Expansion and attraction of RDS: long time behavior of the solution to singular SDE

We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of the SDE can be split into two terms, one of which is singular and the radial component of the other term has a radial component of sufficient strength in the direction of the origin, then the random dynamical system generated by the SDE admits a pullback attractor.


I
Regularization by noise, i.e. existence and uniqueness of solutions under the assumption of nondegenerate noise, has been established for a large class of singular stochastic di erential equations (SDEs).It was shown recently that these equations also generate a random dynamical system (RDS), see [18], and like in the classical (non-singular) case it therefore seems natural to establish asymptotic properties of these RDS for large times, like expansion rates of bounded sets and the existence of attractors or even synchronization (meaning that the attractor is a single random point).
First we analyse the linear expansion rate of the ow generated by a singular SDE.In classical results, see e.g.[21], [7], Lipschitz continuity or one-sided Lipschitz continuity of the coe cients of the SDE is assumed to obtain bounds on the expansion rate.Obviously we lack these properties in our current setting.Instead, we assume the noise to be non-degenerate, so we can apply the Zvonkin transformation to get an SDE which has Lipschitz-like coe cients and this SDE is (in an appropriate sense) equivalent to the original one (1.1).The Zvonkin transformation was invented by A. K. Zvonkin in [32] for  = 1 and then generalized by A. Yu.Veretennikov in [23] to  1.It has become a rather standard tool to study well-posedness of singular SDEs, see e.g.[27], [25] and [24].This tool heavily relies on regularity estimates of the solution to Kolmogorov's equation corresponding to (1.1) which can be found for instance in [13] in the classical setting.In this paper we adapt the method to the study of the RDS induced by singular SDEs.We show that the ow expands linearly (see Theorem 5.4), a property which was established for non-singular SDEs with not necessarily non-degenerate noise in [3,4,19,20,21].Our proof mainly depends on stability estimates (see Theorem 5.2).These kind of estimates were studied before, see for instance [12], [27] and [28], but the dependence of the constants on the coe cients was not speci ed.We give a formula in Theorem 5.2 which states this dependence explicitly.It also yields the expansion rate constant in Theorem 5. 4.
Secondly, we aim at conditions which guarantee the existence of an attractor for the RDS generated by a singular SDE.Clearly, one can not expect that an attractor exists without further conditions (an example without attractor is the case in which the drift is zero and the di usion is constant).Since [6], numerous papers appeared in which the existence of attractors for various nite and in nite dimensional RDS was shown, e.g.[2], [8], [9], [10], [11], [7], [15] and [31].A common way to prove the existence of an attractor is to show the existence of a random compact absorbing set and then to apply the criterion from [6,Theorem 3.11].Just like [7], we will use a di erent and more probabilistic criterion from [5] (Proposition 2.8).Roughly speaking, all one has to show is that the image of a very large ball will be contained inside a xed large ball after a (deterministic) long time with high probability.In [7] this was shown under the assumption that the di usion is bounded and Lipschitz and the drift  () has a component of su cient strength (compared to the di usion) in the direction of the origin for large | |.In our set-up, this condition is too restrictive.Instead, we assume that the drift can be written in the form  =  1 +  2 , in which  1 is singular and  2 has a component of su cient strength (compared to the di usion and the localized   -norm of  1 ) in the direction of the origin for large | |.
Structure of the paper.We introduce notation and the main results in Section 2. In Section 3 we study the expansion rate of the diameter of the image of a bounded set under a ow under rather general conditions.These results are minor modi cations of results contained in [21] which are proved by chaining techniques.Section 4 contains estimates on functionals of the solution to the singular SDE, namely quantitative versions of Krylov's estimates and Khasminskii's lemma.The rst part of the main results of this paper is presented in Section 5, i.e. the linear expansion rate of the diameter of the image of a bounded set under the ow generated by the solution to a singular SDE.In Section 6 we show the existence of an attractor of the RDS generated by the singular SDE.In Appendix A we study regularity estimates of elliptic partial di erential equations with emphasis on the dependence on the coe cients.We believe that these estimates are of independent interest.
From [24, Section 2] and [30,Proposition 4.1] we know that the space H, does not depend on the choice of  and , but the norm does, of course.More precisely, by [30,Proposition 4.1], for the L -norms with di erent , say  1 and  2 and  1 <  2 , if we use the notation ( L )  to denote the L space with support radius  for localization, then where  1 ,  2 are constants independent of  1 ,  2 .For convenience we take  = 1 in the following.
For further properties of these spaces we refer to [24].In the following, all derivatives should be interpreted in the weak sense.Occasionally we will use Einstein's summation convention (omitting the summation sign for indices appearing twice).We will often use the notation  + = max{, 0} for the positive part of  ∈ ℝ,  ∨  := max{, } and  ∧  := min{, }.

