Approximation of heavy-tailed distributions via stable-driven SDEs
aa r X i v : . [ m a t h . P R ] J u l APPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS VIA STABLE-DRIVENSDES
LU-JING HUANG MATEUSZ B. MAJKA JIAN WANG
Abstract.
Constructions of numerous approximate sampling algorithms are based on the well-knownfact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations(SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shownthat the associated SDE is not exponentially ergodic and that related sampling algorithms may performpoorly. A natural idea that has recently been explored in the machine learning literature in this contextis to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paperwe provide a rigorous theoretical framework for studying the problem of approximating heavy-taileddistributions via ergodic SDEs driven by symmetric (rotationally invariant) α -stable processes. Keywords: stochastic differential equations, symmetric α -stable processes, invariant measures, heavy-tailed distributions, approximate sampling, fractional Langevin Monte Carlo. MSC 2020: Introduction
Suppose we are given a probability distribution µ on R d defined via(1.1) µ ( dx ) = Z − exp ( − V ( x )) dx , where V : R d → R is the potential, and Z := R R d exp ( − V ( x )) dx is the normalizing constant. Thegoal in approximate sampling is to generate a sequence of probability measures ( µ k ) k ≥ such that forsufficiently large k the measure µ k constitutes a good approximation of µ . This can be achieved e.g.by utilizing a stochastic process with the unique stationary distribution µ . If we can show that thisprocess is exponentially ergodic, then we can use it to construct an algorithm for approximate samplingfrom µ that, under some assumptions on V in (1.1), converges exponentially fast regardless of its initialcondition.A commonly used example of such a process is the solution ( X t ) t ≥ to the (overdamped) LangevinSDE(1.2) dX t = −∇ V ( X t ) dt + √ dB t , where ( B t ) t ≥ is the standard Brownian motion in R d . If the potential V is sufficiently regular, itcan be easily shown that µ given by (1.1) is a stationary distribution of ( X t ) t ≥ . Moreover, there aremany results on the exponential ergodicity of (1.2) under relatively weak dissipativity conditions on V , see e.g. [17] and the references therein for approaches based on Lyapunov-type drift conditions,the monographs [1, 4, 40] for methods based on functional inequalities, and [4, 40] for probabilisticcoupling techniques (in particular, [12, 13] for a recent study on this topic).There are numerous sampling algorithms in the literature that are based on Euler discretizationsof (1.2), cf. [14, 26] and the references therein. The analysis of their performance is often carried outby bounding the discretization error between the Euler scheme and the SDE, and then by directlyemploying ergodicity results for SDEs, see e.g. [8, 9, 11, 30]. Hence the analysis of convergence of theSDE is an important first step towards evaluating performance of such algorithms, and one usuallycannot expect fast convergence of the algorithm without fast convergence of the associated SDE, see[34] (with some possible exceptions discussed in [15]). L-J Huang:
College of Mathematics and Informatics, Fujian Normal University, 350007 Fuzhou, P.R. China. [email protected] . M. B. Majka:
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK. [email protected] . J. Wang:
College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applic-ations (FJKLMAA) & Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, 350007Fuzhou, P.R. China. [email protected] . However, in [34] (see Theorem 2.4 and Section 2.3 therein) it has been shown that the solution to(1.2) may not be exponentially ergodic if the distribution µ defined in (1.1) is heavy-tailed. Indeed, itis known that the Langevin SDE (1.2) has the generator Lf := ∆ f − ∇ V · ∇ f which is a symmetricoperator on L ( R d ; µ ) , and that the Poincaré inequality for L (which is equivalent to the exponentialergodicity of the SDE (1.2)) implies exponential tails of µ ; see [40, Theorems 1.1.1 and 1.2.5]. However,for heavy-tailed µ , one can only expect weak-Poincaré inequalities, which indicates that the solution to(1.2) only converges with a polynomial or a subexponential rate; see [40, Chapter 4] for more details.A very natural question to ask in this context is whether instead of (1.2) one could use SDEs drivenby other stochastic processes, with tails better suited for the task of approximating heavy-tailed µ .The first steps in that direction have been taken in [36, 31] (see also [37, 44] for further extensions).The idea there is based on the fact that µ given by (1.1) can be shown to be a stationary distributionof(1.3) dX t = b ( X t ) dt + dZ t , where ( Z t ) t ≥ is the symmetric (rotationally invariant) α -stable process in R d with d ≥ and α ∈ (1 , ,and the drift b ( x ) is given by(1.4) b ( x ) = − C d, − α e V ( x ) Z R d e − V ( y ) ∇ V ( y ) | x − y | d − (2 − α ) dy , where the potential V ∈ C ( R d ) is such that e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) , and C d,α := Γ(( d − α ) / / (2 α π d/ Γ( α/ . Hence, if the SDE (1.3) is exponentially ergodic, one could use an algorithmbased on its discretization to obtain a new alternative way of approximating µ (possibly faster thanalgorithms based on (1.2) if µ is heavy-tailed). The authors of [36, 31] called their approach FractionalLangevin Monte Carlo due to a possible interpretation of the drift (1.4) in terms of the Riesz potential,which is an inverse operator to the fractional Laplacian, see e.g., [21, Section 2.7] and the referencestherein.There are, however, several challenges to this approach, related both to verifying theoretical prop-erties of the SDE (1.3) and to finding its appropriate discrete-time counterpart for use in simulations.In the present paper we focus on the former, in response to some questions that were left unansweredin [36, 31]. Indeed, the exponential ergodicity of (1.3) has been checked in [36, 31] only under somevery special and difficult to verify assumptions. As we will see in Section 2, the drift b ( x ) defined by(1.4) seems to be in general only locally (2 − α ) -Hölder continuous, while in the setting of [36, 31] itis assumed to be Lipschitz continuous and differentiable. Moreover, the authors of [31] assume that b ( x ) satisfies a contractivity at infinity condition h b ( x ) − b ( y ) , x − y i ≤ − K | x − y | for all x , y ∈ R d such that | x − y | > R , with some constants K , R > (cf. [31, Assumption (H5) and Proposition 1]),which also seems to be unverifiable in the general case. The lack of all these properties of b ( x ) makes itimpossible to prove the exponential ergodicity of (1.3) by utilizing results from the existing literature(see e.g. [22] for some recent developments in this topic). Furthermore, because of the unusual form of(1.4), it is not even immediately clear whether (1.3) has a unique, non-explosive strong solution, whichalso has not been verified in [36, 31]. Finally, due to non-differentiability of b ( x ) , the proof that µ given by (1.1) is the unique invariant probability measure for (1.3) cannot be as straightforward as in[36, Theorem 1.1] or [44, Theorem 1.1]. In the present paper we fill all these gaps by carefully derivingappropriate bounds on (1.4), and by proving all the properties of (1.3) mentioned above in a rigorousway. In particular, we study the drift term b ( x ) defined by (1.4) for all d > − α (not only for the caseof d ≥ and α ∈ (1 , ), and we define a new drift term to treat the case of d ≤ − α . To this end,we will use the notion of the fractional Laplace operator (see e.g. [2, 3, 21] and the references therein),which is defined for all f ∈ C b ( R d ) by − ( − ∆) α/ f ( x ) := c d,α lim ε → Z {| y − x | >ε } f ( y ) − f ( x ) | y − x | d + α dy, where c d,α := 2 α Γ(( d + α ) / / ( π d/ | Γ( − α/ | ) = α α − Γ(( d + α ) / / ( π d/ Γ(1 − α/ . See e.g. [2,formulas (1.3) and (1.35)] or [21, Definition 2.5], and note that c d,α = | C d, − α | . Then, in order to coverthe case of d ≤ − α , i.e., d = 1 and α ∈ (0 , , we will work with the drift(1.5) b ( x ) = − e V ( x ) Z x −∞ ( − ∆) α/ e − V ( u ) du, x ∈ R . PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 3
Everywhere in this paper, we will be concerned with the SDE (1.3) driven by a symmetric α -stableprocess ( Z t ) t ≥ on R d with α ∈ (0 , , where the drift term b ( x ) is defined by (1.4) when d > − α , andby (1.5) when d ≤ − α . We will refer to b ( x ) as the fractional drift in both cases. We will comment onsome possible approaches to the problem of discretization of (1.3) in Remark 1.5. However, our focusin this paper is the analysis of the SDE (1.3), and we leave a more detailed discussion of discrete-timealgorithms for future work.For our main result, we require that the following assumption on the potential V is satisfied. Assumption (A) V is a radial function on R d ( and hence, by a slight abuse of notation, we write V ( x ) = V ( | x | ) for all x ∈ R d ) such that (1.6) lim sup r →∞ [ e − V ( r ) r d + α ] < ∞ , and one of the following two conditions is satisfied: (i) when d > − α , V ∈ C ( R d ) , e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) , (1.7) r := sup { r > V ′ ( r ) ≤ } < ∞ , and (1.8) Z ∞ e − V ( r ) | V ′ ( r ) | r d dr < ∞ , Z ∞ e − V ( r ) V ′ ( r ) r d dr > . (ii) when d ≤ − α , V ∈ C ( R ) , e − V ∈ L ( R ; dx ) ∩ C b ( R ) , lim sup x →∞ [ x e − V ( x ) | V ′ ( x ) − V ′′ ( x ) | ] < ∞ , and lim inf x →∞ [ x e − V ( x ) ( V ′ ( x ) − V ′′ ( x ))] ≥ . We have the following result.
