Convergence to Equilibrium in Fokker–Planck Equations

Journal of Dynamics and Differential Equations (2019) 31:1591–1615 https://doi.org/10.1007/s10884-018-9705-8 Convergence to Equilibrium in Fokker–Planck Equations Min Ji1 · Zhongwei Shen2 · Yingfei Yi2 Received: 30 April 2017 / Published online: 14 September 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract The present paper is devoted to the investigation of long-time behaviors of global probability solutions of Fokker–Planck equations with rough coefficients. In particular, we prove the convergence of probability solutions under a Lyapunov condition in terms of the Markov semigroup associated to the stationary one. A generalization of earlier results on the existence and uniqueness of global probability solutions is also given. Keywords Fokker–Planck equation · Probability solutions · Convergence Mathematics Subject Classification Primary 37C40 · 37C75 · 34F05 · 60H10; Secondary 35Q84 · 60J60 1 Introduction Let U ⊂ Rn be an open and connected domain which can be bounded, unbounded, or the whole space Rn . Stochastic differential equations of the form d x = V (x)dt + G(x)dW , x ∈ U , (1.1) Dedicated to the memory of Professor George R. Sell. The first author was partially supported by NSFC Innovation Grant 10421101 and NSFC Grant 11571344. The second author was partially supported by a start-up grant from the University of Alberta and an NSERC Discovery Grant. The third author was partially supported by NSERC Discovery Grant 1257749, a faculty development grant from University of Alberta, and a Scholarship from Jilin University. B Zhongwei Shen [email protected] Min Ji [email protected] Yingfei Yi [email protected] 1 Academy of Mathematics and System Sciences, Chinese Academy of Sciences and University of Chinese Academy of Sciences, Beijing 100080, People’s Republic of China 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada 123 1592 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 where V : U → Rn , G : U → Rn×m and W is a standard m-dimensional Wiener process, are often used to model and describe the uncertainties or impurities in a dynamical system in U . If GG ⊤ is everywhere positive definite, V and G are locally Lipschitz continuous, and solutions of (1.1) exist globally in the sense of distributions, then it is well-known that (1.1) generates a Markov semigroup on Bb (U ), the space of bounded measurable functions on U , which is strongly Feller and irreducible. Consequently, if the Markov semigroup admits a unique invariant measure, then it follows from classical convergence results of Markov semigroups (see e.g. [13,18,22]) that the transition probability functions of (1.1) converge to the invariant measure as t → ∞. We note that the existence and uniqueness of an invariant measure for the Markov semigroup requires certain dissipative or Lyapunov conditions in U (see e.g. [1,18,22,25]). The present paper aims at studying a similar convergence problem when the coefficients of (1.1) are rough, which is actually the case in many applications. To do so, we consider the following Fokker–Planck equation ∂t u = Lu := ∂i2j (a i j u) − ∂i (V i u), t > 0, x ∈ U , (1.2) where ∂i = ∂xi , ∂i2j = ∂x2i x j and the usual summation convention is used. We note that, though the Fokker–Planck equation has interests in its own right in describing diffusive processes in ⊤ general, it can be generated from the stochastic differential equation (1.1) when (a i j ) := GG2 and V = (V i ) in the sense that the probability distribution of solutions of (1.1) formally satisfies (1.2). We make the following standard hypothesis: 1, p (H) The diffusion matrix A = (a i j ) is pointwise positive-definite, and a i j ∈ Wloc (U ), p V i ∈ L loc (U ), i, j ∈ {1, . . . , n}, where p > n + 2 is fixed. We note that as p > n the diffusion matrix A = (a i j ) is in fact locally uniformly positivedefinite by Sobolev imbedding. Under the regularity conditions in (H), we will consider weak solutions of (1.2) in the sense of measure. Denote L = a i j ∂i2j + V i ∂i as the formal L 2 adjoint of the Fokker–Planck operator L. Definition 1.1 Let ν be a Borel probability measure on U and T > 0 be given. We say that a family of Borel measures μ = (μt )t∈[0,T ) on U is a measure solution of (1.2) in [0, T ) with 1 (U , dμ dt) for any i, j ∈ {1, . . . , n}, initial condition ν if a i j , V i ∈ L loc t μ0 = ν, and the identities  U φdμt =  U φdν + lim  t ǫ→0+ ǫ U (1.3) Lφdμs ds, φ ∈ C0∞ (U ), hold a.e. t ∈ [0, T ). If, in addition, μt (U ) ≤ 1 for a.e. t ∈ [0, T ) (respectively, μt (U ) = 1 for a.e. t ∈ [0, T )), then μ = (μt )t∈[0,T ) is called a sub-probability solution (respectively, probability solution). In the case T = ∞, a measure solution, a sub-probability solution and a probability solution are called a global measure solution, a global sub-probability solution and a global probability solution, respectively. 123 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1593 In literature (see e.g. [2,3,24]), the following equivalent definition has been used: A family of Borel measures μ = (μt )t∈[0,T ) on U is called a measure solution of (1.2) in [0, T ) 1 (U , dμ dt) for any i, j ∈ {1, . . . , n}, satisfying the initial condition (1.3) if a i j , V i ∈ L loc t and   T 0 U (∂t u + Lu)dμt dt = 0, ∀u ∈ C0∞ ((0, T ) × U ), and for any φ ∈ C0∞ (U ), there exists a set Jφ ⊂ (0, T ) with |(0, T )\Jφ | = 0 such that   φdμt = φdν. lim t∈Jφ ,t→0 U U The existence and uniqueness of global probability solutions of Fokker–Planck equations have been studied by many authors (see e.g. [2–4,10,24,26]) even for time-dependent coefficients. In particular, such existence and uniqueness result has been recently shown for (1.2) in [24] under the existence of a Lyapunov-like function U ∈ C 2 (U ) which is non-negative and satisfies lim U (x) = ∞ (1.4) x→∂ U and LU (x) ≤ C1 + C2 U (x), x ∈U (1.5) for some constants C1 , C2 ≥ 0. As opposed to the issue of global existence and uniqueness, not much results on the convergence of global probability solutions of (1.2) are available or explicitly stated even in the case of U = Rn . For the