2.2.
Preliminaries.In the following, all random processes will be de ned on a given probability space (Ω, F , ℙ).
De nition 2.3 (RDS, [1]).A (global) random dynamical system (RDS) (, ) on a Polish space (, ) over an MDS  is a mapping satis es the following (perfect) cocycle property: for all ,  0,  ∈  , Clearly, an RDS  induces a ow via  , () :=   (,   .).We say that an SDE generates a ow resp.an RDS if its solution map has a modi cation which is a ow resp.an RDS.The following study is based on the ow generated by the solution to the SDE with singular drift.Therefore we state the result from [18, Theorem 4.5, Corollary 4.10] on the existence of a global semi-ow and a global RDS for singular SDEs under the following condition.
). () There exist  1 ,  2 > 0 such that for  :=  * we have 5. Note that L ⊂ L whenever  >  .Therefore, if Assumption 2.4 holds with di erent values of  and , then it also holds with the larger of the two numbers replaced by the smaller one.In particular, the following result which was formulated for  =  can still be applied.We will often write   () instead of  0, ().Abusing notation we will sometimes say "Let   () (or just  ) be a ow ..." instead of "Let  , (),  ∈ ℝ  , 0   < ∞ be a ow and   () :=  0, (),  0,  ∈ ℝ  ... ".
De nition 2.7 (Attractor, [6]).Let  be an RDS over the MDS  = (Ω, F , ℙ, {  }  ∈ℝ ).The random set () is a (pullback) attractor if (1) measurability: () is a random element in the metric space of nonempty compact subsets of  equipped with the Hausdor distance, (2) invariance property: for  > 0 there exists a set Ω  with full measure such that (3) pull-back limit: almost surely, for all bounded closed sets  ⊂  , lim One way to verify the existence of an attractor is the following criterion.Theorem 2.9.If Assumption 2.4 holds, then there exists a constant  > 0 such that for the ow  generated by the solution to (1.1) we have, for any compact X ⊂ ℝ  , lim sup The precise statement including a formula for  will be given in Theorem 5.4.There, we can see that  → ∞ as  1 → 0 (when all other parameters remain unchanged).The following example explains this fact: as the noise becomes more and more degenerate, the linear bound on the dispersion of a bounded set under the ow approaches in nity, so our non-degeneracy assumption on the noise cannot be avoided.
and  1 , 2 are two independent 1-dimensional Brownian motions.Notice that for  (, ) := ((), ((−) ∨ (−1)) ∧ 1) * , we have  ∈ L (ℝ 2 ) for  ∈ (4, 1  ).Clearly there exists a unique solution (,  ) to (2.4) and By the ergodic theorem, almost surely, where   is the invariant probability measure of  .Since   converges to the point measure  0 weakly as  ↓ 0, we see that the linear expansion rate of (,  ) converges to ∞ when  ↓ 0. In particular, we can not expect to have a linear expansion rate for the solution to a singular SDE with degenerate noise in general.
We will now assume that the singular drift  in (1.
In particular,  has a random attractor.
Correspondingly the detailed results are presented in Theorem 6.2 and Theorem 6.3.
In the end we give the following example on the special case that the drift is bounded ( i.e.  = ∞ ) to conclude the results on the expansion rate and attractors.Example 2.13 (A case study: bounded coe cients).We consider the ow (  ())  0 generated by the solution to (1.1) when , ∇ are simply bounded, i.e., Assumption 2.4 holds with arbitrary  =  ∈ (1, ∞).
1. Expansion rate of the ow: Theorem 5.4 shows that for each  > 0 there exist constants  1 (depending on  and ) such that for each compact subset X ⊂ ℝ  lim sup , where  > 0 and  2 > 0 is an appropriate function depending on  and  only, then from Theorem 6.3 we know that  has an attractor.

E
In this section, we assume that  : [0, ∞) × ℝ  × Ω → ℝ  is measurable such that  ↦ →   (, ) is continuous for every  ∈ ℝ  and  ∈ Ω (we do not require that  has any kind of ow property).Lemma 3.1.Assume that there exists  > 0 and a constant  1 > 0 such that for each  > , there exists  =  ( ) > 0 such that for all ,  ∈ ℝ  and  > 0, we have Then  has a modi cation (which we denote by the same symbol) which is jointly continuous in (, ) and for each  > 0 and  > 0, lim sup where sup   , means that we take the supremum over all cubes   , in ℝ  with side length e − , and Let X be a compact subset of ℝ  with box (or upper entropy) dimension Δ > 0. Then where otherwise, with Remark 3.4.In addition to the assumptions of the previous theorem, let us assume that   () =  0, () where  is a ow (later, we will only consider this case).Let X ⊂ ℝ  be any compact set and let  be a ball in ℝ  containing X. Clearly, the boundary  of  has box dimension  − 1.The ow property of  implies that for each  0, the boundary of  0, () is contained in  0, () and therefore any almost sure upper bound  for the linear expansion rate of the set  is at the same time an upper bound for the linear expansion rate of the set  and hence of X.This means that in the case of a ow, the formula for  in the theorem always holds with Δ replaced by  − 1 (or the minimum of Δ and  − 1).