Theorem 1.1.
Under Assumption (A) , the
SDE (1.3) with the fractional drift b ( x ) given by (1.4) when d > − α , and by (1.5) when d ≤ − α , has a unique non-explosive strong solution X := ( X t ) t ≥ such that the process X is exponentially ergodic with the unique invariant probability measure µ givenby (1.1) . More explicitly, for any β ∈ [0 , α ) , there is a constant λ > such that for any X ∼ µ withfinite β -moment and any t > , k L ( X t ) − µ k Var ,V := sup | f |≤ V (cid:12)(cid:12)(cid:12)(cid:12)Z R d E x f ( X t ) µ ( dx ) − µ ( f ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( µ ) e − λt , where V ( x ) = (1 + | x | ) β , C ( µ ) is a positive constant, and L ( X t ) denotes the distribution of X t forevery t > . Note that the weighted total variation distance k · k
Var ,V from Theorem 1.1 dominates both thestandard total variation and the L β -Wasserstein distance (see e.g. [13, Remark 2.3]). Therefore wehave the following immediate corollary. Corollary 1.2.
Under Assumption (A) , the process X := ( X t ) t ≥ solving (1.3) is exponentially ergodicwith the unique invariant probability measure µ given by (1.1) in the total variation norm for all d ≥ and α ∈ (0 , , and in the L -Wasserstein distance when d ≥ and α ∈ (1 , . Let us make some comments on Assumption (A) and Theorem 1.1, as well as the fractional driftsdefined by (1.4) when d > − α and by (1.5) when d ≤ − α . The most important conclusion fromTheorem 1.1 is that the SDE (1.3) with α -stable noise is exponentially ergodic for a large class ofpotentials, for which the corresponding SDE (1.2) with Brownian noise is not. Remark 1.3.
Theorem 1.1 is concerned with rotationally symmetric measures µ (since V is a radialfunction on R d ). Condition (1.6) is a relatively weak condition that we need in order to prove theexponential ergodicity of the process X (indeed, it seems to be optimal as indicated by the exponentialergodicity for Ornstein–Uhlenbeck processes driven by symmetric α -stable processes, cf. [24, 41]). It issatisfied, for example, by all potentials V ( x ) = (1 + | x | ) β for any β > , and by V ( x ) = log β (1 + | x | ) for any β > , as well as by V ( x ) = β log(1 + | x | ) for any β ≥ ( d + α ) / . We remark that it hasbeen shown in [34] that for the latter two large classes of potentials, as well as for the potentials V ( x ) = (1 + | x | ) β with β < / , the SDE (1.2) driven by the Brownian motion is not exponentially LU-JING HUANG MATEUSZ B. MAJKA JIAN WANG ergodic. It is also easy to see that assumption (ii) for d ≤ − α , as well as the first condition in(1.8) for d > − α , are satisfied for all the potentials above. Moreover, when d > − α , we alsorequire condition (1.7), which means that the measure µ is log-concave at infinity. The most restrictivecondition is the second condition in (1.8), which is essentially an assumption about sufficiently heavytails of µ in relation to its mass in the region where V ′ ≤ , i.e., where µ is not log-concave. In otherwords, if r is not too large and if µ has heavy tails, then R ∞ r e − V ( r ) V ′ ( r ) r d dr can be large enough sothat the second condition in (1.8) holds. Obviously, if µ is log-concave everywhere, then the secondcondition in (1.8) is always satisfied. Remark 1.4.
Let us informally discuss how the form of the fractional drifts given by (1.4) and (1.5)is motivated by the requirement that the associated SDE (1.3) has an invariant probability measuregiven by (1.1). Suppose first that d > − α . Note that the generator of the process X solving SDE(1.3) is Lf = − ( − ∆) α/ f + b · ∇ f . Hence, informally, its dual operator enjoys the expression L ∗ f = − ( − ∆) α/ f + div( bf ) ; see Remark 3.4. Roughly speaking, the density function e − V ( x ) of the invariantprobability measure (1.1) is the fundamental solution to L ∗ u = 0 ; that is, div( be − V ) = − ( − ∆) α/ e − V .If we write − ( − ∆) α/ e − V = ∆[( − ∆) − (1 − α/ e − V ] = div ∇ [( − ∆) − (1 − α/ e − V ] , then a right choice forthe drift can be b ( x ) = e V ( x ) ∇ ( − ∆) − (1 − α/ e − V ( x ) , which is equivalent to (1.4); see the discussion in thebeginning of Subsection 2.1. When d ≤ − α , ( − ∆) − (1 − α/ is not well defined, but we can informallywrite ∇ ( − ∆) − (1 − α/ = ∇ (∆) − [ − ( − ∆) α/ ] and understand ∇ (∆) − as an integral operator. Withthis in mind, we can see the intuition behind the formula for (1.5). A fully rigorous proof that theprobability measure given by (1.1) is invariant for (1.3) will be given in Proposition 3.3. Remark 1.5.
As we will see in the sequel, the drift term b ( x ) defined by (1.4) when d ≥ and α ∈ (0 , or by (1.5) when d ≤ − α , belongs to C ( R d ) ; however, when d ≥ − α and α ∈ [1 , , b ( x ) defined by (1.4) seems to be only Hölder continuous; cf. Lemma 2.2. This may lead to some issueswhen one wants to consider discretizations of (1.3) in the latter case. When d = 1 and α ∈ (1 , , in[36] some numerical experiments were carried out by employing an Euler discretization of (1.3) thatinvolved approximating the drift (1.4) via a series representation from [32], see Section 4 and formula(7) in [36]. However, in order to rigorously analyse convergence of discretized (1.3) in this case, onecannot rely on classical results for Euler discretizations that utilize the Lipschitz property of the drift,or even results based on taming such as [10, 20], where the one-sided Lipschitz property is required.Nevertheless, there has been some recent work [18, 29] on discretizations of Lévy-driven SDEs withbounded Hölder continuous drifts that could be applicable in our setting after an extension to theunbounded case (cf. Lemma 2.2 below for a proof of the local Hölder property of b ( x ) given by (1.4)).This, however, falls beyond the scope of the present paper and will be considered in a future project.The remaining part of this paper is organised as follows. In Section 2, we obtain some explicitestimates for the fractional drift given by (1.4) when d > − α and by (1.5) when d ≤ − α , underAssumption (A). In particular, under a mild additional assumption, we get that h b ( x ) , x i ≍ − e V ( x ) | x | d + α | x | for | x | large enough. We also claim that the fractional drift term is locally (2 − α ) -Hölder continuouswhen α ∈ (1 , , locally (1 − ε ) -Hölder continuous for any ε > when α = 1 , and belongs to C ( R d ) when α ∈ (0 , . Section 3 is devoted to properties of the SDE (1.3) with the fractional drift terms.We prove that the SDE (1.3) with these drifts has a unique strong solution, and show that µ given by(1.1) is the unique invariant measure for (1.3). Finally, we conclude by proving Theorem 1.1.2. Properties of the fractional drift
The case of d > − α . In this subsection, we always assume that d ≥ and α ∈ (0 , with d > − α . Let V ∈ C ( R d ) such that e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) . We first note that for the driftterm b ( x ) defined by (1.4), it holds that(2.1) b ( x ) = e V ( x ) ∇ (( − ∆) − (1 − α/ e − V )( x ) , where ( − ∆) − (1 − α/ is the Green operator corresponding to the symmetric (rotationally invariant) (2 − α ) -stable process on R d , cf. [2, 21] and the references therein. Since d > − α , the symmetric (2 − α ) -stable process is transient on R d , and so ( − ∆) − (1 − α/ is well defined; moreover, ( − ∆) − (1 − α/ f ( x ) = C d, − α Z R d f ( y ) | x − y | d − (2 − α ) dy, f ∈ L ( R d ; dx ) , PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 5 see [21, Definition 2.11]. Indeed, because V ∈ C ( R d ) and e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) , by thedominated convergence theorem, for any x ∈ R d ,(2.2) ∇ (( − ∆) − (1 − α/ e − V )( x )= C d, − α ∇ (cid:20) Z R d e − V ( y ) | · − y | d − (2 − α ) dy (cid:21) ( x ) = C d, − α ∇ (cid:20) Z R d e − V ( ·− z ) | z | d − (2 − α ) dz (cid:21) ( x )= − C d, − α Z R d e − V ( x − z ) ∇ V ( x − z ) | z | d − (2 − α ) dz = − C d, − α Z R d e − V ( y ) ∇ V ( y ) | x − y | d − (2 − α ) dy. Remark 2.1.
