the approximation is expected to be valid.
3.3. The remaining cases: n ≥ 3. If 0 ≤ α < 1, then 3 . Hence to analyze L (s 3 ,φ 3 ) around this other potentially dangerous curve, we make another substitution (s 3 , φ 3 ) → (s 4 , φ 4 ) given by where n+2 . As before, we use the new variables (s 4 , φ 4 ) to define the boundary curve b precisely by setting ij are constants, all sums above are finite sums and, since φ 4 will be bounded in any subsequent region, R P 4 and R Q 4 satisfy Similarly, we can now uncover all asymptotic operators and their associated regions when n = 3 and n = 4 because the angular noise term (IV ) 4 is of sufficient strength in λ.

3.3.1.
Remaining regions when n = 3. Analogous to the case when n = 1, satisfies L (s 4 ,φ 4 ) ≈ T 4 as s 4 → ∞ when paths to infinity are restricted to a region of the form where the parameters η * , φ * > 0 have been chosen according to Remark 3.20. Also, can be used to approximate L (s 4 ,φ 4 ) asymptotically for large s 4 in a region of the form where we have again picked the parameters as described in Remark 3.20. Thus, when n = 3, we obtain the approximating operators T 1 , T 2 , T 3 , T 4 , A with corresponding regions S 1 , S 2 , S 3 , S 4 , S 5 where the approximation is expected to be valid.

3.3.2.
Remaining regions when n = 4. Similar to the case when n = 2, the region where is in good approximation to L (s 4 ,φ 4 ) for r = s 4 > 0 large is given by where the parameters have been chosen appropriately. Also, contains the dominant, large s 4 behavior corresponding to L (s 4 ,φ 4 ) in Thus when n = 4, we obtain the asymptotic operators T 1 , T 2 , T 3 , T 4 , A and their regions S 1 , S 2 , S 3 , S 4 , S 5 of approximation.
3.3.3. The remaining cases: (n ≥ 5). We continue until this process stops. More precisely, if n = 2j + 1 or n = 2j + 2 for some j ≥ 0, then the inductive procedure yields the asymptotic operators T 1 , T 2 , . . . , T j+3 , A and respective regions To write each of them explicitly, set s m = s 2 := r, φ 2 := θ for m ≥ 3 and let We see that T 1 , . . . , T j+3 are given by If n = 2j + 1, then the diffusive operator A satisfies On the other hand if n = 2j + 2, we have Choosing the parameters θ * 1 , φ * , η * , r * according to Remark 3.20, we may write all corresponding regions as follows. Note first that S 1 , S 2 , . . . , S j+2 are given by On the other hand if n = 2j + 2, then S j+3 and S j+4 are given by . It is also important to notice that L, when written in the variables (s m , φ m ) for m = 3, . . . , j + 3, satisfies are constants, all sums are finite sums, and by the choice of J > n 2 + 6 the remainders satisfy the following bounds on S m Here the constants C Pm and C Qm may depend on θ * 1 , φ * and η * , but they do not depend on r * .

The Construction of Ψ on R in the general case
Employing the asymptotic analysis of the previous section, we now define our candidate Lyapunov function Ψ on the principal wedge R.
As in Part I [6], to initialize the propagation procedure used to define all of the ψ i 's we need one more region S 0 (hence the i = 0 above) defined by , and we define the initial function ψ 0 on S 0 by ψ 0 (r, θ) = r p for some p ∈ (0, n). We recall that this choice is made because the radial dynamics along T 1 is decreasing in S 0 .
4.1. The construction in the transport regions. We first turn our attention to defining the functions ψ 1 , ψ 2 , . . . , ψ j+3 respectively on the regions S 1 , S 2 , . . . , S j+3 as solutions to boundary-value problems involving the asymptotic operators T 1 , . . . , T j+3 . Because the number of indices will be daunting otherwise, we adopt the following conventions.
Convention 4.1. When it is clear which coordinate system in which we are working, (s m , φ m ) will be written more simply as (s, φ). For example, ψ m (s m , φ m ) for m = 3, . . . , j + 3 will often be written as ψ m (s, φ).