K
We will show a quantitative version of Krylov estimates (4.1).One can nd similar results in the literature with implicit constants, for instance [14], [27] and [24], which however do not t our needs since some proofs in later sections rely on the explicit dependence of the constants on the coe cients of the SDE.In the following lemma, a constant  Kry appears which depends on , , ,  only.While we will regard , ,  as xed throughout, we will apply the formula with di erent values of  and we will therefore write  Kry () for clarity.Lemma 4.1.If Assumption 2.4 holds and (  )  0 solves (1.1), then, for  ∈ L (ℝ  ) with  ∈ (, ∞], there exists a constant  Kry () > 0 depending on , , ,  only such that for 0  , where Proof.It is su cient to show the estimate for positive  .(4.1) clearly holds when  = ∞, so we assume  ∈ (, ∞).All positive constants   ,  = 0, • • • , 7 appearing in the proof only depend on , , , .
The following corollary is a quantitative version of Khasminskii's lemma.The constant  Kry () appearing in there is the same as in the previous lemma.
Proof.The second inequality is an application of the general inequality ( + ) 2 2 2 + 2 2 .Lemma 4.1 shows that there exists some positive integer  such that, for and the proof of [26,Lemma 3.5] shows that for any such  we have (see also [17,Lemma 3.5]).By Lemma 4.1, any  such that 1 2 satis es (4.9).In particular, we can take Here,  is the largest integer that is smaller than or equal to  ∈ ℝ. Therefore (4.8) holds.
Remark 4.3.Note that the right hand side of our version of Krylov's estimate contains the factor , where  ( ) depends on the nal time  .Further, we require the condition  >  instead of  > /2 in [29, Theorem 3.4 (3.8)]).The reason for our restriction to  >  is that we use (4.4) which only holds for  > .Since we will later apply Krylov's estimate to  := | * •  −1 | 2 which is in L/2 we will have to assume  > 2.
Remark 4.4.More general versions of the quantitative Khasminskii's Lemma (but with less explicit constants) can be found in [16].

U SDE
Depending on the regularity of the SDE's coe cients we show upper bounds for the dispersion of sets under the ow generated by the solution in the following two cases.
Remark 5.3.If σ is even globally Lipschitz continuous with Lipschitz constant , then there is no need to use Khasminskii's Lemma for the integral over σ and we easily get (5.2) with where Proof.The idea is to apply Theorem 3.3.All constants  * 1 , ... depend on , ,  only.
Proof.This follows easily from (5.18) and Lemma 3.1.