When α = 2 , by (2.1) the drift term b ( x ) becomes −∇ V ( x ) . Moreover, Z t becomes √ B t , and hence the SDE (1.3) is reduced to (1.2).Recall that for any θ ≥ , the Hölder-Zygmund space C θb ( R d ) is defined by C θb ( R d ) = ( f ∈ C b ( R d ) : k f k C θb ( R d ) := k f k ∞ + sup x ∈ R d ,h =0 ∆ [ θ ]+1 h f ( x ) | h | θ < ∞ ) , where ∆ h f ( x ) = f ( x + h ) − f ( x ) , ∆ jh f ( x ) = ∆ h (∆ j − h f )( x ) , j ≥ . Note that when θ ∈ (0 , ∞ ) \ Z + , C θb ( R d ) coincides with the classical Hölder space C θb ( R d ) equippedwith the norm k f k C θb ( R d ) := k f k ∞ + [ θ ] X j =1 X β ∈ Z d and | β | = j k ∂ β f k ∞ + max β ∈ Z d and | β | =[ θ ] sup x = y | ∂ β f ( x ) − ∂ β f ( y ) || x − y | θ − [ θ ] , where Z + = { , , · · · } , Z = Z + ∪ { } , | β | = | β | + · · · + | β d | for β = ( β , β , · · · , β d ) ; see [39, Theorem1 in Section 2.7.2, p. 201]. However, when θ ∈ Z + , the Hölder-Zygmund space C θb ( R d ) is strictly largerthan C θb ( R d ) . In particular, when θ = 1 , C b ( R d ) is strictly larger than the space of bounded Lipschitzcontinuous functions (see [38, Example in Section 4.3.1, p. 148]), which is, in turn, strictly larger than C b ( R d ) . Note also that C b ( R d ) ⊂ C − εb ( R d ) for any ε > .We have the following statement. Lemma 2.2.
Assume that V ∈ C ( R d ) such that e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) . Then, the driftterm b ( x ) defined by (1.4) is locally (2 − α ) -Hölder continuous when α ∈ (1 , , is locally (1 − ε ) -Höldercontinuous for any ε > when α = 1 , and is in C ( R d ) when α ∈ (0 , .Proof. Suppose first that α ∈ (1 , . By V ∈ C ( R d ) and e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) , it is easy tosee that b ( x ) defined by (1.4) is locally bounded. Since V ∈ C ( R d ) , from (2.2), to prove the desiredassertion it suffices to verify that ( − ∆) − (1 − α/ f ∈ C − αb ( R d ) for all f ∈ L ( R d ; dx ) ∩ B b ( R d ) . Indeed,let p ( t, x, y ) = p ( t, x − y ) and ( P t ) t ≥ be the transition density function and the semigroup of the (2 − α ) -symmetric stable process, respectively. It is known that there is a constant c > such that k∇ P t f k ∞ ≤ c t − / (2 − α ) k f k ∞ , t > , f ∈ B b ( R d ) , which is equivalent to saying that there is a constant c > such that for all t > ,(2.3) Z R d |∇ p ( t, · )( x ) | dx ≤ c t − / (2 − α ) ; see [35, Example 1.5 and Theorem 3.2] or [18, Lemma 4.1 and the proof of Corollary 2.5]. Recall that,for any f ∈ L ( R d ; dx ) ∩ B b ( R d ) , ( − ∆) − (1 − α/ f ( x ) = C d, − α Z R d f ( y ) | x − y | d − (2 − α ) dy = Z R d f ( y ) Z ∞ p ( t, x − y ) dt dy = Z ∞ Z R d f ( y ) p ( t, x − y ) dy dt. Thus, when α ∈ (1 , , for any f ∈ L ( R d ; dx ) ∩ B b ( R d ) and x, h ∈ R d , | ( − ∆) − (1 − α/ f ( x ) − ( − ∆) − (1 − α/ f ( x + h ) |≤ k f k ∞ Z ∞ Z R d | p ( t, x − y ) − p ( t, x + h − y ) | dy dt LU-JING HUANG MATEUSZ B. MAJKA JIAN WANG ≤ k f k ∞ Z | h | − α Z R d ( p ( t, x − y ) + p ( t, x + h − y )) dy dt + k f k ∞ | h | Z ∞| h | − α Z Z R d |∇ p ( t, x + ηh − y ) | dy dη dt ≤ k f k ∞ | h | − α + c k f k ∞ | h | Z ∞| h | (2 − α ) t − / (2 − α ) dt ≤ c k f k ∞ | h | − α , where in the last inequality we used the fact that − α ∈ (0 , due to α ∈ (1 , . In particular, forany f ∈ L ( R d ; dx ) ∩ B b ( R d ) , ( − ∆) − (1 − α/ f ∈ C − αb ( R d ) = C − αb ( R d ) .Next, we consider the case of α ∈ (0 , . According to (2.3) and [18, Lemma 4.1(3)] as well as theiterating procedure, there is a constant c > such that for all t > , Z R d |∇ p ( t, · )( x ) | dx ≤ c t − / (2 − α ) . Then, for any f ∈ L ( R d ; dx ) ∩ B b ( R d ) and x, h ∈ R d , | ∆ h ( − ∆) − (1 − α/ f ( x ) | = | ( − ∆) − (1 − α/ f ( x + 2 h ) − − ∆) − (1 − α/ f ( x + h ) + ( − ∆) − (1 − α/ f ( x ) |≤ k f k ∞ Z ∞ Z R d | p ( t, x + 2 h − y ) − p ( t, x + h − y ) + p ( t, x − y ) | dy dt ≤ k f k ∞ Z | h | − α Z R d ( p ( t, x + 2 h − y ) + 2 p ( t, x + h − y ) + p ( t, x − y )) dy dt + c k f k ∞ | h | Z ∞| h | − α Z (1 − η ) Z R d |∇ p ( t, x + ηh − y ) | dy dη dt ≤ k f k ∞ | h | − α + c k f k ∞ | h | Z ∞| h | − α t − / (2 − α ) dt ≤ c k f k ∞ | h | − α , where in the second inequality we used the Taylor formula. Hence, ( − ∆) − (1 − α/ f ∈ C − αb ( R d ) ,thanks to the fact that ( − ∆) − (1 − α/ f is bounded for any f ∈ L ( R d ; dx ) ∩ B b ( R d ) . The proof iscompleted. (cid:3) Remark 2.3.
From expression (1.4), one may expect that the drift term b ( x ) does not belong to C ( R d ) when α ∈ (1 , . Informally, since the integral Z R d | f ( y ) || x − y | d − (2 − α )+1 dy may diverge for f ∈ L ( R d ; dx ) ∩ B b ( R d ) with α ∈ (1 , , we cannot take the derivative inside theintegral in (1.4).In the rest of this part, we will further assume that V is a radial function. We will present someexplicit estimates for the drift term b ( x ) defined by (1.4), i.e., b ( x ) = − C d, − α e V ( x ) Z R d e − V ( y ) ∇ V ( y ) | x − y | d − (2 − α ) dy = − C d, − α e V ( | x | ) Z R d e − V ( | y | ) V ′ ( | y | ) y | y || x − y | d − (2 − α ) dy . In particular, it holds that b ( x ) = − b ( − x ) and b (0) = 0 , i.e., b ( x ) is an anti-symmetric function on R d .With a slight abuse of notation, in the following we write V ( x ) = V ( | x | ) for all x ∈ R d . Lemma 2.4.
Let V ( x ) = V ( | x | ) for all x ∈ R d such that V ∈ C ( R d ) and e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) . Suppose that r := sup { r > V ′ ( r ) ≤ } < ∞ , (2.4) Z ∞ e − V ( r ) | V ′ ( r ) | r d dr < ∞ PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 7 and (2.5) Z ∞ e − V ( r ) V ′ ( r ) r d dr > . Then, there exist constants c , c > and r ≥ such that for all x ∈ R d , (2.6) h x, b ( x ) i ≤ c {| x |≤ r } − c e V ( | x | ) (1 + | x | ) d + α | x | {| x | >r } . Proof.