case of uniformly Hölder continuous and bounded coefficients of (1.2) in U = Rn , certain strong dissipative condition is assumed in [21] to yield the convergence of global probability solutions of (1.2) to the stationary one (also see [14] and references therein). In this paper, we would like to give some results on the convergence of global probability solutions of (1.2) in a general domain U in Rn which can be bounded, unbounded, or the whole space Rn . We will do so under the assumption (H) and certain Lyapunov type of conditions imposed only near ∂ U . It turns out that we are also able to obtain the existence and uniqueness of global probability solutions of (1.2) by imposing conditions like (1.5) but only near ∂ U . For any non-negative continuous function U on U satisfying (1.4), we note that all sublevel sets ρ = {x ∈ U : U (x) < ρ} , ρ > 0 are pre-compact. In fact, such a function is a special compact function on U defined in [19,20]. We refer the reader to [19,20] for the meaning of x → ∂ U when U is unbounded. In particular, when U = Rn , x → ∂Rn simply means |x| → ∞. Making use of such compactness, we will work with Lyapunov types of functions as follows. Definition 1.2 Let U ∈ C 2 (U ) be a non-negative function. (1) U is said to be an unbounded Lyapunov-like function associated to L if (1.4) holds and there are constants C1 , C2 , ρm > 0 such that LU (x) ≤ C1 + C2 U (x), x ∈ U \ρm . (1.6) 123 1594 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 (2) U is said to be an unbounded Lyapunov function associated to L if (1.4) holds and there are constants γ , ρm > 0 such that LU (x) ≤ −γ , x ∈ U \ρm . (1.7) (3) U is said to be an unbounded strong Lyapunov function associated to L if (1.4) holds and there are constants C1 , C2 , ρm > 0 such that LU (x) ≤ C1 − C2 U (x), x ∈ U \ρm . (1.8) The condition (1.6) is weaker than (1.5) under the assumption (H) on the drift term V = (V i ). They are of course equivalent if V is continuous on U . Due to the condition (1.4), it is clear that a function of class (3) is stronger than that of (2) which is further stronger than that of (1). Our result concerning the existence and uniqueness of global probability solutions of (1.2) states as follows. Theorem A Suppose (H) and that there exists an unbounded Lyapunov-like function associated to L. Let ν be a Borel probability measure on U . Then, the following statements hold. (1) The equation (1.2) admits a global probability solution (μt )t∈[0,∞) with initial condition  ν satisfying the property that the function t → U φdμt is continuous on [0, ∞) for any φ ∈ C0∞ (U ). (2) If (μ̃)t∈[0,∞) is a global sub-probability solution of (1.2) with initial condition ν satisfying  the function t → U φd μ̃t being continuous on [0, ∞) for any φ ∈ C0∞ (U ), then μ̃t = μt for all t > 0. Among probability solutions of (1.2), the stationary ones, defined as follows, are of particular importance. Definition 1.3 Under the assumption (H), a Borel probability measure μ on U is called a stationary measure of the Fokker–Planck equation (1.2) if Lμ = 0 in the sense that  1 V i ∈ L loc (U , μ), i = 1, . . . , n, and Lφdμ = 0, ∀φ ∈ C0∞ (U ). U We note that the above definition actually only requires p > n in (H). The existence and uniqueness of stationary measures of (1.2) under Lyapunov type of conditions have been extensively studied in literature (see e.g. [8,9,11,12,18,20,28]). In particular, it is shown in [20] that stationary measures of (1.2) exist under the condition (H) with p > n + 2 replaced by p > n and the existence of a Lyapunov function associated to L which needs not be unbounded. The unboundedness of the Lyapunov function however guarantees the uniqueness of stationary measures (see also [11] for the case U = Rn ). As mentioned at the beginning of this section, if (1.1) generates a Markov semigroup which admits a unique invariant measure, then the transition probability functions of (1.1), i.e., the global probability solutions of (1.2), converge to the invariant measure which is necessarily a stationary measure of (1.2). Under the condition (H) and in the presence of an unbounded Lyapunov function associated to L, a natural question is whether one can have a similar theory for the convergence of global probability solutions of (1.2) to the unique stationary measure. Thanks to the recent works [12,27] which show the existence of a unique strongly continuous Markov semigroup associated to the unique stationary measure of (1.2), we have the following convergence results for global probability solutions of (1.2) which also indicate certain exponential mixing property of the unique stationary measure. 123 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1595 Theorem B Suppose (H). Then, the following statements hold. (1) If there exists an unbounded Lyapunov function associated to L, then, as t → ∞, any global probability solution (μt )t∈[0,∞) of (1.2) strongly converges to the unique stationary measure μ∗ , i.e., for any Borel set B ⊂ U , there holds μt (B) → μ∗ (B) as t → ∞. (2) If there exists an unbounded strong Lyapunov function U associated to L, then, as t → ∞, any global probability solution (μt )t∈[0,∞) of (1.2) with initial condition ν satisfying U ∈ L 1 (U , ν) exponentially converges to the unique stationary measure μ∗ which is exponentially mixing, i.e., there are constants C, r > 0 such that μt − μ∗ TV ≤ Ce−r t , t ≥ 0, where · T V denotes the total variation distance, and     1/2  1/2    −r t    ψ Tt φdμ∗ − (φ − μ∗ , φ)2 U ψ 2 U , t ≥ 0, ψdμ φdμ ∗ ∗  ≤ Ce  U U U for any pair of measurable functions φ and ψ with φ 2 U < ∞ and ψ 2 μ∗ , φ = U φdμ∗ and      φ   φ     φ U :=  = . 1 + U  ∞ 1 + U  ∞ L (U ,μ∗ ) L (U ,d x) U < ∞, where The rest of the paper is organized as follows. In Sect. 2, we study the existence and uniqueness of global probability solutions and prove Theorem A. In Sect. 3, we investigate the long-time behaviors of global probability solutions and prove the results of convergence as stated in Theorem B. In Sect. 4, some examples are given as applications of Theorems A, B. In Appendix A, we collect some facts from [12,27] about Markov semigroups associated to stationary measures. Through the rest of the paper, for short, we refer a measure solution, a sub-probability solution or a probability solution of (1.2) with initial condition ν as that of the initial value problem (1.2)–(1.3), or Cauchy problem (1.2)–(1.3), or simply, (1.2)–(1.3). 