Convention 4.2.
In the expressions we will derive for ψ 1 , . . . , ψ j+3 , there will be several parameters with double indices, e.g. see p l,m and q l,m below in Lemma 4.11. The second index m simply corresponds to the function ψ m . Thus when it is clear that we are working with ψ m , we will often write p l,m and q l,m more compactly as p l and q l respectively.
It is convenient for our analysis that the boundary conditions given for Poisson equation defining ψ j+4 on the inner most region S j+4 (the one dominated by diffusion) are symmetric under reflection in the angular coordinate φ in S j+3 . Thanks to this symmetry, the value of ψ j+4 at the time of exit of the diffusion from S j+4 depends only on the time of exit and not on which side of the boundary it exits (each being possible since the dynamics in S j+4 is diffusive). As we will see below, this allow a clean reduction in the dimension of problem used to define ψ j+4 . We accomplish the desired symmetry of the boundary conditions of ψ j+3 by making the penultimate function ψ j+3 satisfy the desired symmetry The cost of producing this symmetry in the penultimate region can be seen in the need to delicately choose constants in the all of the preceding regions since both the regions and the corresponding dominant operators are inherently asymmetric in the angular coordinate.
Since we will have to make different choices of defining problems above and below φ = 0 in each region to produce the symmetry described above, we will break up the definition of ψ m in two pieces as follows The construction in S 1 . Let ψ ± 1 satisfy the following PDEs on S 1 : where q ∈ ( p n , 1) is fixed and h + 1 , h − 1 > 0 will be determined later (to produce the reflective symmetry).
Since θ * 0 > π 2 , we recall from Section 7.3 of Part I [6] that the equations above are not well-defined with the given boundary data because some characteristics along T 1 cross r = r * before reaching the lines θ = ±θ * 0 . This can be easily remedied by enlarging the domain of definition of the equation (4.4) to to ensure that all characteristics hit the boundaries θ = ±θ * 0 prior to crossing r = r * .
The construction in S 2 . In a similar fashion, let ψ ± 2 solve (4.6) Note that we may solve (4.6) explicitly using the method of characteristics to obtain the following expression for ψ ± 2 : Observe that ψ ± 2 consists of two terms, each of which is homogeneous under S λ α for every α ≥ 0. Moreover, The inductive construction in the remaining transport regions. We can now use the same idea employed above for ψ 2 to define ψ 3 , . . . , ψ j+3 inductively. Thus for m = 3, . . . , j + 3, define ψ m as the solution of (4.9) (T m ψ ± m )(s, φ) = −h ± m s pm |φ| −qm ψ ± m (s, ±φ * ) = ψ m−1 (s m−1 , φ m−1 (±φ * )) for all (r(s, φ), θ(s, φ)) ∈ S m where p m , q m , h ± m > 0. We again recall that we have suppressed the index m in (s, φ) using our convention. We have also suppressed the second index m in p m and q m above; that is, p m,m = p m and q m,m = q m .
Inductively, p m and q m are chosen to satisfy These choices may at the outset seem mysterious but they ar all determined by the exit distribution of the diffusion generated by A from S j+4 and by the scaling relationships of the ψ i 's along common boundaries. See the discussion in Section 7.4 of Part I [6]. We now prove a lemma which gives an expression for ψ m which is convenient for further analysis. Although we will need them, at first glance it is important to ignore the many relations that the constants in the statement of the result satisfy. The basic form of ψ ± m is what is most important. Moreover, the constants d ± l = d ± l,m are such that ψ m > 0 on S m and ψ m (s, φ) → ∞ (4.14) as s → ∞, (r(s, φ), θ(s, φ)) ∈ S m . Remark 4.15. Notice that (4.13) and (4.10) imply the following ordering of the constants p l,m , q l,m for m > 2 p 1,m < p 2,m < · · · < p m−1,m = p m,m (4.16) q 1,m < q 2,m < · · · < q m−1,m < q m,m < 1.