For any x ∈ R d , by changing the variables, we find that C − d, − α h x, b ( x ) i = − e V ( | x | ) Z R d e − V ( | y | ) V ′ ( | y | ) h y, x i| y || x − y | d − (2 − α ) dy = − e V ( | x | ) Z {h x,y i≥ } e − V ( | y | ) V ′ ( | y | ) h y, x i| y | (cid:18) | x − y | d − (2 − α ) − | x + y | d − (2 − α ) (cid:19) dy = − e V ( | x | ) Z { V ′ ( | y | ) ≤ , h x,y i≥ } e − V ( | y | ) V ′ ( | y | ) h y, x i| y | (cid:18) | x − y | d − (2 − α ) − | x + y | d − (2 − α ) (cid:19) dy − e V ( | x | ) Z { V ′ ( | y | ) ≥ , h x,y i≥ } e − V ( | y | ) V ′ ( | y | ) h y, x i| y | (cid:18) | x − y | d − (2 − α ) − | x + y | d − (2 − α ) (cid:19) dy = : J + J . Note that, for any x, y ∈ R d , we have | x − y | d − (2 − α ) − | x + y | d − (2 − α ) = (cid:0) | x | + | y | − h x, y i (cid:1) − ( d + α − / − (cid:0) | x | + | y | + 2 h x, y i (cid:1) − ( d + α − / , and that for the function ψ ( r ) := r − ( d + α − / , we have ψ ( r − δ ) − ψ ( r + δ ) ≤ − δψ ′ ( r − δ ) , ≤ δ ≤ r, thanks to ψ ′′ ≥ and the mean value theorem. Hence, taking r = | x | + | y | and δ = 2 h x, y i ≥ , weget J ≤ − d + α − e V ( | x | ) Z { V ′ ( | y | ) ≤ , h x,y i≥ } e − V ( | y | ) V ′ ( | y | ) h y, x i | y | | x − y | d + α dy ≤ − d + α − e V ( | x | ) Z { V ′ ( | y | ) ≤ , h x,y i≥ } e − V ( | y | ) V ′ ( | y | ) h y, x i | y | || x | − | y || d + α dy = − ( d + α − e V ( | x | ) Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) h y, x i | y | || x | − | y || d + α dy = − ( d + α − e V ( | x | ) | x | Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) y | y | || x | − | y || d + α dy = − d + α − d e V ( | x | ) | x | Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y ||| x | − | y || d + α dy. Since r := sup { r > V ′ ( r ) ≤ } < ∞ , for any C > and any x ∈ R d with | x | ≥ Cr , we have J ≤ − d + α − d (1 − C − ) − d − α e V ( | x | ) | x | | x | d + α Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy. On the other hand, for any x, y ∈ R d with h x, y i ≥ ,(2.7) | x − y | d − (2 − α ) − | x + y | d − (2 − α ) ≥ d + α − | x | + | y | ) − ( d + α ) / h x, y i . Here we used the fact that for the function ψ ( r ) = r − ( d + α − / , it holds that ψ ( r − δ ) − ψ ( r + δ ) ≥ − ψ ′ ( r ) δ, ≤ δ ≤ r, LU-JING HUANG MATEUSZ B. MAJKA JIAN WANG thanks to the mean value theorem again and the fact that ψ ′′′ ≤ . Combining (2.7) with the fact V ′ ( r ) ≥ for all r ≥ r , we get that for any a > and any x ∈ R d , J ≤ − d + α − e V ( | x | ) Z { V ′ ( | y | ) ≥ , h y,x i≥ } e − V ( | y | ) V ′ ( | y | ) h x, y i | y | ( | x | + | y | ) ( d + α ) / dy = − ( d + α − e V ( | x | ) Z { V ′ ( | y | ) ≥ } e − V ( | y | ) V ′ ( | y | ) h x, y i | y | ( | x | + | y | ) ( d + α ) / dy = − ( d + α − e V ( | x | ) Z { V ′ ( | y | ) ≥ } e − V ( | y | ) V ′ ( | y | ) | x | y | y | ( | x | + | y | ) ( d + α ) / dy = − d + α − d e V ( | x | ) | x | Z { V ′ ( | y | ) ≥ } e − V ( | y | ) V ′ ( | y | ) | y | ( | x | + | y | ) ( d + α ) / dy ≤ − d + α − d e V ( | x | ) | x | Z { V ′ ( | y | ) ≥ , | y |≤ a | x |} e − V ( | y | ) V ′ ( | y | ) | y | ( | x | + | y | ) ( d + α ) / dy ≤ − d + α − d (1 + a ) − ( d + α ) / e V ( | x | ) | x | | x | d + α Z { V ′ ( | y | ) ≥ , | y |≤ a | x |} e − V ( | y | ) V ′ ( | y | ) | y | dy. According to both estimates above for J and J , we find that for any x ∈ R d with | x | ≥ Cr , C − d, − α h x, b ( x ) i ≤ − (1 − C − ) − d − α (cid:16) d + α − d (cid:17) e V ( | x | ) | x | | x | d + α × (cid:20) (1 + a ) − ( d + α ) / (1 − C − ) − d − α Z { V ′ ( | y | ) ≥ , | y |≤ a | x |} e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy (cid:21) . Note that, under (2.5), Z R d e − V ( | y | ) V ′ ( | y | ) | y | dy > . Then, by (2.4), there is a constant R > r such that Z {| y |≤ R } e − V ( | y | ) V ′ ( | y | ) | y | dy > . This implies that(2.8) Z { V ′ ( | y | ) ≥ , | y |≤ R } e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy > , where we used the facts that r := sup { r > V ′ ( r ) ≤ } < ∞ and r < R . Furthermore, by (2.8),we can choose ε ∈ (0 , small enough so that M := (1 − ε ) Z { V ′ ( | y | ) ≥ , | y |≤ R } e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy > . Now for these fixed R and ε , we find C > large enough and a > small enough such that (1 + a ) − ( d + α ) / (1 − C − ) − d − α ≥ − ε, aCr ≥ R . Then, for any x ∈ R d with | x | ≥ Cr , (1 + a ) − ( d + α ) / (1 − C − ) − d − α Z { V ′ ( | y | ) ≥ , | y |≤ a | x |} e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy ≥ (1 − ε ) Z { V ′ ( | y | ) ≥ , | y |≤ aCr } e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy ≥ (1 − ε ) Z { V ′ ( | y | ) ≥ , | y |≤ R } e − V ( | y | ) V ′ ( | y | ) | y | dy + Z { V ′ ( | y | ) ≤ } e − V ( | y | ) V ′ ( | y | ) | y | dy = M > PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 9 and so(2.9) h x, b ( x ) i ≤ − C d, − α M (1 − C − ) − d − α ( d + α − d e V ( | x | ) | x | | x | d + α . Furthermore, by V ∈ C ( R d ) and e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) , b ( x ) is locally bounded; seeLemma 2.2. Then, for any x ∈ R d with | x | ≤ l , one can find a constant C ( l ) > such that | b ( x ) | ≤ C ( l ) ,and so(2.10) h x, b ( x ) i ≤ | x || b ( x ) | ≤ lC ( l ) . Therefore, by (2.9) and (2.10), we can choose the constants c , c > and r > so that (2.6)holds. (cid:3) The following statement indicates that the estimate (2.6) for | x | large enough is indeed optimal,under a mild additional assumption. Lemma 2.5.
Let V ( x ) = V ( | x | ) for all x ∈ R d such that V ∈ C ( R d ) , e − V |∇ V | ∈ L ( R d ; dx ) ∩ C b ( R d ) ,and (2.4) is satisfied. If (2.11) lim sup r →∞ [ e − V ( r ) V ′ ( r ) r d +1 ] < ∞ , then there exists a constant c > such that for all x ∈ R d , | b ( x ) | ≤ ce V ( | x | ) (1 + | x | ) d + α − . Proof.