2 Existence and Uniqueness of Global Probability Solutions In this section, we prove the existence and uniqueness of global probability solutions of the initial value problem (1.2)–(1.3). We first construct a global sub-probability solution, which is further shown to be a global probability solution in the presence of an unbounded Lyapunovlike function, which also ensures the uniqueness. In particular, Theorem A is proven by combining Propositions 2.1, 2.2 and 2.3 below. 2.1 Existence of Global Sub-probability Solutions We first prove the existence of global sub-probability solutions of the Cauchy problem (1.2)– (1.3). Proposition 2.1 Assume (H). Then, the Cauchy problem (1.2)–(1.3) admits a global subprobability solution (μt )t∈[0,∞) in the sense of Definition 1.1. Moreover, the following properties of (μt )t∈[0,∞) hold. 123 1596 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 (i) μt (U ) ≤ 1 for all t ∈ (0, ∞); p (ii) There exists a positive and locally Hölder continuous function u ∈ L loc ((0, ∞), 1, p p −1, p Wloc (U )) with ∂t u ∈ L loc ((0, ∞), W  loc (U )) such that dμt dt = u(t, x)d xdt; ∞ (iii) For any φ ∈ C0 (U ), the map t → U φdμt is a continuous function on [0, ∞), and   t  Lφdμτ dτ, 0 < s < t < ∞. φdμt = φdμs + U U s U Proof This theorem is more or less proven in [2,3,24] by means of the regularity theory established in [5]. For completeness, we recall the essential arguments below. By [3, Theorem 3.1], [5, Section 3] and [24, Theorem 2.3], for any T > 1, there exists a sub-probability solution (μtT )t∈[0,T ) of (1.2)–(1.3) on [0, T ) such that (i)-(iii) with [0, ∞) replaced by [0, T ) hold. Let u T be the function on (0, T ) × U given in (ii), so dμtT dt = u T (t, x)d xdt. We then apply the regularity theory established in [5, Section 3] to conclude that there are a sequence {Tk }k≥1 with Tk → ∞ as k → ∞ and a nonnegative function u on (0, ∞)× U such that u Tk (t, x) converges to u locally uniformly as k → ∞. Then, the measures (μt )t∈[0,∞) defined by u, i.e., dμt dt = u(t, x)d xdt, are a global measure solution of (1.2)–(1.3). Now, (i) follows readily, (ii) follows from the regularity theory (see [5, Corollary 3.9]), Harnack’s inequality, Sobolev imbedding(see e.g. [23, Section 7]) and the connectivity of U , and (iii), except the continuity of t → U φdμt at t = 0, is a consequence of Lebesgue dominated convergence theorem and the locally uniform convergence of u Tk to u as  k → ∞.  It remains to show the continuity of t → U φdμt at t = 0, namely, U φdμt → U φdν as t → 0+ . As in the proof of [24, Theorem 2.3], we can prove the existence of some C > 0, α > 0 and ǫ0 > 0 such that for any T > 0 there holds      T  φu (t, x)d x − φdν  ≤ Ct α , t ∈ (0, ǫ0 ],  U U ⊔ ⊓ which leads to the result. Denote by Cc (U ) the space of compactly supported continuous functions on U . Lemma 2.1 Assume (H). Let (μt )t∈[0,∞) be the global sub-probabilitysolution of (1.2)–(1.3) given in Proposition 2.1. Then, for any φ ∈ Cc (U ), the function t → U φdμt is continuous on [0, ∞). If, in addition, μt (U ) = 1 for allt > 0, then for any φ ∈ C(U ) that is constant outside some compact set, the function t → U φdμt is continuous on [0, ∞). Proof Fix φ ∈ Cc (U ) and t0 ∈ [0, ∞). Let ∗ be an open bounded set such that supp(φ) ⊂⊂ ∗ . By smooth approximation and the continuity of φ, we can find a sequence (φn )n≥1 ⊂ C0∞ (U ) with supp(φn ) ⊂ ∗ for all n such that φn → φ as n → ∞ locally uniformly on U . In particular, φn → φ as n → ∞ uniformly on ∗ . It then follows that for any n ≥ 1,        φdμ − φdμ t t 0  U U                     ≤ φdμt − φn dμt  +  φn dμt − φn dμt0  +  φn dμt0 − φdμt0  U U U U U  U    φn dμt − φn dμt0 . ≤ 2 sup |φ − φn | +  ∗ U U The result now follows by first setting t → t0 and then n → ∞. 123 ⊔ ⊓ Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1597 2.2 Existence of Global Probability Solutions In this subsection, we prove that the global sub-probability solution (μt )t∈[0,∞) of (1.2)–(1.3) constructed in Proposition 2.1 is actually a global probability solution in the presence of an unbounded Lyapunov-like function. The following result generalizes [3, Theorem 3.1] and [24, Theorem 2.7] in the case of time-independent coefficients. Proposition 2.2 Suppose (H) and that there exists an unbounded Lyapunov-like function associated to L. Then, the global sub-probability solution (μt )t∈[0,∞) of (1.2)–(1.3) given in Proposition 2.1 is a global probability solution of (1.2)–(1.3). Proof Let U ∈ C 2 (U ) be the unbounded Lyapunov-like function associated to L as in the statement and C1 , C2 , > 0, ρm > 0 be as in (1.6) for the present U . We may assume, without loss of generality, that U ∈ L 1 (U , ν). In fact, arguing exactly the same as in the proof of [3, Corollary 2.3], an C 2 function θ (depending on ν) satisfying θ (0) = 0, θ (∞) = ∞, 0 ≤ θ ′ ≤ 1, θ ′′ ≤ 0 and θ ◦ U ∈ L 1 (ν). (2.1) can be found. Then, for x ∈ U \ρm , L(θ ◦ U ) = θ ′ (U )LU + θ ′′ (U )a i j ∂i U ∂ j U ≤ C1 + C2 θ ′ (U )U ≤ C1 + C2 θ ◦ U , where we used a i j ∂i U ∂ j U ≥ 0 due to the positive-definiteness, and θ ′ (t)t ≤ θ (t) due to the concavity. Hence, U ∈ L 1 (U , ν) is assumed in the rest of the proof. To proceed, we fix ρ0 ∈ (ρm , ∞), and for ρ ≥ ρ0 + 1, we define ηρ ∈ C 2 [0, ∞) such that it is non-decreasing and satisfies ⎧ ⎪ t ∈ [0, ρm ], ⎨0, ηρ (t) = t, t ∈ [ρ0 , ρ], ⎪ ⎩ ρ + 1, t ∈ [ρ + 2, ∞). We can further fix the shape of ηρ (t) for t ∈ [ρm , ρ0 ] ∪ [ρ, ρ + 2] so that C := sup sup max{ηρ′ (t), |ηρ′′ (t)|} < ∞, ηρ′′ (t) ρ≥ρ0 +1 t≥0 (2.2) ≤ 0 for all t ∈ [ρ, ρ + 2] and ρ ≥ ρ0 + 1. Set η(t) = limρ→∞ ηρ (t). By Proposition 2.1, there holds    t φdμt = φdμs + Lφdμτ dτ, 0 < s < t < ∞ U U s U C0∞ (U ). By assumptions on the coefficients of C02 (U ). In particular, since ηρ (U ) − (ρ + 1) ∈ for any φ ∈ L, the above equality also holds for any φ ∈ C02 (U ), we have   t  [ηρ (U ) − (ρ + 1)]dμt = [ηρ (U ) − (ρ + 1)]dμs + L(ηρ (U ))dμτ dτ. U U s U 123 1598 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 Denote ρ as ρ-sub-level set of U for each ρ > 0. To treat the second term on the right-hand side of the above equality, we write  t L(ηρ (U ))dμτ dτ s U  t  t ηρ′′ (U )a i j ∂i U ∂ j U dμτ dτ ηρ′ (U )LU dμτ dτ + = s s U U  t  t (2.3) ′ = ηρ (U )LU dμτ dτ + ηρ′′ (U )a i j ∂i U ∂ j U dμτ dτ ρ+2 \ρm s +  t s ρ+2 \ρ ρ0 \ρm s ηρ′′ (U )a i j ∂i U ∂ j U dμτ dτ. For the first term on the right-hand side of the second equality in (2.3), we find from (1.6) and the fact μt (U ) ≤ 1 that  t  t ′ ηρ (U )LU dμτ dτ ≤ C1 (t − s) + ηρ′ (U )U dμτ dτ s ρ+2 \ρm ≤ C1 (t − s) + Cρ0 (t − s) + = C1 (t − s) + +  t s ρ+2 \ρm s ρ \ρ0  t s s ηρ (U )dμτ dτ ρ+2 \ρ0 ηρ′ (U )U dμτ dτ ρ0 \ρm U dμτ dτ +  t  t s ≤ C1 (t − s) + Cρ0 (t − s) + = C1 (t − s) + Cρ0 (t − s) + ρ+2 \ρ  t s ρ \ρ0  t s ηρ′ (U )U dμτ dτ U dμτ dτ +  t s ηρ (U )dμτ dτ ρ+2 \ρ ηρ (U )dμτ dτ, ρ+2 \ρ0 where C > 0 is as in (2.2). In the above estimates, we used the facts that a i j ∂i U ∂ j U ≥ 0 and ηρ′′ (t) ≤ 0 for t ∈ (ρ, ρ + 2), ηρ′ (t)t ≤ η(t) for t ∈ (ρ, ρ + 1), which are simple consequences of the construction of ηρ . Setting C3 := C1 + Cρ0 + C sup a i j ∂i U ∂ j U , ρ0 \ρm we arrive at  t s U L(ηρ (U ))dμτ dτ ≤ C3 (t − s) +  t s ηρ (U )dμτ dτ, U and hence,   t  [ηρ (U ) − (ρ + 1)]dμt ≤ [ηρ (U ) − (ρ + 1)]dμs + C3 (t − s) + ηρ (U )dμτ dτ. U U s U Since ηρ (U ) − (ρ + 1) ∈ C02 (U ), Lemma 2.1 ensures that   [ηρ (U ) − (ρ + 1)]dμs = [ηρ (U ) − (ρ + 1)]dν. lim s→0+ U 123 U (2.4) Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1599 Therefore, passing to the limit s → 0+ in (2.4) yields    t [ηρ (U ) − (ρ + 1)]dμt ≤ [ηρ (U ) − (ρ + 1)]dν + C3 t + ηρ (U )dμτ dτ, U 0 U U which leads to    t ηρ (U )dμt + (ρ + 1)[1 − μt (U )] ≤ ηρ (U )dμτ dτ. (2.5) ηρ (U )dν + C3 t + U 0 U U Since μt (U ) ≤ 1, Grönwall’s inequality gives   ηρ (U )dμt ≤ ηρ (U )dν + C3 t et . U Setting ρ → ∞, we find U  U η(U )dμt ≤  U η(U )dν + C3 t et . (2.6) Hence, for fixed t > 0, the right-hand side of (2.5) is bounded uniformly in ρ ≥ ρ0 + 1, and therefore, if μt (U ) < 1, we set ρ → ∞ in (2.5) to deduce a contradiction. Thus, μt (U ) = 1 for all t > 0. Remark 2.1 Note that we obtained the estimate (2.6) in the proof of Proposition 2.2. This in particular implies that under the assumption of Proposition 2.2, if U is an unbounded Lyapunov-like function associated to L and the initial condition ν satisfies U ∈ L 1 (U , ν), then the global probability  solution (μt )t∈[0,∞) of (1.2)–(1.3) given in Proposition 2.1 and Proposition 2.2 satisfies U U dμt < ∞ for all t > 0. 2.3 Uniqueness of Global Probability Solutions In this subsection, we prove the uniqueness of global probability solutions of the Cauchy problem (1.2)–(1.3) in the class of global sub-probability solutions. Proposition 2.3 Suppose (H) and that there is an unbounded Lyapunov-like function associated to L. Let (μt )t∈[0,∞) be the global probability solution of (1.2)–(1.3) given in Propositions 2.1 and 2.2. If (μ̃)t∈[0,∞) is a global sub-probability solution of (1.2)–(1.3), then μ̃t = μt for all t > 0. Proof We adapt the proof of [24, Theorem 3.5]. Let U be the unbounded Lyapunov-like function associated to L and C1 , C2 , ρm be as in (1.6) for the present U . Let u(t, x) and ũ(t,x) ũ(t, x) be the densities of (μt )t∈[0,∞) and (μ̃t )t∈[0,∞) , respectively, and set v(t, x) = u(t,x) . By the regularity theorem established in [5], v(t, x) is continuous and positive. It is proven in [26, Lemma 2.2] (see also [24, Lemma 3.4]) that for any φ ∈ C02 (U ) with φ ≥ 0 and λ > 0, there holds  eλ(1−v(t,x)) − eλ φ(x)dμt U (2.7)   t eλ(1−v(s,x)) − eλ Lφ(x)dμs ds, t > 0. ≤ (1 − eλ ) φdν + U 0 U Fix ρ0 ∈ (ρm , ∞). Let η ∈ C 2 ([0, ∞)) be a non-decreasing function satisfying  0, t ∈ [0, ρm ], η(t) = t, t ∈ [ρ0 , ∞). 123 1600 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 Let ζ ∈ C 2 ([0, ∞)) be a non-increasing and convex function satisfying ζ (0) = 1 and ζ (t) = 0 for t ≥ 1. For N ≫ 1, an application of (2.7) with φ = ζ (N −1 η(U )) yields  eλ(1−v(t,x)) − eλ ζ (N −1 η(U (x)))dμt U   t ≤ (1 − eλ ) [eλ(1−v(s,x)) − eλ ]L(ζ (N −1 η(U )))dμs ds , ζ (N −1 η(U ))dν + 0 U U    (I ) (2.8) where L(ζ (N −1 η(U )) is given by L(ζ (N −1 η(U ))) = ζ ′ (N −1 η(U ))N −1 η′ (U )LU + ζ ′′ (N −1 η(U ))[N −1 η′ (U )]2 a i j ∂i U ∂ j U + ζ ′ (N −1 η(U ))N −1 η′′ (U )a i j ∂i U ∂ j U . It follows from the dominated convergence theorem and the fact ζ (N −1 η(U (x))) → 1 pointwise as N → ∞ that   eλ(1−v(t,x)) − eλ dμt eλ(1−v(t,x)) − eλ ζ (N −1 η(U (x)))dμt → U U and (1 − eλ )  U ζ (N −1 η(U ))dν → 1 − eλ , as N → ∞. To treat the second term on the right hand side of (2.8), we write (I ) = (I 1) + (I 2) + (I 3), where  t eλ(1−v(s,x)) − eλ ζ ′ (N −1 η(U ))N −1 η′ (U )LU dμs ds, (I 1) = 0 U  t eλ(1−v(s,x)) − eλ ζ ′′ (N −1 η(U ))[N −1 η′ (U )]2 a i j ∂i U ∂ j U dμs ds, (I 2) = 0 U  t eλ(1−v(s,x)) − eλ ζ ′ (N −1 η(U ))N −1 η′′ (U )a i j ∂i U ∂ j U dμs ds. (I 3) = 0 U As [eλ(1−v(s,x)) − eλ ]ζ ′ (N −1 η(U )) ≥ 0, we find   sup η′ t (I 1) ≤ eλ(1−v(s,x)) − eλ ζ ′ (N −1 η(U ))(C1 + C2 U (x))dμs ds N 0 U \ρm   C sup η′ t (C1 + C2 U (x))dμs ds, ≤ N 0 {η(U )≤N } where     C = sup  eλ(1−v(s,x)) − eλ ζ ′ (N −1 η(U )) . [0,t]×U Since 123   1   χ{η(U )≤N } (C1 + C2 U (x)) ≤ C1 + C2  N Journal of Dynamics and Differential Equations (2019) 31:1591–1615 and 1601 1 χ{η(U )≤N } (C1 + C2 U (x)) → 0 as N → ∞ pointwise, N the dominated convergence theorem yields lim sup N →∞ (I 1) ≤ 0. It is clear that (I 2) ≤ 0. For (I 3), the fact η′′ (t) = 0 for t ∈ [0, ρm ] ∪ [ρ0 , ∞) implies that  t (I 3) = eλ(1−v(s,x)) − eλ ζ ′ (N −1 η(U ))N −1 η′′ (U )a i j ∂i U ∂ j U dμs ds. 0 ρ 0 \ρm The dominated convergence theorem then yields lim N →∞ (I 3) = 0. Hence, we have shown lim sup N →∞ (I ) ≤ 0, which then implies that  eλ(1−v(t,x)) − eλ dμt ≤ 1 − eλ , U i.e.,  U eλ(1−v(t,x)) dμt ≤ 1, ∀λ > 0. If there is a Boreal set B in U with positive Lebesgue measure and δ > 0 such that v(t, x) ≤ 1 − δ for x ∈ B, then  eλ(1−v(t,x)) dμt ≤ 1, eλδ |B| ≤ B which leads to a contradiction by setting λ → ∞. Therefore, v(t, x) ≥ 