The above relations will be especially helpful later when we do asymptotic analysis of ψ m .
Remark 4.17. In the proof of Lemma 4.11 we will, in addition, derive some properties of the constants d ± l = d ± l,m . These will be collected in the statement of Corollary 4.18 below, and they will be used, in particular, to show that we can choose the constants h ± m > 0 in a natural way so that the symmetry property (4.3) is satisfied.
Before proving the lemma and corollary above, we state another lemma which shows that, assuming the conclusions of Lemma 4.11 and Corollary 4.18, we can pick the constants h ± m in a reasonable way so as to have (4.3).
Lemma 4.19. Fixing a constant K 0 > 0, for all ǫ > 0 there exists a constant K 1 > 0 so that the following holds. If h + 1 , h + 2 , . . . , h + j+3 is a collection of positive parameters with h + i ≤ K 0 for all i then for any Remark 4.20. Recall that φ * > 0 is as in the definition of the regions S i . The content of the Lemma 4.19 is the following. Later, we will need to use the parameters h + i to ensure that the fluxes across the boundaries between osculating regions have the desired sign just as we did in Section 8.1 of Part I [6]. We then need to choose the h − i to both satisfy the boundary flux condition and make the ψ j+3 have the desired symmetry. Since as φ * → ∞ the regions become increasingly symmetric in the angular variable φ, it is intuitively clear that the h − i which produce a symmetric φ are close the h + i which were already chosen. Hence, the h − i which produce symmetry also satisfy the needed boundary flux condition. To check (4.14) is valid for ψ ± 3 , consider the dynamics along T 3 s = s andφ = (n + 1)φ, and let τ * = inf t>0 {|φ t | = φ * }. Using the inequality (4.8), notice for all (s, φ) such that (r(s, φ), θ(s, φ)) ∈ S 3 for some constant c > 0. Hence we now see that ψ 3 > 0 on S 3 and ψ 3 → ∞ as s → ∞, (r(s, φ), θ(s, φ)) ∈ S 3 . Now assume all conclusions are valid for some m − 1 ≥ 3. Using the method of characteristics and the inductive hypothesis, we can obtain the claimed expression for ψ m as well as all relationships between constants in the statements of Lemma 4.11 and Corollary 4.18. To obtain (4.14), we may assume inductively that for all (s, φ) with (r(s, φ), θ(s, φ)) ∈ S m−1 where c > 0 is a constant (which is in general different from the one used above). As before, consider the dynamics along T m : s = s andφ = (n + m − 2)φ and let, recycling notation, τ * = inf t>0 {|φ t | = φ * }. Then we similarly obtain for some c > 0 which is in general different from the c used above. This now finishes the proof of the result.
Proof of Lemma 4.19. Let h + 1 , h + 2 , . . . , h + j+3 be a bounded collection of positive parameters and fix ǫ > 0. We will see that there is a unique for all m = 1, . . . , j + 3. By Corollary 4.18 and Lemma 4.11, the first conclusion of the lemma will then follow immediately since the ψ ± j+3 are a linear combination functions with coefficients d ± m,j+3 respectively. The closeness of the h's will follow by inspection of the choice of the h − j 's giving (4.22) for all m = 1, . . . , j + 3. We proceed inductively and begin by analyzing the equality implying that we must pick .