For convenience, we set ˜ b ( x ) = C − d, − α e − V ( | x | ) b ( x ) . Then, for any x ∈ R d , | ˜ b ( x ) | = d X i =1 (cid:16) Z R d e − V ( | y | ) V ′ ( | y | ) y i | y || x − y | d − (2 − α ) dy (cid:17) =: d X i =1 I i . For fixed i , assume that x i ≥ . Then,I i = (cid:16) Z { y i > } e − V ( | y | ) V ′ ( | y | ) y i | y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) ≤ (cid:16) Z { y i > } e − V ( | y | ) | V ′ ( | y | ) | y i | y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) ≤ (cid:16) Z { y i > , | x − y |≤| x | / } e − V ( | y | ) | V ′ ( | y | ) | y i | y | (cid:20) | x − y | d + α − + 1( | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) + 3 (cid:16) Z { y i > , | x − y |≥ | x |} e − V ( | y | ) | V ′ ( | y | ) | y i | y | (cid:20) | x − y | d + α − + 1( | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) + 3 (cid:16) Z { y i > , | x | / ≤| x − y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) | y i | y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) ≤ (cid:16) Z {| x − y |≤| x | / } e − V ( | y | ) | V ′ ( | y | ) || y i || y | | x − y | d + α − dy (cid:17) + 6 (cid:16) Z {| x − y |≥ | x |} e − V ( | y | ) | V ′ ( | y | ) || y i || y | | x − y | d + α − dy (cid:17) + 3 (cid:16) Z { y i > , | x | / ≤| x − y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) | y i | y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) =: 6 I i + 6 I i + 3 I i . When x i < , similarly we haveI i = (cid:16) Z { y i < } e − V ( | y | ) V ′ ( | y | ) y i | y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) ≤ (cid:16) Z { y i < } e − V ( | y | ) | V ′ ( | y | ) || y i || y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) ≤ i + 6I i + 3 (cid:16) Z { y i < , | x | / ≤| x − y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) || y i || y | (cid:20) | x − y | d + α − − | x − y | + 4 x i y i ) ( d + α − / (cid:21) dy (cid:17) =: 6 I i + 6 I i + 3˜ I i . Next, we estimate the above terms respectively. For I i , we have that for x ∈ R d with | x | largeenough, (cid:16) d X i =1 I i (cid:17) / ≤ √ d Z {| x − y |≤| x | / } e − V ( | y | ) | V ′ ( | y | ) | | x − y | d − (2 − α ) dy ≤ √ d sup {| x − y |≤| x | / } { e − V ( | y | ) | V ′ ( | y | ) |} Z | x − y |≤| x | / | x − y | d − (2 − α ) dy ≤ c | x | α − sup | y |≥| x | / { e − V ( | y | ) | V ′ ( | y | ) |} ≤ c d +1 | x | d + α − sup | y |≥| x | / { e − V ( | y | ) | V ′ ( | y | ) || y | d +1 }≤ c | x | d + α − , where the last inequality follows from (2.11).For I i , we have that for x ∈ R d with | x | large enough, (cid:16) d X i =1 I i (cid:17) / ≤ √ d Z {| x − y |≥ | x |} e − V ( | y | ) | V ′ ( | y | ) | | x − y | d − (2 − α ) dy ≤ √ d − d − α | x | d − (2 − α ) Z {| y |≥| x |} e − V ( | y | ) | V ′ ( | y | ) | dy ≤ c | x | d − (2 − α ) Z {| y |≥| x |} | y | − d − dy ≤ c | x | d + α − , where in the third inequality we used (2.11) again.To estimate I i , define f ( r ) = 1( | x − y | + r ) ( d + α − / , r ≥ . By the Lagrange mean value theorem, for any y ∈ R d with | x | / ≤ | x − y | ≤ | x | and y i > , and any x i ≥ , there exists θ i ∈ [0 , x i y i ] such that f (0) − f (4 x i y i ) = − x i y i f ′ ( θ i ) = d + α −
22 4 x i y i ( | x − y | + θ i ) ( d + α ) / ≤ d + α − | x | y i | x − y | d + α ≤ c y i | x | d + α − . Note that it always holds that f (0) − f (4 x i y i ) > . Therefore, for all x ∈ R d , according to (2.4),I i ≤ c | x | d + α − (cid:16) Z { y i > , | x | / ≤| x − y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) || y | dy (cid:17) ≤ c | x | d + α − (cid:16) Z {| x | / ≤| x − y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) || y | dy (cid:17) ≤ c | x | d + α − (cid:16) Z {| y |≤ | x |} e − V ( | y | ) | V ′ ( | y | ) || y | dy (cid:17) ≤ c | x | d + α − and so (cid:16) d X i =1 I i (cid:17) / ≤ c | x | d + α − . Similarly, we also can prove that for all x ∈ R d , (cid:16) d X i =1 ˜ I i (cid:17) / ≤ c | x | d + α − . PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 11
Combining all the estimates above, we can obtain that there exists a constant c > such that forall x ∈ R d with | x | ≥ large enough, | ˜ b ( x ) | ≤ c | x | d + α − ; that is, | b ( x ) | ≤ ce V ( | x | ) | x | d + α − . The proof is completed, since b ( x ) is locally bounded. (cid:3) Remark 2.6. (1) If condition (2.11) is strengthened into lim sup r →∞ [ e − V ( r ) V ′ ( r ) r d +1 ] = 0 , then bothterms (cid:16) P di =1 I i (cid:17) / and (cid:16) P di =1 I i (cid:17) / are o (cid:16) | x | d + α − (cid:17) for all x ∈ R d with | x | large enough. Hence,the remaining term (cid:16) P di =1 I i (cid:17) / or (cid:16) P di =1 ˜ I i (cid:17) / plays the lead role in the estimates above.(2) Under the assumptions of Lemmas 2.4 and 2.5, it holds that, for | x | large enough, h x, b ( x ) i ≍ − e V ( | x | ) (1 + | x | ) d + α | x | . The case of d ≤ − α . In this part, we will consider the case of d ≤ − α , i.e., d = 1 and < α ≤ . Let V ∈ C ( R ) be such that e − V ∈ L ( R ; dx ) ∩ C b ( R ) , and let b ( x ) be defined by (1.5).We first show that Lemma 2.7.
Let V ∈ C ( R ) be such that e − V ∈ L ( R ; dx ) ∩ C b ( R ) . If (2.12) lim sup | x |→∞ [ | x | e − V ( x ) | V ′ ( x ) − V ′′ ( x ) | ] < ∞ , then b ( x ) given by (1.5) is well defined.Proof. Since e − V ∈ C b ( R ) , we know that − ( − ∆) α/ e − V ( x ) ∈ C b ( R ) , and so − ( − ∆) α/ e − V ( x ) is locallyintegrable on R . Next, we will estimate ( − ∆) α/ e − V ( x ) for x < − small enough. For x < − , − ( − ∆) α/ e − V ( x ) = Z R (cid:0) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z {| z |≤ } (cid:1) c ,α | z | α dz = Z {| z | < − x/ } (cid:0) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z (cid:1) c ,α | z | α dz + Z {| z |≥− x/ } ( e − V ( x + z ) − e − V ( x ) ) c ,α | z | α dz =: I ( x ) + I ( x ) . Since | I ( x ) | ≤ Z {| z |≥− x/ } e − V ( x + z ) c ,α | z | α dz + Z {| z |≥− x/ } e − V ( x ) c ,α | z | α dz ≤ c (cid:16) | x | − − α + e − V ( x ) (cid:17) , by e − V ∈ L ( R ; dx ) we know that R R | I ( x ) | dx < ∞ . On the other hand, by the mean value theorem, | I ( x ) | ≤ Z {| z | < − x/ } | e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z | c ,α | z | α dz ≤ c ,α h sup x/ ≤ u ≤ x/ | e − V ( u ) ( V ′ ( u ) − V ′′ ( u )) | i Z − x/ z − α dz ≤ c ( − x ) − α sup x/ ≤ u ≤ x/ (cid:2) e − V ( u ) | V ′ ( u ) − V ′′ ( u ) | (cid:3) ≤ c ( − x ) − − α sup x/ ≤ u ≤ x/ (cid:2) | u | e − V ( u ) | V ′ ( u ) − V ′′ ( u ) | (cid:3) ≤ c ( − x ) − − α , (2.13)where in the last inequality we used (2.12). Note that analogous estimates hold also for x > largeenough, and hence we arrive at the desired assertion. (cid:3) Remark 2.8.
From the proof above, we can see that under the assumptions of Lemma 2.7,(2.14) Z R | ( − ∆) α/ e − V ( u ) | du < ∞ and hence Z ∞ x ( − ∆) α/ e − V ( u ) du is also well defined for any x ∈ R .In the following, we always assume that (2.12) holds. We further suppose that V ( x ) = V ( − x ) forall x ∈ R . Then, we claim that Lemma 2.9.