1 a.e.. But, since   ũ(t, x)d x ≤ 1 = u(t, x)d x, we conclude v(t, x) = 1 a.e., and hence, v(t, x) = 1 by U U continuity. ⊔ ⊓ 3 Convergence to Equilibrium In this section, we study the convergence of global probability solutions of (1.2)–(1.3) and prove, in particular, Theorem B. We first recall the following result concerning the existence and uniqueness of stationary measures. Proposition 3.1 Suppose (H) with p > n + 2 replaced by p > n and that there exists an unbounded Lyapunov function associated to L. Then, (1.2) admits a unique stationary 1, p measure μ∗ with density function lying in Wloc (U ). The existence part in Proposition 3.1 is proven in [20] even without the unboundedness assumption of a Lyapunov function (see [19,20] for definition of a general Lyapunov function). The uniqueness part in Proposition 3.1 follows from [12, Example 5.1] and [20, Theorem A] in which it is actually shown that the unique stationary measure μ∗ ∈ Mmd , where Mmd is defined in (A.1). Hence, by Proposition A.2, the semigroup (Tt )t≥0 given in Proposition A.1 (1) is a Markov semigroup and the only C0 -semigroup in L 1 (U , μ∗ ) whose generator (L, D(L)) extending (L, C0∞ (U )), and moreover, μ∗ is the unique invariant measure of (Tt )t≥0 . If (H) is assumed, then (Tt )t≥0 is joint continuous (see Proposition A.1 (2)), and hence, 123 1602 Tt φ(x) = Journal of Dynamics and Differential Equations (2019) 31:1591–1615  U φ(y) p(t, x, y)dy =  U φ(y) pt (x, y)dμ∗ (y), φ ∈ L 1 (U , μ∗ ), t > 0, (3.1) where p(t, x, y) and pt (x, y) satisfy properties described in Remark A.1 (3). In what follows in this section, we assume (H) and the existence of an unbounded Lyapunov function associated to L, and let (μt )t∈[0,∞) be the unique global probability solution of (1.2)–(1.3) obtained in Theorem A (or, in Propositions 2.1, 2.2 and 2.3) for a given Borel probability measure ν on U . We use B(U ) to denote the Borel σ -algebra of U . 3.1 Strong Convergence to Equilibrium To study the convergence of the global probability solution (μt )t∈[0,∞) as t → ∞, we imbed it into the dual semigroup of (Tt )t≥0 . To do so, we note that for each t > 0, φ → U Tt φdν defines a bounded linear functional on Cb (U ), and therefore, by Riesz representation theorem, for any t > 0, there exists a unique Borel probability measure, denoted by Tt∗ ν, such that     ∗ φdTt∗ ν = Tt φdν, ∀φ ∈ Cb (U ). Tt ν, φ := U U The next two lemmas summarize some properties of Tt∗ ν. Lemma 3.1 Tt∗ ν strongly converges to μ∗ as t → ∞, i.e., for any B ∈ B(U ), there holds Tt∗ ν(B) → μ∗ (B) as t → ∞. Proof Note that Tt∗ ν(B) = Tt∗ ν, χ B  =  U Tt χ B dν, t ≥ 0. By Remark A.1 (2), for any B ∈ B(U ), there holds lim Tt χ B (x) = μ∗ (B), ∀x ∈ U . t→∞ It follows that Tt∗ ν(B) =  U Tt χ B dν → μ∗ (B) as t → ∞, i.e., Tt∗ ν converges to μ∗ strongly as t → ∞. ⊔ ⊓ Lemma 3.2 For any φ ∈ C0∞ (U ), the function t → Tt∗ ν, φ is continuous on [0, ∞), and  t  ∗      ∗ Tt ν, φ = Ts∗ ν, φ + Tτ ν, Lφ dτ (3.2) s for all 0 < s < t < ∞. Proof The continuity of t → Tt∗ ν, φ at t  = 0 is obvious. The continuity at t = 0 follows from Remark A.1 (4). More precisely, since Tt φ(x) → φ(x) as t → 0+ locally uniformly in x ∈ U , the dominated convergence theorem then yields   ∗    T ν, φ − ν, φ ≤ |Tt φ − φ|dν → 0 as t → 0+ . t U 123 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1603 The identity (3.2) follows from Fubini’s theorem, the fact Lφ = Lφ and the formula  t Tt φ − Ts φ = Tτ Lφdτ, s ⊔ ⊓ which is a general fact of strongly continuous semigroups. The following result implies Theorem B (1). Theorem 3.1 Suppose (H) and that there exists an unbounded Lyapunov function associated to L. Then, μt strongly converges to μ∗ as t → ∞. Proof We note from Lemma 3.2 that Tt∗ ν is a global probability solution of (1.2)–(1.3) with additional continuity properties. Hence, by Proposition 2.3, μt = Tt∗ ν for all t > 0. The theorem now follows from Lemma 3.1. ⊔ ⊓ 3.2 Exponential Convergence to Equilibrium We now study the convergence of (μt )t∈[0,∞) to μ∗ under stronger conditions. We start with the following result giving convergence results of the Markov semigroup (Tt )t≥0 under appropriate assumptions. Lemma 3.3 Suppose (H) and that there exists an unbounded Lyapunov function U associated to L. Also, suppose that there exist constants t0 , c, b > 0 and κ ∈ (0, 1) such that for any t ∈ (0, t0 ), and Tt U (x) ≤ c(1 + U (x)), x ∈ U (3.3) Tt0 U (x) ≤ κU (x) + b, x ∈ U . (3.4) Then, there exist constants C, r > 0 such that Tt φ − μ∗ , φ U ≤ Ce−r t φ − μ∗ , φ U, t ≥ t0 for every measurable function φ : U → R with φ U < ∞, where μ∗ , φ =      φ   φ      = . φ U :=  1 + U  L ∞ (U ,μ∗ )  1 + U  L ∞ (U ,d x)  U φdμ∗ and Proof We apply a version of Harris’s theorem stated in [17, Theorem 3.6] to Ttn0 = Tnt0 . To do so, we need to verify two conditions. The first condition involved in that theorem is just (3.4). The second condition involved is that for every R > 0, there exists 0 < α = α(R) < 1 such that sup Tt0 (x, ·) − Tt0 (y, ·) T V ≤ 2(1 − α), (3.5) where the supremum is taken over all x, y ∈ U such that U (x) + U (y) ≤ R. To verify this condition, we fix R > 0 and set U R = {(x, y) ∈ U × U : U (x) + U (y) ≤ R} ⊂  R ×  R , where  R = {x ∈ U : U (x) < R} is the R-sub-level set of U . Note that Tt0 (x, ·) − Tt0 (y, ·) TV = 2 sup |Tt0 (x, B) − Tt0 (y, B)| B∈B(U )      = 2 sup  [ p(t0 , x, z) − p(t0 , y, z)]dz  . B∈B(U ) B 123 1604 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 We claim that Tt0 (x, ·) − Tt0 (y, ·) TV < 2, ∀x, y ∈ U . (3.6) Obviously, Tt0 (x, ·) − Tt0 (y, ·) T V ≤ 2 for any x, y ∈ U . If there are x, y ∈ U such that Tt0 (x, ·) − Tt0 (y, ·) T V = 2, then there exists a sequence (Bn )n ⊂ B(U ) such that either Tt0 (x, Bn ) → 1 and Tt0 (y, Bn ) → 0 as n → ∞, or Tt0 (x, Bn ) → 0 and Tt0 (y, Bn ) → 1 as n → ∞. Suppose, without loss of generality, that the former is true. Let us fix some compact set B0 ⊂ U such that Tt0 (x, B0 ) = 21 . Since p(t0 , y, z) is positive and continuous, c y := minz∈B0 p(t0 , y, z) > 0. Since maxz∈B0 p(t0 , x, z) < ∞, the Lebesgue measure of B0 ∩ Bn , denoted by |B0 ∩ Bn |, does not go to 0 as n → ∞, for