Using the expressions given in Corollary 4.18 for the b's, a simple argument employing Taylor's theorem gives the following asymptotic formula as φ * → ∞: Therefore as φ * → ∞ since the constants h ± j+2 are bounded functions of φ * . In particular, this implies that the unique choice of h + j+2 (which is positive for φ * large enough) determined by the relation d + j+2,j+3 = d − j+2,j+3 has the desired property of part (2) of the lemma. To continue by induction, we need one more step to see how to proceed in general. Notice that this is only necessary if j ≥ 1 where n = 2j + 1 or n = 2j + 2. By Corollary 4.18, observe that and, by the right most equality, d + j+1,j+3 = d − j+1,j+3 is equivalent to . By (4.23) and Corollary 4.18, we have as φ * → ∞. Putting these formulas together, since h + 1 , h + 2 , . . . , h + j+3 were assumed to be bounded and q j+1 − q j+2 < 0 by (4.10) we obtain This finishes the result in this case. To see in general when d + m,j+3 = d − m,j+3 for general m = 2, . . . , j, assume by induction that Similarly, using Taylor's theorem and the asymptotic formulas above, we see that realize that it is equivalent to the relation as φ * → ∞, one can easily deduce from (4.5) that for fixed θ * 1 , as φ * → ∞ the choice of h − 1 determined by the symmetry condition (4.24) approaches h + 1 . Note that this finishes the proof of the result. Now that we have the desired symmetry we turn to defining the final function ψ j+4 in the region S j+4 where noise does play a role.
To show that ψ j+4 is strictly positive on S j+4 and ψ j+4 (s, φ) → ∞ as s → ∞ with (s, φ) such that (r(s, φ), θ(s, φ)) ∈ S j+4 , using (4.27) we see that for some constant c > 0 where the last inequality follows by the inductive argument proving both Lemma 4.11 and Corollary 4.18. We thus obtain the desired bound for some c ′ > 0.

Boundary flux calculations
Here show how one can choose the positive parameters h + i , θ * 1 , φ * , η * so that the jump conditions of Corollary 6.9 of Part I [6] are also satisfied. We must be careful to see that all choices are consistent with Remark 3.20 and Lemma 4.19. Each boundary has two disjoint parts, implying that we must check two, although very similar, flux conditions. We proceed from boundary to boundary, starting with the: 5.1. Boundary between S 0 and S 1 . We begin on the side of the boundary where θ > 0. We must pick the parameters so that for r ≥ r * . By inspection of the formula (4.5), we first note that ψ + 1 (r, θ) = r p ψ + 1 (1, θ). Using this and the equation (4.4) defining ψ + 1 , observe also that Rearranging this produces Therefore combining ∂ψ 0 ∂θ = 0 with (5.2) gives . Because ψ + 1 (1, θ * 0 ) = 1, sin(nθ * 0 ) > 0 and cos(nθ * 0 ) < 0, picking On the side of the boundary where θ < 0, we must see that this for r ≥ r * . By Lemma 4.19, we have already picked h − 1 and we recall that as φ * → ∞, h − 1 → h + 1 . Using the very same process as above, (5.4) is satisfied provided Therefore, both quantities can be seen to be negative by first picking and then taking φ * > 0 sufficiently large. Note that this is consistent with the flow of choices outlined in Remark 3.20.

5.3.
Boundary between S 2 and S 3 . For illustrative purposes, we perform one more boundary-flux estimate before proceeding on to the general, inductive calculation in the remaining transport regions. We begin on the side of the boundary where φ 3 > 0. Note that for φ * > 0 large, it is also true that θ > 0 on this side. As opposed to the previous cases, it is more convenient to use the explicit expressions obtained for ψ ± 2 and ψ ± 3 . In doing this, we first note that and C 2 (φ * ) is a constant depending on φ * . Our goal is to see that for φ * large enough, C 1 (φ * ) < 0. Hence for r * > 0 large enough, the quantity (5.10) will also be negative. Recalling the dependence of d + 2,3 on φ * in Corollary 4.18 and that q 3 > q 2 , we note that as φ * → ∞. Since d − 2,2 is positive and independent of φ * , q 2,3 = p 2,3 n + 1 = p 2,2 + q 2,2 n + 1 = p + q n + 1 , and q 2 = q ∈ ( p n , 1) we find that q 2 > q 2,3 . Hence, choosing φ * > 0 large enough, C 1 (φ * ) is negative. Thus for r * > 0, the quantity on the left-hand side of (5.10) is also negative. A nearly identical computation will yield the desired result on the other side of the boundary.

The boundaries between the remaining transport regions.