Let V ∈ C ( R ) be such that e − V ∈ L ( R ; dx ) ∩ C b ( R ) . Suppose that (2.12) holds andthat V ( x ) = V ( − x ) for all x ∈ R . Then, b ( x ) given by (1.5) is an anti-symmetric function on R ( i.e., b ( x ) = − b ( − x ) for all x ∈ R ) such that (2.15) b ( x ) = e V ( x ) Z ∞ x ( − ∆) α/ e − V ( z ) dz, x ≥ , − e V ( x ) Z ∞− x ( − ∆) α/ e − V ( z ) dz, x < . In particular, b (0) = 0 . Moreover, b ( x ) ∈ C ( R ) and is locally bounded.Proof. As mentioned in Remark 2.8, under the assumptions of this lemma, we have (2.14). We willshow that this yields(2.16) − Z R ( − ∆) α/ e − V ( u ) du = 0 , and hence b ( x ) = − e V ( x ) Z x −∞ ( − ∆) α/ e − V ( u ) du = e V ( x ) Z ∞ x ( − ∆) α/ e − V ( u ) du, x ≥ . Indeed, for any ε ∈ (0 , and any x ∈ R , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y − x |≥ ε } ( e − V ( y ) − e − V ( x ) ) | y − x | α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| z |≥ ε } (cid:0) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z {| z |≤ } (cid:1) dz | z | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z {| z |≤ } (cid:12)(cid:12) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z (cid:12)(cid:12) dz | z | α + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| z | > } (cid:0) e − V ( x + z ) − e − V ( x ) (cid:1) dz | z | α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k [ e − V ] ′′ k ∞ Z {| z |≤ } | z | | z | α dz + 2 k e − V k ∞ Z {| z | > } | z | α dz ≤ c < ∞ . On the other hand, for any ε ∈ (0 , and any x ∈ R with | x | > large enough, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y − x |≥ ε } ( e − V ( y ) − e − V ( x ) ) | y − x | α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| z |≥ ε } (cid:0) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z {| z |≤| x | / } (cid:1) | z | α dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z {| z |≤| x | / } (cid:12)(cid:12) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z (cid:12)(cid:12) | z | α dz + Z {| z | > | x | / } e − V ( x + z ) | z | α dz + e − V ( x ) Z {| z | > | x | / } | z | α dz ≤ c | x | − (1+ α ) + 2 α | x | α Z R e − V ( z ) dz + c e − V ( x ) , PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 13 where the first term in the last inequality follows from (2.12) and the argument for (2.13). Hence,there is a constant c > such that for all x ∈ R , sup ε ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z {| y − x |≥ ε } ( e − V ( y ) − e − V ( x ) ) | y − x | α dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:16) (1 + | x | ) − − α + e − V ( x ) (cid:17) . Therefore, by using the dominated convergence theorem and changing the order of integration, we findthat − Z R ( − ∆) α/ e − V ( x ) dx = c ,α Z R lim ε → Z {| y − x |≥ ε } ( e − V ( y ) − e − V ( x ) ) | y − x | α dy dx = c ,α lim ε → Z R Z {| y − x |≥ ε } ( e − V ( y ) − e − V ( x ) ) | y − x | α dy dx = − c ,α lim ε → Z R Z {| y − x |≥ ε } ( e − V ( x ) − e − V ( y ) ) | y − x | α dx dy = − c ,α Z R lim ε → Z {| x − y |≥ ε } ( e − V ( x ) − e − V ( y ) ) | x − y | α dx dy = Z R ( − ∆) α/ e − V ( y ) dy, which proves (2.16).On the other hand, − Z −∞ ( − ∆) α/ e − V ( u ) du = Z −∞ Z R (cid:16) e − V ( u + z ) − e − V ( u ) − (cid:2) e − V ( u ) (cid:3) ′ z {| z |≤ } (cid:17) c ,α | z | α dz du = Z −∞ lim ε → Z {| z |≥ ε } (cid:16) e − V ( u + z ) − e − V ( u ) (cid:17) c ,α | z | α dz du = Z ∞ lim ε → Z {| z |≥ ε } (cid:16) e − V ( − u + z ) − e − V ( − u ) (cid:17) c ,α | z | α dz du = Z ∞ lim ε → Z {| z |≥ ε } (cid:16) e − V ( − u + z ) − e − V ( u ) (cid:17) c ,α | z | α dz du = Z ∞ lim ε → Z {| z |≥ ε } (cid:16) e − V ( − u − z ) − e − V ( u ) (cid:17) c ,α | z | α dz du = Z ∞ lim ε → Z {| z |≥ ε } (cid:16) e − V ( u + z ) − e − V ( u ) (cid:17) c ,α | z | α dz du = Z ∞ Z R (cid:16) e − V ( u + z ) − e − V ( u ) − (cid:2) e − V ( u ) (cid:3) ′ z {| z |≤ } (cid:17) c ,α | z | α dz du = − Z ∞ ( − ∆) α/ e − V ( u ) du, (2.17)where in the third and the fifth equalities we changed the variables, and the fourth and the sixthequalities follow from the symmetry V ( x ) = V ( − x ) for all x ∈ R . Combining (2.16) with (2.17), wehave(2.18) Z ∞ ( − ∆) α/ e − V ( u ) du = 0 and so b (0) = 0 . Furthermore, by R ∞ ( − ∆) α/ e − V ( u ) du = 0 and ( − ∆) α/ e − V ( x ) = ( − ∆) α/ e − V ( − x ) for all x ∈ R (which is also due to the symmetry V ( x ) = V ( − x ) for all x ∈ R ), we can get that forany x < , b ( x ) = − e V ( x ) Z x −∞ ( − ∆) α/ e − V ( u ) du = − e V ( x ) Z ∞− x ( − ∆) α/ e − V ( z ) dz. The desired assertion (2.15) follows.As we mentioned in the proof of Lemma 2.7, since e − V ∈ C b ( R ) , ( − ∆) α/ e − V ( x ) ∈ C b ( R ) . By(2.18), we can easily see that b ( x ) ∈ C ( R ) and is locally bounded. (cid:3) The following statement is analogous to Lemma 2.4.
Lemma 2.10.
Let V ∈ C ( R ) be a symmetric function on R such that e − V ∈ L ( R ; dx ) ∩ C b ( R ) and (2.12) holds. Suppose that (2.19) lim x →∞ xe − V ( x ) = 0 and (2.20) lim inf x →∞ [ x e − V ( x ) ( V ′ ( x ) − V ′′ ( x ))] ≥ . Then there exist constants c , c > and r > such that for all x ∈ R , xb ( x ) ≤ c {| x |≤ r } − c e V ( x ) | x | α | x | {| x | >r } . Proof.
Since b ( x ) is anti-symmetric, we only need to consider x ≥ . According to Lemma 2.9, b ( x ) ∈ C ( R ) and is therefore locally bounded. Hence, in order to prove the desired assertion, it issufficient to verify that there exists a constant c > such that for x > large enough(2.21) − ( − ∆) α/ e − V ( x ) ≥ cx α . To this end, for x > we write − ( − ∆) α/ e − V ( x ) = Z R (cid:0) e − V ( x + z ) − e − V ( x ) + e − V ( x ) V ′ ( x ) z {| z |≤ } (cid:1) c ,α | z | α dz = Z {| z | Hence, for x > large enough, − ( − ∆) α/ e − V ( x ) ≥ c x α − c xe − V ( x ) x α + c x α h x inf x/ ≤ z ≤ x/ [ e − V ( z ) ( V ′ ( z ) − V ′′ ( z ))] i . This along with (2.19) and (2.20) yields (2.21). The proof is completed. (cid:3) Lemma 2.11. Let V ∈ C ( R ) be a symmetric function on R such that e − V ∈ L ( R ; dx ) ∩ C b ( R ) and (2.12) holds. If (2.19) is satisfied, then there exist constants c > and r > such that for all x ∈ R with | x | ≥ r , b ( x ) ≥ − c e V ( x ) | x | α . Proof. The assertion follows from the conclusion that there exists a constant c > such that for x > large enough(2.22) − ( − ∆) α/ e − V ( x ) ≤ c x α . For (2.22), one can follow the idea for the argument of (2.21). In particular, under (2.12) it holds that(2.23) lim sup x →∞ [ x e − V ( x ) ( V ′ ( x ) − V ′′ ( x ))] < ∞ . Then we can deduce that I ( x ) ≤ c x α by applying (2.23) instead of (2.20). The details are omitted here. (cid:3) Remark 2.12. Under the assumptions of Lemma 2.10, for | x | large enough, xb ( x ) ≍ − e V ( x ) (1 + | x | ) α | x | . Properties of the SDE with the fractional drift In this section, we will consider the following stochastic differential equation (SDE)(3.1) dX t = b ( X t ) dt + dZ t , where ( Z t ) t ≥ is a symmetric (rotationally invariant) α -stable process on R d with α ∈ (0 , and d ≥ ,and b ( x ) is defined by (1.4) when d > − α and by (1.5) when d ≤ − α . Everywhere below, weassume that Assumption (A) is satisfied.Suppose first that d > − α . According to Lemmas 2.2 and 2.4, for the drift b ( x ) defined by (1.4),we have b ∈ C β ( R d ) with β = 2 − α when α ∈ (1 , , β = 1 − ε for any ε > when α = 1 , and β = 1 when α ∈ (0 , (in particular, b ∈ C β ( R d ) with β ∈ (0 , − α/ for all α ∈ (0 , ), and(3.2) h b ( x ) , x i ≤ K (1 + | x | ) , x ∈ R d for some constant K > , where C β ( R d ) denotes the set of locally β -Hölder continuous functions from R d to R d for β ∈ (0 , . Suppose now that d ≤ − α . Then, by Lemmas 2.9 and 2.10, the drift b ( x ) defined by (1.5) belongs to C ( R ) and satisfies (3.2) as well. Here we used the fact that (2.19)holds under condition (1.6) and hence under Assumption (A), all the conditions required in Lemmas2.9 and 2.10 are satisfied. Consequently, for all d ≥ and α ∈ (0 , , the equation (3.1) has a uniquenon-explosive strong solution ( X t ) t ≥ , which is a strong Markov process with the generator Lf ( x ) = − ( − ∆) α/ f ( x ) + h b ( x ) , ∇ f ( x ) i , f ∈ C b ( R d ) . For the case of d > − α , the reader can be referred to [43, Theorem 2.4 and Lemma 7.1], while for d ≤ − α one can directly apply e.g. [25, Theorem 1.1], since b ∈ C ( R ) obviously implies that b ( x ) satisfies a local Lipschitz condition. Alternatively, for any d ≥ and α ∈ (0 , , we can first apply[33, Theorem 1.1] or [6, Corollary 1.4(i)] (with b ∈ C βb ( R d ) , i.e., with b ( x ) being globally β -Höldercontinuous) to get the locally unique strong solution, and then use the additional global one-sidedlinear growth condition (3.2) to obtain the unique non-explosive strong solution; see the proof of [16,Theorem 1] or [25, Theorem 1.1].In the following, we will prove rigorously that (1.1) is indeed the unique invariant measure for theprocess ( X t ) t ≥ defined as the solution to (1.3) with the drift term b ( x ) defined by (1.4) and (1.5).We begin with the following simple lemma. Lemma 3.1. Under Assumption (A) , for any β ∈ (0 , α ) , there are constants C , C > such that forall x ∈ R d , (3.3) LV ( x ) ≤ C − C e V ( x ) | x | d + α V ( x ) , where V ( x ) = (1 + | x | ) β/ .Proof. According to Lemmas 2.4 and 2.10, we know that under Assumption (A) there are constants λ , λ > such that for all x ∈ R d ,(3.4) h x, b ( x ) i ≤ λ − λ U ( x ) | x | , where U ( x ) = e V ( x ) / (1 + | x | ) d + α . Here, we used again the fact that (2.19) holds true under condition(1.6) .Recall that c d,α | z | d + α is the density function of the Lévy measure for the symmetric α -stable process.Since ∇ V ( x ) = β (1 + | x | ) ( β − / x and k∇ V k ∞ ≤ β (2 − β/ , we find that for all x ∈ R d and l ≥ , LV ( x ) = β (1 + | x | ) ( β − / h x, b ( x ) i + Z {| z |≤ l } ( V ( x + z ) − V ( x ) − h∇ V ( x ) , z i ) c d,α | z | d + α dz + Z {| z | >l } ( V ( x + z ) − V ( x )) c d,α | z | d + α dz ≤ β (1 + | x | ) ( β − / h x, b ( x ) i + ( β/ 2) (2 − β/ c d,α Z {| z |≤ l } | z | d + α − dz + Z {| z | >l } [(1 + 2 | x | ) β/ + (2 | z | ) β/ ] c d,α | z | d + α dz ≤ β (1 + | x | ) ( β − / ( λ − λ U ( x ) | x | ) + c l − α + c l − α (1 + 2 | x | ) β/ + c , where c i (1 ≤ i ≤ are independent of l and x ∈ R d . Here, in the equality above, we used the factthat R { ≤| z |≤ l } z c d,α | z | d + α dz = 0 ; the first inequality follows from the mean value theorem and the factthat V ( x + z ) ≤ (1 + 2 | x | + 2 | z | ) β/ ≤ (1 + 2 | x | ) β/ + (2 | z | ) β/ ; and in the last inequality we used(3.4) and the facts that Z {| z |≤ l } | z | d + α − dz ≤ c l − α , Z {| z | >l } | z | d + α dz ≤ c l − α and Z {| z |≥ } | z | d + α − β dz < ∞ , β ∈ [0 , α ) . From the right hand side of the inequality above, we can see that LV ( x ) is locally bounded, and for | x | large enough, LV ( x ) ≤ − λ β U ( x ) | x | β + c l − α + 4 c l − α | x | β , which is dominated by − λ β U ( x ) | x | β by choosing | x | ≫ l ≫ . Then, (3.3) follows. (cid:3) We also need the following statement. Lemma 3.2. Let ( X t ) t ≥ be the unique strong solution to the SDE (3.1) with b ( x ) defined by (1.4) when d > − α and by (1.5) when d ≤ − α , such that Assumption (A) is satisfied. Then, (i) The process ( X t ) t ≥ is strong Feller and Lebesgue irreducible; (ii) The transition probability function of the process ( X t ) t ≥ is absolutely continuous with respectto the Lebesgue measure.In particular, the process has a unique invariant probability measure µ ( dx ) = ρ ( x ) dx , where ρ ( x ) > for all x ∈ R d .Proof. For simplicity, we only consider the case of d > − α , since the case of d ≤ − α can be provedsimilarly and easily.(i) For any n ≥ , let b n ( x ) = − C d, − α e V ( x ) ∧ K ( n ) Z R d e − V ( y ) ∇ V ( y ) | x − y | d − (2 − α ) dy, PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 17 where K ( n ) = 1 + sup | x |≤ n | V ( x ) | . Then, according to the proof of Lemma 2.2, the function x R R d e − V ( y ) ∇ V ( y ) | x − y | d − (2 − α ) dy is bounded, andglobally (2 − α ) -Hölder continuous when α ∈ (1 , , globally (1 − ε ) -Hölder continuous for any ε > when α = 1 , and belongs to C ( R d ) when α ∈ (0 , , and hence b n ( x ) also shares these properties.Consider the following SDE(3.5) dX ( n ) t = b n ( X ( n ) t ) dt + dZ t . It follows from [33, Theorem 1.1] or [6, Corollary 1.4(i)] that the SDE (3.5) has a unique strong solution,which will be denoted by X ( n ) := ( X ( n ) t ) t ≥ . Note that the infinitesimal generator of the process X ( n ) is given by L ( n ) f ( x ) = h b n ( x ) , ∇ f ( x ) i − ( − ∆) α/ f ( x ) , f ∈ C b ( R d ) . Hence, according to [7, Theorem 1.5] for α ∈ (1 , and [42, Theorem 1.1] for α = 1 as well as [19,Theorem 2.2] for α ∈ (0 , , the process X ( n ) has a continuous and strictly positive transition densityfunction, which implies that X ( n ) is strong Feller (i.e., for any f ∈ B b ( R d ) and t > , x P ( n ) t f ( x ) := E x f ( X ( n ) t ) is continuous) and Lebesgue irreducible (i.e., for any t > and open set O ∈ B ( R d ) withLeb ( O ) > , P x ( X ( n ) t ∈ O ) > ). Here and in what follows, we assume that X and X ( n ) are definedon the same probability space (Ω , F , P ) . Let P x ( · ) = P ( ·| X = x ) or P x ( · ) = P ( ·| X ( n )0 = x ) withoutconfusion. Since b n ( x ) = b ( x ) for all | x | ≤ n , the law of X t ∧ τ n is the same as the law of X ( n ) t ∧ τ n for any t > , where τ n := inf { t > | X t | ≥ n } .Now, let ( P t ) t ≥ be the semigroup of the process X . For any f ∈ B b ( R d ) , x ∈ R d and for anysequence { x k } k ≥ ⊆ R d such that x k → x as k → ∞ , we choose n large enough so that { x k } k ≥ ⊂ B (0 , n ) , and then find that | P t f ( x k ) − P t f ( x ) | = | E x k f ( X t ) − E x f ( X t ) |≤ | E x k ( f ( X t ) { t<τ n } ) − E x ( f ( X t ) { t<τ n } ) | + k f k ∞ (cid:0) P x k ( τ n ≤ t ) + P x ( τ n ≤ t ) (cid:1) = | E x k ( f ( X ( n ) t ) { t<τ n } ) − E x ( f ( X ( n ) t ) { t<τ n } ) | + k f k ∞ (cid:0) P x k ( τ n ≤ t ) + P x ( τ n ≤ t ) (cid:1) ≤ | E x k f ( X ( n ) t ) − E x f ( X ( n ) t ) | + 2 k f k ∞ (cid:0) P x k ( τ n ≤ t ) + P x ( τ n ≤ t ) (cid:1) ≤ | P ( n ) t f ( x k ) − P ( n ) t f ( x ) | + 4 k f k ∞ sup k ≥ P x k ( τ n ≤ t ) . (3.6)Note that, combining Lemma 3.1 with the standard argument (for example, see the proof of [28,Theorem 2.1]), we can see that for any k ≥ and t > , P x k ( τ n ≤ t ) = P x k ( max s ∈ [0 ,t ] | X s | ≥ n ) = P x k (cid:18) max s ∈ [0 ,t ] (1 + | X s | ) β/ ≥ (1 + | n | ) β/ (cid:19) ≤ c (1 + | x k | ) β/ (1 + | n | ) β/ . Since x k → x as k → ∞ , without loss of generality we may and will assume that x k ∈ B ( x , .Hence, lim n →∞ sup k ≥ P x k ( τ n ≤ t ) = 0 . Letting k → ∞ and then n → ∞ in (3.6), we show that lim k →∞ | P t f ( x k ) − P t f ( x ) | = 0 . Hence, for any f ∈ B b ( R d ) and t > , P t f is a continuous function, i.e., the process X is strong Feller.For any x ∈ R d , t > and open set O ∈ B ( R d ) with Leb ( O ) > , choosing n large enough suchthat Leb ( O ∩ B (0 , n )) > , P x ( X t ∈ O ) ≥ P x ( X t ∈ O, τ n > t ) = P x ( X ( n ) t ∈ O ∩ B (0 , n ) , τ n > t ) . According to (the proof of) [5, Corollary 3.6], the Dirichlet heat kernel of the process X ( n ) is positiveeverywhere, and so the right hand side of the inequality above is positive (even though the setting of[5] is restricted to d ≥ , the proof of [5, Corollary 3.6] is based on the global heat kernel estimates and the Lévy system for X ( n ) , both of which are available for d = 1 too, and so [5, Corollary 3.6] holdstrue for all d ≥ ). Hence, P x ( X t ∈ O ) > and thus the process X is Lebesgue irreducible.Therefore, all compact sets are petite for X (cf. [27, Theorem 4.1(i)]), and hence the existence ofthe invariant probability measure µ follows from (3.3), while the uniqueness is a direct consequence ofthe strong Feller property and irreducibility; see [28, Theorems 5.1 and 5.2].