otherwise, Tt0 (x, Bn ) can not converge to 1 as n → ∞. It then follows that Tt0 (y, Bn ) ≥ Tt0 (y, B0 ∩ Bn ) ≥ c y |B0 ∩ Bn | ≥ c y ǫ0 , n ≫ 1, where ǫ0 > 0 is such that |B0 ∩ Bn | ≥ ǫ0 for all n ≫ 1. This leads to a contradiction. Thus (3.6) is true. Denote  | p(t0 , x1 , z) − p(t0 , x2 , z)| dz, x1 , x2 ∈ U . h(x1 , x2 ) = 2 sup B∈B(U ) B Then for any x, x ′ , y, y ′ ∈  R , Tt0 (x, ·) − Tt0 (y, ·) T V    ′  = 2 sup  [ p(t0 , x, z) − p(t0 , x , z)]dz + [ p(t0 , x ′ , z) − p(t0 , y ′ , z)]dz B∈B(U ) B B   ′ + [ p(t0 , y , z) − p(t0 , y, z)]dz  B   ≤ 2 sup | p(t0 , x, z) − p(t0 , x ′ , z)|dz + 2 sup | p(t0 , y ′ , z) − p(t0 , y, z)|dz  + B∈B(U ) B Tt0 (x ′ , ·) − ′ B∈B(U ) B ′ Tt0 (y , ·) TV ′ ≤ h(x, x ) + h(y, y ) + Tt0 (x ′ , ·) − Tt0 (y ′ , ·) TV. By symmetry, we find that   Tt (x, ·) − Tt (y, ·) T V − Tt (x ′ , ·) − Tt (y ′ , ·) 0 0 0 0 TV   ≤ h(x, x ′ ) + h(y, y ′ ). (3.7) We note from Remark A.1 (3) that there is ct0 ,R > 0 such that h(x, x ′ ) ≤ ct0 ,R |x − x ′ |α and h(y, y ′ ) ≤ ct0 ,R |y − y ′ |α . This, together with (3.7) implies the continuity of the map (x, y) → Tt0 (x, ·)− Tt0 (y, ·) T V on  R ×  R . With this continuity, the condition (3.5) now follows from the claim (3.6) and the compactness of  R . It now follows from Harris’s theorem that (Tnt0 )n≥1 admits a unique invariant measure, which must be μ∗ , and there are C1 > 0 and r1 ∈ (0, 1) such that Tnt0 φ − μ∗ , φ 123 U ≤ C1 r1n φ − μ∗ , φ U, n≥1 (3.8) Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1605 holds for every measurable function φ : U → R with φ U < ∞. For any t ≥ t0 , write t = [t] +rt , where [t] is the largest number of the form nt0 not greater than t, and tr ∈ [0, t0 ). Then by (3.8) and (3.3), Tt φ − μ∗ , φ U = Tt (φ − μ∗ , φ) U    Trt T[t] (φ − μ∗ , φ)    =  ∞ 1+U L    Trt (1 + U )   ≤  1 + U  ∞ T[t] (φ − μ∗ , φ) L [t] t ≤ (c + 1)C1 r1 0 φ − μ∗ , φ U U. ⊔ ⊓ This completes the proof. Next, we provide sufficient conditions for the assumptions in Lemma 3.3. Lemma 3.4 Suppose (H) and that there exists an unbounded strong Lyapunov function U associated to L. Then, the following statements hold. (1) U ∈ L 1 (U , μ∗ ). (2) If ν satisfies U ∈ L 1 (U , ν), then conditions of Lemma 3.3 hold. Proof Let C1 , C2 , ρm > 0 be as in (1.8) for the unbounded strong Lyapunov function U as in the statement. (1) Let ν̃ be an Borel probability measure on U such that U ∈ L 1 (U , ν̃). Let (μ̃)t∈[0,∞) be the global probability solution of (1.2)–(1.3) with ν = ν̃. We claim that  sup U d μ̃t < ∞. (3.9) t>0 U Let us assume (3.9) for the moment and show that U ∈ L 1 (U , μ∗ ). Clearly, we can find a sequence of non-negative functions {Un }n ⊂ Cb (U ) satisfying Un ≤ Un+1 and Un (x) → U (x) locally uniformly in x ∈ U as n → ∞. By the monotone convergence theorem,   U dμ∗ = lim Un dμ∗ . U n→∞ U The strong convergence of μ̃t to μ∗ as t → ∞ given in Theorem 3.1 implies that   lim Un d μ̃t = Un dμ∗ , ∀n. t→∞ U U  It then follows from (3.9) and the monotonicity of the sequence {Un }n that supn U Un dμ∗ < ∞, which leads to the result. To finish the proof, it remains to show (3.9). To do so, we fix ρ0 ∈ (ρm , ∞). For ρ ≥ ρ0 +1, we define ηρ ∈ C 2 [0, ∞) such that it is non-decreasing and satisfies ⎧ ⎪ t ∈ [0, ρm ], ⎨0, ηρ (t) = t, t ∈ [ρ0 , ρ], ⎪ ⎩ ρ + 1, t ∈ [ρ + 2, ∞). 123 1606 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 Moreover, the shape of ηρ (t) for t ∈ [ρm , ρ0 ] ∪ [ρ, ρ + 2] can be fixed so that C3 := sup sup max{ηρ′ (t), |ηρ′′ (t)|} < ∞, ηρ′′ (t) ρ≥ρ0 +1 t≥0 (3.10) ≤ 0 for all t ∈ [ρ, ρ + 2] and ρ ≥ ρ0 + 1. We set η(t) := lim  ρ→∞ ηρ (t). We estimate U ηρ (U )d μ̃t . By Proposition 2.1(iii), the equalities  U φd μ̃t =  U  t φd μ̃s + s U Lφd μ̃τ dτ, 0 < s < t < ∞ hold for any φ ∈ C0∞ (U ), which, by the assumption (H), actually hold for any φ ∈ C02 (U ). Since μ̃t (U ) = 1 = μ̃s (U ), they are also true for any φ +c, where φ ∈ C02 (U ) and c ∈ R. That is, they are true for any C 2 function that is constant outside some compact set. In particular, they can be applied to the function ηρ (U ) for any ρ ≥ ρ0 + 1. Hence, we find     d ηρ (U )d μ̃t = L(ηρ (U ))d μ̃t = ηρ′ (U )LU d μ̃t + ηρ′′ (U )a i j ∂i U ∂ j U d μ̃t . dt U U U U (3.11) For the first term on the right-hand side of (3.11), we have   ηρ′ (U )LU d μ̃t = ηρ′ (U )LU d μ̃t ρ+1 \ρm U ≤ C1  ρ+2 \ρm ≤ C1 C3 − C2 = C1 C3 − C2  ηρ′ (U )d μ̃t − C2   ρ+2 \ρm ηρ′ (U )U d μ̃t U d μ̃t ρ \ρ0  U ηρ (U )d μ̃t + C2  ρ0 ηρ (U )d μ̃t + C2 ηρ (U )d μ̃t U \ρ ≤ C4 − C2  U ηρ (U )d μ̃t + C2 (ρ + 1)μ̃t (U \ρ ), where C4 = C1 C3 + C2 ρ0 . For the second term on the right-hand side of (3.11), we deduce from the positivedefiniteness of (a i j ) and the negativity of ηρ′′ (t) on [ρ, ρ + 2] that   ηρ′′ (U )a i j ∂i U ∂ j U d μ̃t ηρ′′ (U )a i j ∂i U ∂ j U d μ̃t = ρ0 \ρm U + ≤  ρ+2 \ρ  ρ0 \ρm ≤ C3 123 ηρ′′ (U )a i j ∂i U ∂ j U d μ̃t ηρ′′ (U )a i j ∂i U ∂ j U d μ̃t sup ρ0 \ρm a i j ∂i U ∂ j U . Journal of Dynamics and Differential Equations (2019) 31:1591–1615 With C5 := C4 + C3 supρ d dt  U 0 \ρm ηρ (U )d μ̃t ≤ −C2 1607 a i j ∂i U ∂ j U , the equality (3.11) gives  U ηρ (U )d μ̃t + C5 + C2 (ρ + 1)μ̃t (U \ρ ), which implies that for any t > t0  U ηρ (U )d μ̃t ≤ e−C2 (t−t0 ) + ≤   U t t0  U ηρ (U )d μ̃t0 +  t e−C2 (t−s) C5 ds t0 e−C2 (t−s) C2 (ρ + 1)μ̃s (U \ρ )ds ηρ (U )d μ̃t0 + C5 + C2  t t0 (3.12) e−C2 (t−s) C2 (ρ + 1)μ̃s (U \ρ )ds.   Since U ηρ (U )d μ̃t0 → U ηρ (U )d ν̃ as t0 → 0+ by Lemma 2.1, we pass to the limit t0 → 0+ in (3.12) to obtain  U ηρ (U )d μ̃t ≤  U ηρ (U )d ν̃ + By Remark 2.1, we have  U  U C5 + C2  t 0 e−C2 (t−s) C2 (ρ + 1)μ̃s (U \ρ )ds. (3.13) U d μ̃s < ∞. Since min{U , ρ}d μ̃s =  ρ U d μ̃s + ρ μ̃s (U \ρ ),    and as ρ → ∞, both U min{U , ρ}d μ̃s and ρ U d μ̃s converge to U U d μ̃s due to the monotone convergence theorem, we find ρ μ̃s (U \ρ ) → 0 as ρ → ∞, which together with the dominated convergence theorem imply that for any fixed t > 0,  t 0 e−C2 (t−s) C2 (ρ + 1)μ̃s (U \ρ )ds → 0 as ρ → ∞. Hence, for any fixed t > 0, we pass to the limit ρ → ∞ in (3.13) to find  U η(U )μ̃t ≤  U η(U )d ν̃ + C5 , C2 which leads to (3.9). (2) Note that L = L on C02 (U ) and the constant function 1 belongs to D(L) with L1 = 0. Hence, Lφ = Lφ for any C 2 function that is constant outside some compact set. Let ηρ be as in the proof of (1). Then ηρ (U ) is a C 2 function that equals ρ + 1 on U \ρ+2 , and therefore, L(ηρ (U )) = L(ηρ (U )). It follows that d Tt ηρ (U ) = Tt Lηρ (U ) dt = Tt Lηρ (U ) = Tt [ηρ′ (U )LU + ηρ′′ (U )a i j ∂i U ∂ j U ]. 