We now consider the flux across the two boundaries between S m and S m+1 where k = 3, . . . , j + 2. As done in the previous case, we focus on the side of the boundary where φ m+1 > 0. Note, too, with φ * > 0 large enough, φ m is also positive on that side of the boundary. Using the expressions derived in Lemma 4.11, realize that where c < p m+1 + m − 1, and C 2 (φ * ) is a constant that depends on φ * . Using Corollary 4.18 to write out d + m,m+1 and recalling that q m+1 > q m , note that as φ * → ∞ Recalling that and q m ∈ ( pm n+m−2 , 1) we see that q m > q m,m+1 giving that C 1 (φ * ) is negative for φ * large enough. Thus for r * > 0 large enough the quantity on the left-hand side of (5.11) is negative. A nearly identical result holds on the other side of the boundary. 5.5. Boundary between S j+3 and S j+4 . In the following computation, we will need to employ Lemma 8.7 of Part I [6] since the expression for ψ j+4 in (4.27) is not explicit. Also, we only show the case when n = 2j + 1, j ≥ 0, as the other case is similar.
Consider the side of the boundary where φ j+3 > 0. Recalling the notation G a,c introduced in Section 8.1 of Part I [6] for the Laplace transform of the exit time, observe that and C 2 (η * ) is a constant which depends on η * . Choosing for some h > 0 and applying the Lemma 8.7 of Part I [6], realize that as η * → ∞ Using (4.26) and the relations q j+3 > p j+3 /(n + j + 1) and n = 2j + 1, we see that 2p j+4 3n + 2 < q j+3 Picking h small enough implies that C 1 (η * ) < 0 for η * > 0 large enough. Therefore choosing r * > 0 large enough implies that the quantity on the left-hand side of (5.12) is negative. A similar result is easily seen to hold on the other side of the boundary.
6. Checking the Global Lyapunov Bounds 6.1. Checking the Local Lyapunov Property. Here we check that the approximating operators T 1 , T 2 , . . . , T j+3 , A were chosen correctly so that ψ 0 , ψ 1 , . . . , ψ j+4 are actually locally Lyapunov functions on their respective domains S 0 , S 1 , . . . , S j+4 . This simply involves replacing each asymptotic operator with L and estimating the remainder locally on each region. Factoring in the time change, the required bound for Lψ i on S i will then follow easily.
Hence we may pick K > 0 large enough so that for φ * ∧ r * ≥ K Lψ ± 2 (r, θ) ≤ −c 2 r p |θ| q + d 2 for all (r, θ) ∈ S 2 . Undoing the time change, we then determine the existence of positive constants C 2 , D 2 such that on S 2 Lψ ± 2 ≤ −C 2 r p+n |θ| q + D 2 (6.4) Remark 6.5. Before proceeding onto the remaining regions, it is important to note that Corollary 4.18 and the relations (4.16) imply that for m ∈ {3, 4, . . . , j + 3} and l ∈ {1, 2, . . . , m − 1}: where the constant in the asymptotic formula above is independent of θ * 1 , η * and r * . The above fact will be helpful when controlling remainder terms in what follows.