(ii) As we already established in the first part of the proof, according to [7, Theorem 1.5], for any t > , the law of X ( n ) t is absolutely continuous with respect to the Lebesgue measure. We will claimthat the law of X t is also absolutely continuous with respect to the Lebesgue measure. Indeed, for anyopen set O ∈ B ( R d ) such that Leb ( O ) = 0 , any t > , x ∈ R d and n large enough, P x ( X t ∈ O ) = P x ( X t ∈ O, τ n > t ) + P x ( X t ∈ O, τ n ≤ t )= P x ( X ( n ) t ∈ O, τ n > t ) + P x ( X t ∈ O, τ n ≤ t ) ≤ P x ( X ( n ) t ∈ O ) + 2 P x ( τ n ≤ t ) = 2 P x ( τ n ≤ t ) . As mentioned above, for any x ∈ R d and t > , P x ( τ n ≤ t ) → as n → ∞ . Hence, P x ( X t ∈ O ) = 0 for any x ∈ R d and t > .Let P ( t, x, · ) be the transition function of the process X . By the argument for the Lebesgue irredu-cibility above, we know that P ( t, x, · ) and the Lebesgue measure are equivalent, so that P ( t, x, A ) = R A p ( t, x, y ) dy for any A ∈ B ( R d ) and p ( t, x, y ) can be chosen to be strictly positive everywhereon R d × R d for any fixed t > . Hence, for the invariant probability measure µ , since µ ( A ) = R R d P ( t, x, A ) dµ ( x ) for A ∈ B ( R d ) and t > , µ is also absolutely continuous with respect to theLebesgue measure and the associated density function can be chosen to be strictly positive every-where. (cid:3) Proposition 3.3. Let X := ( X t ) t ≥ be the unique strong solution to the SDE (1.3) with b ( x ) definedby (1.4) when d > − α and by (1.5) when d ≤ − α such that Assumption (A) is satisfied. Then, µ ( dx ) := Z − e − V ( x ) dx with Z = R R d e − V ( x ) dx is the unique invariant probability measure for theprocess X .Proof. Recall that the infinitesimal generator of the process ( X t ) t ≥ is given by Lf ( x ) = − ( − ∆) α/ f ( x ) + h b ( x ) , ∇ f ( x ) i . Let D ( L ) be the domain of the operator under the norm k · k ∞ . Then, if µ is an invariant measure for ( P t ) t ≥ , for any f ∈ D ( L ) ,(3.7) µ ( Lf ) = µ (cid:18) lim t → P t f − ft (cid:19) = lim t → µ ( P t f ) − µ ( f ) t = 0 . Actually, (3.7) is equivalent to saying that µ is an invariant probability measure of the process X andthis is still true if we replace D ( L ) with a core; see e.g. [23, Theorem 3.37].According to [7, Theorem 1.5], C b ( R d ) is contained in the domain of the infinitesimal generator ofthe process X ( n ) given by the SDE (3.5). Then, by the localization argument that we used in the proofof the strong Feller property above, we can check that C ∞ c ( R d ) ⊂ D ( L ) . In the following, we take µ ( dx ) := Z − e − V ( x ) dx with Z = R R d e − V ( x ) dx , and verify that for any f ∈ C ∞ c ( R d ) , µ ( Lf ) = 0 .Let us first suppose that d > − α . Then, for b ( x ) defined by (1.4) and for any f ∈ C ∞ c ( R d ) , Z R d Lf ( x ) e − V ( x ) dx = − Z R d e − V ( x ) ( − ∆) α/ f ( x ) dx + Z R d e − V ( x ) h∇ f ( x ) , b ( x ) i dx = − Z R d e − V ( x ) ( − ∆) α/ f ( x ) dx + C d, − α Z R d * ∇ f ( x ) , ∇ "Z R d e − V ( y ) | · − y | d − (2 − α ) dy ( x ) + dx. (3.8)On the other hand, by the integration by parts, we find that for any f ∈ C ∞ c ( R d ) , C d, − α Z R d * ∇ f ( x ) , ∇ "Z R d e − V ( y ) | · − y | d − (2 − α ) dy ( x ) + dx PPROXIMATION OF HEAVY-TAILED DISTRIBUTIONS 19 = C d, − α Z R d ( − ∆) f ( x ) Z R d e − V ( y ) | x − y | d − (2 − α ) dy dx = C d, − α Z R d ( − ∆) − α/ (cid:2) ( − ∆) α/ f (cid:3) ( x ) Z R d e − V ( y ) | x − y | d − (2 − α ) dy dx = C d, − α Z R d ( − ∆) α/ f ( x ) · ( − ∆) − α/ "Z R d e − V ( y ) | · − y | d − (2 − α ) dy ( x ) dx = Z R d e − V ( x ) ( − ∆) α/ f ( x ) dx, where in the second equality we used the fact that ( − ∆) = ( − ∆) α/ ( − ∆) − α/ (which can be checkedby the standard Fourier analysis), the third equality follows from the symmetry of ( − ∆) − α/ on L ( R d ; dx ) , and in the fourth equality we used the fact that C d, − α | x − y | d − (2 − α ) is the Green function for thesymmetric (2 − α ) -stable process, and hence for all x ∈ R d , ( − ∆) − α/ "Z R d C d, − α e − V ( y ) | · − y | d − (2 − α ) dy ( x ) = e − V ( x ) , cf. [21, Proposition 7.2]. The equality above along with (3.8) yields that R R d Lf ( x ) e − V ( x ) dx = 0 , andso the desired assertion follows.Now, we consider the case that d ≤ − α ; i.e., d = 1 and α ∈ (0 , . For b ( x ) defined by (1.5), using(2.15), we have for any f ∈ C ∞ c ( R ) , Z R Lf ( x ) e − V ( x ) dx = − Z R e − V ( x ) ( − ∆) α/ f ( x ) dx + Z ∞ f ′ ( x ) Z ∞ x ( − ∆) α/ e − V ( z ) dz dx − Z −∞ f ′ ( x ) Z ∞− x ( − ∆) α/ e − V ( z ) dz dx = − Z R e − V ( x ) ( − ∆) α/ f ( x ) dx + Z ∞ f ( x )( − ∆) α/ e − V ( x ) dx + Z −∞ f ( x )( − ∆) α/ e − V ( − x ) dx. = − Z R e − V ( x ) ( − ∆) α/ f ( x ) dx + Z R e − V ( x ) ( − ∆) α/ f ( x ) dx = 0 , where in the second equality we used the fact that R ∞ ( − ∆) α/ e − V ( z ) dz = 0 (cf. (2.18)) and the thirdequality follows from the fact that ( − ∆) α/ e − V ( − x ) = ( − ∆) α/ e − V ( x ) for all x ≤ due to the symmetryof V ( x ) .Therefore, according to both conclusions above and Lemma 3.2, we prove that µ ( dx ) := Z − e − V ( x ) dx is the unique invariant probability measure of the process X . (cid:3) Remark 3.4. When d > − α , by some elementary calculations, the dual of the operator L on L ( R d ; dx ) is given by L ∗ f ( x ) = − ( − ∆) α/ f ( x ) − h b ( x ) , ∇ f ( x ) i − div b ( x ) f ( x )= − ( − ∆) α/ f ( x ) − div ( bf )( x ) . Arguing informally, we havediv ( be − V )( x ) = div[ ∇ ( − ∆) − (1 − α/ e − V ]( x ) = [∆( − ∆) − (1 − α/ e − V ]( x )= − ( − ∆)( − ∆) − (1 − α/ e − V ( x ) = − ( − ∆) α/ e − V ( x ) , and so, by (2.1), L ∗ e − V ( x ) = 0 for x ∈ R d , which would imply the infinitesimal invariance of µ givenby (1.1) for the process ( X t ) t ≥ defined by (1.3), cf. the proof of [36, Theorem 1.1]. However, since wedo not know whether ∇ ( − ∆) − (1 − α/ e − V belongs to C ( R d ) or not when α ∈ (1 , (cf. Remark 2.3),div ∇ ( − ∆) − (1 − α/ e − V may be not well defined. Hence the argument above is informal and, in orderto rigorously prove that µ is the unique invariant measure of ( X t ) t ≥ , it is necessary to argue as in theproof of Proposition 3.3. Using Lemma 3.2 and Proposition 3.3, we can now easily prove Theorem 1.1. Proof of Theorem . . From Lemma 3.2, we know that the process X := ( X t ) t ≥ obtained as theunique solution to the SDE (3.1) is strong Feller and irreducible. Hence, due to [27, Theorem 4.1(i)],all compact sets are petite for X . Moreover, according to Lemma 3.1, we have the Lyapunov condition(3.3). As a consequence, [28, Theorem 6.1] applies, and so there is a constant λ > such that for any x ∈ R d and t > , k P ( t, x, · ) − µ k Var ,V ≤ C ( x ) V ( x ) e − λt , where V ( x ) = (1 + | x | ) β/ with β ∈ (0 , α ) , C ( x ) is a non-negative and locally bounded function on R d , and µ is the unique invariant probability measure for X . Finally, from Proposition 3.3 we knowthat µ is given by (1.1), and the proof is concluded. (cid:3) Acknowledgement. Mateusz B. Majka would like to thank Aleksandar Mijatović for discussionsregarding Fractional Langevin Monte Carlo, and Jian Wang would like to thank Professor RenmingSong and Dr. Longjie Xie for helpful comments on heat kernel estimates for SDEs with Lévy jumps.The research of Lu-Jing Huang is supported by the National Natural Science Foundation of China (No.11901096). A part of this work was completed while Mateusz B. Majka was affiliated to the Universityof Warwick and supported by the EPSRC grant no. EP/P003818/1. The research of Jian Wang issupported by the National Natural Science Foundation of China (No. 11831014), the Program forProbability and Statistics: Theory and Application (No. IRTL1704), and the Program for InnovativeResearch Team in Science and Technology in Fujian Province University (IRTSTFJ). 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