123 1608 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 Clearly, ηρ′′ (U )a i j ∂i U ∂ j U ≤ C3 supρ \ρ a i j ∂i U ∂ j U , where C3 > 0 is as in (3.10). m 0 Since −C2 ηρ′ (U )U + C2 ηρ (U ) = −C2 ηρ′ (U )U [χρ + C2 ηρ (U )[χρ ≤ C2 ηρ (U )[χρ 0 \ρm 0 \ρm 0 \ρm + χU \ρ ] − C2 U χρ \ρ + χU \ρ ] + C2 U χρ \ρ 0 0 + χU \ρ ] ≤ C2 ρ0 + C2 (ρ + 1)χU \ρ , we have ηρ′ (U )LU ≤ C1 ηρ′ (U ) − C2 ηρ′ (U )U ≤ C1 C − C2 ηρ′ (U )U + C2 ηρ (U ) − C2 ηρ (U ) ≤ C1 C + C2 ρ0 + C2 (ρ + 1)χU \ρ − C2 ηρ (U ). Thus, setting C6 := C3 we find sup ρ0 \ρm a i j ∂i U ∂ j U + C1 C3 + C2 ρ0 , d Tt ηρ (U ) ≤ C6 + C2 Tt [(ρ + 1)χU \ρ ] − C2 Tt ηρ (U ), dt which leads to  t e−C2 (t−s) Ts [(ρ + 1)χU \ρ ]ds e−C2 (t−s) C6 ds + C2 0 0  t C6 −C2 t ηρ (U ) + e−C2 (t−s) Ts [(ρ + 1)χU \ρ ]ds. + C2 ≤e C2 0 (3.14) Multiplying the above inequality by a function φ ∈ C0∞ (U ) with φ ≥ 0, and then integrating the resulting one over U with respect to μ∗ yields    C6 Tt ηρ (U )φdμ∗ ≤ e−C2 t ηρ (U )φdμ∗ + φdμ∗ C2 U U U   (3.15)  t  −C2 (t−s) e Ts [(ρ + 1)χU \ρ ]φdμ∗ ds. + C2 Tt ηρ (U ) ≤ e−C2 t ηρ (U ) +  t 0 Since U ∈ L 1 (U , μ∗ ), L 1 (U , μ∗ ) U by (1), we have (ρ + 1)μ∗ (U \ρ ) < ∞, i.e., (ρ + 1)χU \ρ ∈ and hence (ρ + 1)μ∗ (U \ρ ) → 0 as ρ → ∞. Using the invariance of μ∗ with respect to (Tt )t≥0 , we have    0≤ Ts [(ρ + 1)χU \ρ ]φdμ∗ ≤ sup φ Ts [(ρ + 1)χU \ρ ]dμ∗ U U U   (ρ + 1)χU \ρ dμ∗ = sup φ U U   = sup φ (ρ + 1)μ∗ (U \ρ ). U It follows that  U 123 Ts [(ρ + 1)χU \ρ ]φdμ∗ → 0 as ρ → ∞ uniform for s > 0. Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1609 Setting ρ → ∞ in (3.15), with η(t) = limρ→∞ ηρ (t), we find    C6 Tt η(U )φdμ∗ ≤ e−C2 t η(U )φdμ∗ + φdμ∗ , C2 U U U which then implies Tt η(U ) ≤ e−C2 t η(U ) + C6 . C2 Since η(U ) is also an unbounded strong Lyapunov function, we can simply take a t0 > 0 so that e−C2 t0 < 1, for which conditions of Lemma 3.3 are satisfied. ⊔ ⊓ Remark 3.1 If we would use (3.1) to treat the term Ts [(ρ + 1)χU \ρ ] on the last line of the inequality (3.14), some singularity would occur because p(s, x, y) is singular at s = 0. This is why we put the inequality (3.14) into the space L 1 (U , μ∗ ) in the proof of the lemma. Finally, we prove the following theorem which implies Theorem B (2). Theorem 3.2 Suppose (H) and that there exists an unbounded strong Lyapunov function U associated to L. If ν satisfies U ∈ L 1 (U , ν), then there exist t0 > 0 and constants C, r > 0 such that the followings hold for all t ≥ t0 : μt − μ∗ T V = Tt∗ ν − μ∗ T V ≤ Ce−r t ,       −r t  ψ Tt φdμ∗ −  ψdμ φdμ (φ − μ∗ , φ)2 ∗ ∗  ≤ Ce  U U U for any pair of measurable functions φ and ψ with φ 2 U (3.16) 1/2 U < ∞ and ψ 2 ψ2 U 1/2 U , (3.17) < ∞. Proof By Lemma 3.4 (2), we can apply Lemma 3.3 to deduce that Tt∗ ν − μ∗ TV = sup | Tt∗ ν, φ − μ∗ , φ| φ:|φ|≤1 = sup | ν, Tt φ − μ∗ , φ| φ:|φ|≤1 ≤ sup  φ:|φ|≤1 U ≤  U |Tt φ − μ∗ , φ| (1 + U )dν 1+U (1 + U )dν sup ≤ Ce−r t φ:|φ|≤1  U Tt φ − μ∗ , φ (1 + U )dν sup φ:|φ|≤1 U φ − μ∗ , φ U. This proves (3.16). √ Since p(t, x, y)dy is a probability measure and a → a is a concave function, Jensen’s inequality yields   √  √ Tt U (x) = U (y) p(t, x, y)dy = Tt U (x). U (y) p(t, x, y)dy ≤ U U By Lemma 3.4 (2), conditions of Lemma 3.3 hold. It follows that there are constants c̃ > 0, κ̃ ∈ (0, 1) and b̃ > 0 such that √ √ Tt U (x) ≤ c̃(1 + U (x)), x ∈ U 123 1610 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 for any t ∈ (0, t0 ), and √ √ Tt0 U (x) ≤ κ̃ U (x) + b̃, x ∈ U . Applying Lemma 3.3 concludes that there exist constants C̃, r̃ > 0 such that Tt φ − μ∗ , φ √ U ≤ C̃e−r̃ t φ − μ∗ , φ √ , U √ U holds for every measurable function φ : U → R with φ φ 2 U < ∞. Clearly, (3.18) implies that √ |Tt φ(x) − μ∗ , φ| ≤ C̃e−r̃ t (1 + U (x)) φ − μ∗ , φ It then follows that         ψ T φdμ − ψdμ φdμ t ∗ ∗ ∗   U U U  ≤ |ψ||Tt φ − μ∗ , φ|dμ∗ U  ≤ C̃e ≤ √ −r̃ t φ − μ∗ , φ √ U 2C̃e−r̃ t φ − μ∗ , φ U √ U |ψ|2 dμ∗ ψ2 1/2 U   U U t ≥ t0 < ∞, or equivalently, √ , U (1 + (3.18) x ∈ U , t ≥ t0 . √ U )2 dμ∗ (1 + U )dμ∗ . ⊔ ⊓ This proves (3.17). 4 Examples In this section, we apply Theorems A and B to some examples. Example 4.1 Consider the following Itô stochastic differential equation  d x = bxdt + 2σ (x 2 + 1)dW , x ∈ R, where σ > b > 0. The corresponding Fokker–Planck equation reads ∂t u = Lu := ∂x2x (σ (x 2 + 1)u) − ∂x (bxu), t > 0, x ∈ R, (4.1) and the adjoint Fokker–Planck operator is simply L = σ (x 2 + 1)∂x2x + bx∂x . Consider U (x) = ln(x 2 + 1). Since LU (x) = − 2(σ − b)x 2 − 2σ , x2 + 1 LU (x) ≤ −(σ − b) for |x| ≫ 1, i.e., U is an unbounded Lyapunov function. Hence, Proposition 3.1 ensures the existence of a unique stationary measure μ∗ of the Fokker–Planck equation (4.1). It follows from Theorem A that for any initial Borel probability measure, (4.1) has a unique global probability solution (μt )t∈[0,∞) . Moreover, Theorem B (1) asserts the strong convergence of μt to μ∗ as t → ∞. Example 4.2 Consider the Itô stochastic differential equation √ d x = b(x)dt + 2dW , x ∈ R, 123 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1611 whose corresponding Fokker–Planck equation is given by ∂t u = Lu := ∂x2x u − ∂x (b(x)u), t > 0, x ∈ R, where b : R → R is continuous and satisfies b(x) = − Fokker–Planck operator is simply L = ∂x2x + b(x)∂x . Consider a C 2 function U : R → [0, ∞) such that x 3 |x| 2 (4.2) for |x| ≫ 1. The adjoint 1 4 U (x) = e|x| , |x| ≫ 1. Simple calculation yields that LU (x) = e|x| 1 4 3 7 5 1 3 1 |x|− 2 − |x|− 4 − |x|− 4 , |x| ≫ 1, 16 16 4 which implies that LU (x) → −∞ as |x| → ∞. Hence, U is an unbounded Lyapunov function. It follows from Proposition 3.1 that (4.2) admits a unique stationary measure μ∗ , and from Theorem A that for any initial Borel probability measure, (4.2) admits a unique global probability solution (μt )t∈[0,∞) . Theorem B (1) then guarantees that μt converges strongly to μ∗ as t → ∞. Example 4.3 For given