Region S m , m = 3, . . . , j+2. In the following computations, it is helpful to consult (3.32) and the remainder estimate immediately below it. For lack of better notation, we will also use ψ ± m to denote the function of (r, θ) determined by ψ ± m = ψ ± m (s, φ). Let and write where each equality above is valid on S m . We first focus on estimating (L (s,φ) − T m − N)ψ ± m (s, φ) for all (s, φ) such that (r(s, φ), θ(s, φ)) ∈ S m . Using the simple nature of the expression derived for ψ ± m as well as Remark 6.5, we note that the bound s pm |φ| qm holds on S m where C 1 (r * , φ * ) is a constant which can be chosen to be a small as we wish by first picking φ * > 0 large and then picking r * > 0 large. Therefore, making such choices we see that To estimate the remaining term Nψ ± m (s, φ), first recall that s = r φ = r m−2 θ + r m−3 c 2 + · · · + c m−1 and so we may write where Y is a polynomial in s of degree at most s m−3 . Using this allows us to obtain a similar bound where C 2 (r * , φ * ) is a constant satisfying the same property as C 1 (r * , φ * ) above. Hence, we may choose φ * > 0 large enough and then r * > 0 large enough so that for some positive constants c m , d m for all (s, φ) with (r(s, φ), θ(s, φ)) ∈ R m . Undoing the time change, we see that there exist positive constants C m , D m such that on S m Lψ ± m (r, θ) ≤ −C m s pm+n |φ| qm + D m . (6.8) Region S j+3 . Here, we again use ψ ± j+3 to also denote the function of (r, θ) defined by ψ ± j+3 (s, φ). The estimate in this region is nearly identical to the one that precedes it, except that the lower bound in the definition of S j+3 is slightly different depending on the parity of n. Nevertheless, we may essentially trace through the inequalities in the previous case to see that the same estimates hold, except that the constants C 1 and C 2 in this case depend on, in addition to φ * and r * , η * . We may, however, still pick the parameters according to Remark 3.20 to arrive at the desired estimate on S j+3 s p j+3 +n |φ| q j+3 + D j+3 (6.9) for some positive constants C j+3 , D j+3 .
Region S j+4 . The estimates in this case are also very similar to the previous ones, but in fact they are a little easier since we include more terms of L in the approximating operator A. In what follows, we again use ψ j+4 to denote the function of (r, θ) determined by ψ j+4 (s, φ) where (s, φ) = (s j+3 , φ j+3 ). Notice that for all (s, φ) with (r(s, φ), θ(s, φ)) ∈ S j+4 Lψ j+4 (r, θ) = L (s,φ) ψ j+4 (s, φ) where the operator N was defined in (6.7). Using the very same ideas in the previous two regions and recalling that γ (j+3) 1 s −1 ∂ φ is included in A when n = 2j +2, we note that for all (s, φ) with (r(s, φ), θ(s, φ)) ∈ S j+4 where C(r * , η * , φ * ) is a constant which can be made arbitrarily small by picking r * , η * , φ * according to Remark 3.20. Thus choosing these parameters accordingly, we see that there exist constants c j+4 , d j+4 > 0 such that for all (s, φ) with (r(s, φ), θ)s, φ)) ∈ S j+4 . Undoing the time change, we determine the existence of positive constants C j+4 , D j+4 such that on S j+4 Lψ j+4 (r, θ) ≤ −C j+4 s p j+4 +n + D j+4 . (6.10) 6.2. Checking the specific Lyapunov bounds. Here we show that we can pick parameters so that the conditions of Proposition 6.13 of Part I [6] are satisfied when γ ∈ (n, 2n) is arbitrary. By the estimates of the previous section, all we need is the following proposition.
In the proof of Theorem 2.4 we will need the following result which is a corollary of the proof of Theorem 2.2. The result gives us uniform bounds in the initial condition on return times to large compact sets of the process, time-changed to accommodate the "substitution" w t = 1/z n t , determined by the adjoint L * . Corollary 7.1. Consider the stochastic differential equation on C\{0} where P(z,z) = z n+1 + F (z,z), and F , n, σ and B t are as in equation Then the stopped process z * t∧Sγ is non-explosive and for each γ > 0 sufficiently large we have: Additionally, for each t, ǫ > 0 there exists γ > 0 large enough so that Proof of Corollary 7.1. Let a ∈ C be such that a n = −1 and consider the process az * t∧Sγ . Our goal is to show that az t∧Sγ has a Lyapunov pair (Ψ, Ψ 1+δ ) for some δ > 0. Non-explosivity will follow from Lemma 4.4 of Part I [6], and the remaining conclusions concerning the entrance times S γ , γ > 0, will follow from Proposition 4.7 of [6].
Notice first that the generator M of az * t∧Sγ is of the following form when written in polar coordinates (r, θ): where L is of the form (3.1). Hence, because of the form of M, our Lyapunov function Ψ will be the same one constructed in Section 4. Upon replacing n by 1 in the inequalities (6.2), (6.3), (6.4), (6.8), (6.9), and (6.10), and then applying Proposition 6.8, we see that the chosen Ψ has the required local Lyapunov estimate for (r, θ) ∈ S m , m = 0, 1, . . . , j + 4: where C, D and δ are positive constants. Since the boundary flux contributions will have the appropriate sign, the result now follows.