E SDE
Inspired by the work [7], we are interested in the question whether there exists a random attractor of the RDS generated by the solution to the singular SDE.We start with estimates of the one-point motion (items 1-5 of the following lemma) and then move to estimates for the dispersion of sets (items 6 and 7).Lemma 6.1.Let Assumption 2.4 hold.Further assume that there exist vector elds  1 and  2 such that  =  1 +  2 with  1 ∈ L (ℝ  ).Let (  ())  0 be the ow generated by the solution to (1.1).Let 2. If  2 satis es Assumption 2.11 (  ) for some  < 0 and  0 > 1 is such that  * ( 0 ) 0 where  * ( 0 ) is from (6.1), then for every    0 and every  ∈ ℝ  , we have 3. If  2 satis es Assumption 2.11 (  ) for some  < 0 and  0 > 1 such that  * ( 0 ) 0 where  * ( 0 ) is from (6.1) and if   0 , then for every | | = , ,  1 > 0, we have 4. Let 1  , and  1 ,  2 >  .If  2 satis es Assumption 2.11 (  ) for some  ∈ ℝ, then for each 2 ) and  > 0 with  +  <  −  0 .For  > 2, de ne  := ℎ(),  = (1 − ) and  1 :=  + ℎ().Then 7. Assume that  2 satis es Assumption 2.11 (  ) for Let ℎ() =   for some  ∈ (0, 1  3 ).Let  ∈ (0, 1 2 ) and  > 0 with  +  < − −  0 .For  > 2, de ne  := ℎ(),  = (1 − ) and  1 :=  + ℎ().Then Proof.Let us explain the idea of the proof of parts 1 to 5: we express the probabilities on the left side by the corresponding ones for the ow  2 generated by the SDE with drift  replaced by  2 by applying Girsanov's theorem.This is possible since  1 ∈ L .The required estimates for  2 can then be obtained from results in [7].Notice that strictly speaking the SDE generating  2 cannot be applied since the assumptions in [7] require the coe cients to be one-sided Lipschitz continuous which is not necessarily true in our set-up.It is easy to check however that the estimates of the one-point motion in Propositions 4.2 to 4.6 in [7] hold without additional Lipschitz-type assumptions.Therefore, we divide the proof into two steps: a Girsanov argument and then estimates for the ow  2 . Let Therefore, (  )  0 is a martingale.Fix  > 0 and let ℙ  :=   ℙ. Girsanov's theorem and Hölder's inequality show for each measurable set  ∈  ([0, ], ℝ  ) If   denotes the set inside ℙ on the left side of item i in the Lemma (i= 1, ..., 5), then   (   ) −1 balls centered on   for any  ∈ (0, ].Here we take  = exp(−ℎ()) for some  > 0 which will be chosen later and we label the balls by Case 4 gives us the following upper bound (note Furthermore, by Proposition 5.5, where  1 is taken from (5.23) with  replaced by  2 .Therefore, by (6.5), (6.6) and (6.7), it follows that, for  > 4 1  3 , lim sup Therefore case 6 holds.
Finally, we state the following theorem on the existence of random attractors.Theorem 6.3.Let Assumption 2.4 hold.Further assume that there exist vector elds  1 and  2 such that  =  1 +  2 with  1 ∈ L (ℝ  ).Let (  ())  0 denote the ow generated by the solution to (1.1).Let where  Kry ( In particular,  has a random attractor. Proof.The existence of an attractor is an easy observation from Proposition 2.8 if we have (6.13).So we only need to show (6.13).The argument is essentially the same as [7, Proof of Theorem 3.1 a)].
We give the outline of the proof emphasising those arguments which are di erent.
To estimate  2,1 (, ), for xed 0   , denote  0 :=  +  2 , we cover   0 by     −1 0   (−1) balls of radius  − centered on   0 with  <  1  2 3(−1) (the same choice as in the proof of Lemma 6.1 case 7. Label the balls by  1 , • • • ,   and their centers correspondingly by  1 , • • • ,   .Then for a number  2 such that  * ( 2 ) < 0 where  * ( 2 ) is from (6.1), we have By the same argument from Lemma 6.1 case 7 (6.11) with ℎ() =   =  , and Lemma 6.1 case 2, and Proposition 5.5 we get lim sup Up to here, in order to get (6.15),we only need to show lim sup In our setting, we already showed the second and the third statements: these are Lemma 6.1 case 2 and case 3 correspondingly.Therefore it is su cient to show the estimate corresponding to [7, (4.7)] in our setting.In order to do so we rst apply Girsanov Theorem as we did in Lemma 6. Therefore (  )  0 is a martingale.Let ℙ  :=  1 ℙ.As we already did in (6.4), by Girsanov theorem and Hölder's inequality, for  > 0, for any ,  ∈ ℝ  , In this section we will show estimates of the solution of the elliptic PDE above.Such estimates were obtained in [29,Theorem 3.3] in the case where  is uniformly elliptic and uniformly continuous and  ∈   1 for some  1 > .These estimates were, however, not explicit in terms of the coe cients ,  and  .We prove the following theorem which shows this dependence since we need it in the main text but it may also be of independent interest.Theorem A.3.Suppose Assumption A.2 holds.There exists a constant  0 > 0 depending on ,  1 ,  and  only, such that for   0  1 and for any  ∈ L (ℝ  ) with  ∈ (/2 ∨ 1,  1 ], there is a unique solution  ∈ H 2, to (A.1).Further, for  ∈ [1, ∞] there exists a constant  depending on , , ,  and  1 only, such that Proof.Assume  ∈ H 2, is a solution to (A.1).We rst show the a priori estimates (A.4).Then the continuity method, as shown in [13], is a standard way to conclude the existence and uniqueness of the solution to (A.1) for those  for which (A.4) holds.We divide the proof into three steps.Note that all positive constants   ,  = 1, • • • appearing in the proof only depend on , ,  1 ,  ,  (and not on ,  , , , and   ()).
Step 1. Assume that  is a constant (positive de nite) matrix,  = 0 and  ∈   .
For  > 0, let  ∈  2, be the solution to the following equation where  is the unique positive de nite matrix satisfying  * = .Then  = ( − Δ) −1 f is the unique solution in  2, .From [29, (3.3)] we know that, for each  ∈ [1, ∞], there are constants  1 ,  2 ,  3 such that and hence  solves (A.1).Uniqueness of a solution under the conditions of Step 1 holds since the map  ↦ →  is a bijection between solutions of the corresponding PDEs.Considering 1 Step 2.  satis es Assumption A.2 (  ),  = 0 and  ∈ L .