constant b < 0, consider the Itô stochastic differential equation √ d x = bxdt + 2(1 − x 2 )dW , x ∈ U = (−1, 1), whose corresponding Fokker–Planck equation is given by ∂t u = Lu := ∂x2x [(1 − x 2 )2 u] + ∂x (bxu), t > 0, x ∈ (−1, 1). (4.3) The adjoint Fokker–Planck operator reads L = (1 − x 2 )2 ∂x2x + bx∂x . Let U (x) = − ln(1 − x 2 ), x ∈ (−1, 1). Then U (x) → ∞ as |x| → 1− . Since LU (x) = 2 + 2x 2 + 2bx 2 , x ∈ (−1, 1), 1 − x2 we can find constants C1 , C2 , ρm > 0 such that LU (x) ≤ C1 − C2 U (x), x ∈ (−1, 1)\ρm , where ρm = {x ∈ (−1, 1) : U (x) < ρm }. Hence, U is an unbounded strong Lyapunov function of (4.3). Applying Proposition 3.1, Theorems A and B (1), we conclude that if ν is a Borel probability measure on (−1, 1), then the unique global probability solution (μt )t∈[0,∞) of (4.3) with initial condition ν converges strongly to the unique stationary measure of (4.3) as t → ∞. If, in addition, ν satisfies U ∈ L 1 ((−1, 1), ν), then Theorem B (2) ensures that (μt )t∈[0,∞) converges exponentially fast in the total variation distance to the unique stationary measure of (4.3) as t → ∞. Example 4.4 Consider the Itô stochastic differential equation  π  d x = tan − x + sign(x) dt + |1 − |x||α dW , x ∈ (−1, 1) 2 123 1612 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 studied in [16], where α > 0. The corresponding Fokker–Planck equation reads   π   1 ∂t u = Lu := ∂x2x (|1−|x||2α u)−∂x tan − x + sign(x) u , t > 0, x ∈ (−1, 1). 2 2 (4.4) Hence, the adjoint Fokker–Planck operator is given by  π  1 L = (|1 − |x||2α )∂x2x + tan − x + sign(x) ∂x . 2 2 Note that the drift coefficient is discontinuous at x = 0 and the diffusion coefficient is not Hölder continuous with exponent 21 if α < 21 . It was shown in [24, Example 3.10] that U (x) = function and satisfies U (x) → ∞ as |x| → 1− and LU (x) U (x) 2−x 2 , 1−x 2 x ∈ (−1, 1) is a positive C 2 → −∞ as |x| → 1. In particular, U is an unbounded strong Lyapunov function, and therefore, the assumptions in Proposition 3.1, Theorems A and B are satisfied. Hence, if ν is a Borel probability measure on (−1, 1), then the unique global probability solution (μt )t∈[0,∞) of (4.4) with initial condition ν converges  strongly to the unique stationary measure of (4.4) as t → ∞. If, in addition, ν satisfies (−1,1) U dν < ∞, then the convergence is exponentially fast in the total variation distance. Acknowledgements We would like to thank Professors Wen Huang and Zhenxin Liu for some preliminary discussions. Appendix A. Stationary Measures and Associated Markov Semigroups In this section, we recall some results obtained in [12] (also see [6,7]) concerning Markov semigroups associated to stationary measures. Let P be the set of Borel probability measures on U . Let M := {μ ∈ P : Lμ = 0 in the sense of Definition 1.3} be the set of stationary measures of (1.2). The following results describe the existence and some properties of sub-Markov semigroups associated to a given stationary measure of (1.2). Proposition A.1 Let μ ∈ M. (1) If (H) is assumed with p > n + 2 replaced by p > n, then there exists a closed extension (L, D(L)) of (L, C0∞ (U )) generating a sub-Markov contractive C0 -semigroup (Tt )t≥0 on L 1 (U , μ) such that μ is sub-invariant for (Tt )t≥0 , i.e.,   Tt φdμ ≤ φdμ, t ≥ 0 U L ∞ (U , μ) U for all φ ∈ with φ ≥ 0. (2) If (H) is assumed, then there exist unique sub-probability kernels K t (·, dy), t > 0, on U such that K t (x, dy) = p(t, x, y)dy, 123 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 1613 where p(t, x, y) is locally Hölder continuous and positive on (0, ∞) × U × U , and for each φ ∈ L 1 (U , μ), the function  x → K t φ(x) := φ(y) p(t, x, y)dy, U → R U is a μ-version of Tt φ such that (t, x) → K t φ(x) is continuous on (0, ∞)× U . In addition, if μ̃ ∈ P is invariant for (K t )t≥0 , i.e.,  μ̃ = K t∗ μ̃(dy) := K t (x, dy)dν(x), t ≥ 0, U then μ̃ = μ. Proof See [12, Theorem 2.3] or [26] for (1), and [12, Theorem 4.4] or [5, Theorem 4.1, Corollary 4.3] for (2). ⊔ ⊓ Here are some remarks, implied by Proposition A.1(2), about the semigroup (K t )t≥0 given in Proposition A.1(2) (see [6, Remark 1.7.6]). Remark A.1 Assume (H). (1) The semigroup (K t )t≥0 is strongly Feller and stochastically continuous. (2) The probability measures  B → K t χ B (x) := p(t, x, y)dy, t > 0, x ∈ U B are equivalent. In particular, if μ is invariant for (K t )t≥0 , i.e.,   K t φdμ = φdμ, t ≥ 0, U for all φ ∈ L 1 (U , μ), U Doob’s theorem (see e.g. [15, Theorem 4.2.1]) yields lim K t χ B (x) = μ(B), ∀x ∈ U t→∞ for any Borel set B ⊂ U . (3) By the proof of [5, Theorem 4.1], the transition density function p(t, x, y) of (K t )t≥0 is given by p(t, x, y) = pt (x, y)̺(y), (t, x, y) ∈ (0, ∞) × U × U , 1, p where ̺ ∈ Wloc (U ) is the density of μ, and (t, x, y) → pt (x, y) is continuous on (0, ∞) × U × U and satisfies sup sup x,y∈K z∈U | pt (x, z) − pt (y, z)| <∞ |x − y|α for any t > 0 and any compact set K ⊂ U , where α > 0 is some constant. (4) By [12, Theorem 2.3(iii)], or [6, Theorem 1.5.7(iii)], for any φ ∈ C0∞ (U ), Tt φ has a continuous modification, which must be K t φ, such that K t φ(x) → φ(x) as t → 0+ locally uniformly in x ∈ U . In the next result, sufficient conditions for the sub-Markov semigroup (Tt )t≥0 in Proposition A.1 being a Markov semigroup are provided. 123 1614 Journal of Dynamics and Differential Equations (2019) 31:1591–1615 Proposition A.2 Assume (H) with p > n + 2 replaced by p > n. Let μ ∈ M. Then, the following two assertions are equivalent: (1) For some (and therefore all) λ > 0, there holds L 1 (U , μ) = (L − λI )(C0∞ (U )); (2) There exists a unique C0 -semigroup in L 1 (U , μ) whose generator extending (L, C0∞ (U )). If one of the above two equivalent assertions holds, then the semigroup (Tt )t≥0 given in Proposition A.1(1) is a Markov semigroup and μ is invariant for (Tt )t≥0 . Proof See [12, Proposition 2.6]. Set  ⊔ ⊓  Mmd = μ ∈ M : L 1 (U , μ) = (L − I )(C0∞ (U )) . The following result holds. (A.1) Proposition A.3 Assume (H) with p > n + 2 replaced by p > n. If Mmd  = ∅, then #M = 1. Proof See [12, Theorem 4.1]. ⊔ ⊓ References 1. 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