Proof of Theorem 2.4. First note that the generator L of the process z t has the following form when written in the variables (z,z): L = P(z,z)∂ z + P(z,z)∂z + σ 2 ∂ z ∂z where P(z,z) = z n+1 + F (z,z). Let L * denote the formal adjoint of L . Motivated by the discussion of the substitution w t = 1/z n t at the beginning of the section, define c(z,z) = |z| 2n+2 ρ(z,z) where ρ is the invariant probability density function. Since L * is elliptic and L * ρ = 0, observe that c is a smooth function everywhere since ρ is smooth everywhere. To see which equation c satisfies, write ρ = c|z| −2n−2 and use the fact that L * ρ = 0 to see that where M is of the form and the potential f satisfies f (z,z) = −∂ z (P(z,z)) − ∂z(P(z,z)) In particular, we also note that c solves the equation (|z| −n−1 Mc)(z,z) = 0 for z = 0. (7.2) Using (7.2), we will now apply Feynman-Kac to obtain an expression for c(z,z) that can be analyzed as |z| → ∞. Now consider the time-changed process z * t∧Sγ , γ > 2, introduced in Corollary 7.1. Observe that the generator of z * t∧Sγ constitutes every term in |z| −(n−1) M except for multiplication by the potential function |z| −(n−1) f (z,z) which is smooth in (z,z) for z = 0 and satisfies |z| −(n−1) f (z,z) = O(1) as |z| → ∞.
Hence, in particular, |z| −(n−1) f (z,z) is bounded on the set {z ∈ C : |z| ≥ γ} for all γ > 0. Let S γ,n be the first exit time of z * t from the annulus A γ,n = {γ < |z| < n}. By Feynman-Kac, we have for γ ≥ 2 By Corollary 7.1, we have that z * t∧Sγ is non-explosive. Thus by Fatou's lemma, taking the lim inf n→∞ of both sides of the above we obtain for |z| ≥ γ ≥ 2 Applying Corollary 7.1 again, for γ > 0 large enough S γ < ∞ almost surely. Hence, applying Fatou's lemma and taking the lim inf t→∞ of both sides of the previous inequality we see that We now bound the right-hand side of (7.3) from below. Since (|z| −n−1 Mc)(z,z) = 0 for z = 0 and the operator |z| −n−1 M is elliptic for z = 0, there exists a constant C(γ) > 0 such that Moreover, since |z| −(n−1) f is bounded for |z| ≥ γ, there exists a constant D(γ) > 0 such that Hence, we obtain where the inequality above holds for all γ, t > 0. Applying Corollary 7.1 once more, we see that for each t > 0 there exists γ > 0 such that inf |z|≥γ c(z,z) > 0 finishing the result.

Generalized Itô's Formula
In this section, we give a differently packaged proof of a weaker version Peskir's extension of Tanaka's formula [7] which still affords the structure needed to build the Lyapunov functions contained in this paper and Part I [6]. Instead of making use of Tanaka's formula as in [7], we opt to mollify along interfaces where the function is not C 2 and then take limits. For convenience, we deal solely with the case of a time-homogeneous diffusion process ξ t on R d with generator where f i , g ij are locally Lipschitz and the matrix (g ij ) is non-negative. Furthermore, assume that ϕ ∈ C(R d : R) is such that where the ϕ i 's are C 2 on the domains above and b ∈ C 2 (R d−1 : R). The case of finitely many non-intersecting boundaries is a simple consequence of the following result.
Proof of Theorem 8.1. Because we will stop the process ξ t at time τ n , without loss of generality we may assume that ϕ has compact support (e.g. in a ball centered at the origin with radius much larger than n). Let χ : R d → R be a smooth mollifier, set χ ǫ (ξ) = ǫ −d χ(ǫ −1 ξ) and define Applying Dynkin's formula we have To obtain the desired formula, we begin computing partial derivatives of ϕ ǫ . To keep expressions compact, we will use ∂ ξ j to denote ∂ ∂ξ j . Write Integrate by parts once and use the fact that ϕ 1 and ϕ 2 agree on the boundary Γ = {x ∈ R d : Using the equality on the previous line, apply ∂ ξ i to both sides and then integrate by parts in the same fashion to obtain where Γ = {x : x d = b(x 1 , . . . , x d−1 )} and ν i is the i-th component of the unit surface normal (−∇b(x), 1)/ 1 + |∇b(x)| 2 of Γ. We now claim that for x ∈ Γ and j = 1, . . . , d − 1 To prove this claim, for x ∈ R d define h(x 1 , . . . , x d−1 ) = ϕ(x 1 , . . . , x d−1 , b(x 1 , . . . , x d−1 )).
We now see what this does for us. Letting * denote convolution, we can now write (8.3) as g ij (ξ s )ν i ν j dS Γ (x)ds Since f i , g ij are locally bounded, by dominated convergence the limit as ǫ ↓ 0 of the left-hand side of (8.6) exists and is precisely Moreover, this implies that the limit as ǫ ↓ 0 on the right-hand side of (8.6) also exists. We denote this limit by Flux(ξ, t, n). To see that Flux(ξ, t, n) has the claimed properties, since the matrix (g ij ) is nonnegative, note that finishing the proof of the result as the surface measure is also nonnegative.

Conclusions
The techniques developed in this and its accompanying work provide a general framework for constructing a Lyapunov function well adapted to the dynamics of a particular problem. This systematic approach began in [1]. Here, however, a significant number of advances have been made, allowing us to both cover a much larger class of problems and simplify many details in the analysis. In particular, the use of a generalized Tanaka formula [7] greatly simplifies the patching together of the piecewise-defined Lyapunov functions when compared to the treatments of the similar situations in [1,2,4].
A few considerations remain uncomplete by this work. Section 9.2 in Part I [6] makes a compelling argument supported by numerical simulations for the scaling of large excursions. It would be interesting to add the missing details, producing a rigorous argument. Related to this, it would be also interesting to see if one could scale space and time so that in the limit of vanishing noise, the system converges to a random distribution on the loops of the underlying deterministic system. The limiting loop system would be in the spirit of the random spider/graph processes considered in [3] and subsequent works.
There also is a number of possible directions for generalization. Here we have only considered complex polynomials whose highest order term is the monomial az n+1 . More generally, one could consider leadingorder monomial terms of the form az kzj where k + j = n + 1. If k > j + 1, then the system with noise added can be proven to be stable by essentially the same arguments used in this and its companion paper [6]. In this case, the invariant measure will again have polynomial decay at infinity. If k = j, then the system is trivially stable if a < 0 and trivially unstable if a > 0. In this case, the norm-squared |z| 2 is easily shown to be a Lyapunov function if a < 0. If j < k then the deterministic flow rotates towards the unstable directions and not away from them as was the case when k < j. Here one expects to be able to prove that they system blows-up with probability one.
More interesting is the case when the leading order monomial is replaced by a polynomial made up of terms which all have homogeneity n + 1 under the radial scaling z → λz. This will produce a richer collection of possibilities. Nonetheless, we expect the ideas contained in these notes to be very useful in determining and proving stability properties when noise is added.
Another possible direction of generalization would be to consider state-dependent noise. In some cases these may be transformed into equations with additive noise by time changes. A more difficult direction of generalization would be to consider higher dimensional unstable ODEs under the addition of noise. Here the geometry of the underlying, deterministic dynamics can be quite complicated, if not chaotic. In this work, we relied on the simplicity of the underling dynamics in our analysis. Understanding how different regions patch together could be much more complicated in higher dimensions, if not intractable. The most interesting and wide-open direction to pursue would be to consider a PDE which was unstable under the addition of noise and show that it was stabilized.