Invariant Probability Measure for Mckean-Vlasov Sdes with Singular

arXiv:2108.05802v2 [math.PR] 20 Sep 2021 nain rbblt esr steeulbimsaei hsc.Ther physics. semigroup in linear for state measure probability equilibrium invariant the the on is results measure probability Invariant Introduction 1 inequality. Log-Sobolev’s entropy, Invar Information distance, variation Total SDEs, McKean-Vlasov 60G44. Keywords: 60H10, Classification: subject AMS cenVao Dswt iglrDrifts Singular with SDEs McKean-Vlasov ∗ upre npr yNSC(11801406). NNSFC by part in Supported uefrMKa-lsvSE r bandb aahsfie th fixed Banach’s by presented. obtained are are examples inva SDEs some stationa of McKean-Vlasov two uniqueness for and comparing sure existence by the Furthermore, de equations, estimate the Kolmogorov proved. Sobolev’s using and are addition, entropy measure relative In as McKean such investigated. for regularity well-posedness is weak drift the integrable theorem, point fixed nach nti ae,b tlzn agsHrakieult ihp with inequality Harnack Wang’s utilizing by paper, this In nain rbblt esr for Measure Probability Invariant )etrfrApidMteais ini nvriy ini 30007 Tianjin University, Tianjin Mathematics, Applied for a)Center )eateto ahmtc,Saga nvriy hnhi2 Shanghai University, Shanghai Mathematics, of b)Department igHuang Xing [email protected] [email protected] [email protected], a ) hnWang Shen , etme 1 2021 21, September [email protected] Abstract 1 a ) e-e Yang Fen-Fen , fivratprobability invariant of ope ehd some method, coupled in rbblt mea- probability riant atpoaiiymeasure, probability iant P Vao Dswith SDEs -Vlasov t yFokker-Planck- ry 04,China 00444, wradteBa- the and ower soitdt classical to associated oe.Finally, eorem. ,China 2, r plentiful are e b ) ∗ SDEs. Among them the existence of invariant probability measure can be studied by investigating the tightness of the sequence of probability measures 1 n Pt∗δxdt n Z0 with respect to n, see [15]. Moreover, Lyapunov’s condition is an useful sufficient condition to obtain the existence of invariant probability measure, i.e. there exists a function V with lim x V (x)= and constants C,R > 0 such that | |→∞ ∞ LV C, x R, ≤ − | | ≥ see, for instance [9]. For the uniqueness, the classical principle is strong Feller property together with irreducibility, see [15, Theorem 4.2.1]. By Wang’s Harnack inequality [29, Theorem 1.4.1], the uniqueness can also be ensured. Furthermore, using couplings or generalized couplings, [28] proved the uniqueness of the invariant measures. Recently, in [31], the existence and uniqueness as well as the regularity such as relative entropy and Sobolev’s estimate are investigated. However, all the above methods are invalid to obtain the existence and uniqueness for

distribution dependent SDEs, where the associated semigroup Pt∗ is nonlinear. In [30], Wang obtained the existence and uniqueness of invariant probability measure by Chauchy sequence method, see [20] for the path-distribution dependent case. One can also refer to [24] for the McKean-Vlasov SDEs with Lévy noise. Quite recently, [34] investigated the existence of invariant probability measure by Schauder’s fixed point theorem, see also [4] for the existence of invariant probability measure of functional McKean-Vlasov SDEs by Kakutani’s fixed point theorem. In addition, [33] proved the existence and uniqueness of invariant probability measure for (reflecting) McKean-Vlasov SDEs by exponential ergodicity and Banach’s fixed point theorem. For more results, one can see [16, 18, 19] and references therein. Moreover, by using log-Sobolev’s inequality, Poincáre’s inequality or Poisson equations, [7, 8] investigated the existence and uniqueness of the solution to stationary nonlinear and non-degenerate Fokker-Planck-Kolmogorov equations. In this paper, we will extend the results in [7] to the degenerate case, where the diffusion coefficients may be not invertible. More precisely, we will give the estimate of information entropy between two invariant probability measures of two different SDEs. Accordingly, the existence and uniqueness of invariant probability measure for McKean- Vlasov SDEs are proved by Banach’s fixed point theorem. Moreover, the regularity of invariant probability measure with integrable drift is also obtained by decoupled technique and the existed results in [31]. To do so, let P(Rd) be the space of all probability measures on Rd equipped with the weak topology. Consider the following distribution dependent SDE on Rd: (1.1) dX = Z (X )+ σ(X )Z(X , L ) dt + σ(X )dW , t { 0 t t t Xt } t t where (Wt)t 0 is an n-dimensional Brownian motion on a complete filtration probability F≥ F L space (Ω, , t t 0, P), Xt is the law of Xt, { } ≥ Z : Rd P(Rd) Rn, Z : Rd Rd, σ : Rd Rd Rn × → 0 → → ⊗ 2 are measurable. Compared with [31], Z can depend on the distribution of the solution, see (Ab) below for the condition of Z on the measure component. When a different probability measure P˜ is concerned, we use Lξ P˜ to denote the law of a random variable ξ under the ˜ | ˜ probability P, and use EP˜ to stand for the expectation under P.

d Deﬁnition 1.1. (1) An adapted continuous process (Xt)t 0 on R is called a solution ≥ of (1.1), if X0 is F0-measurable,

T 2 (1.2) E Z0(Xt) + σ(Xt)Z(Xt, LX ) + σ(Xt) dt< , T> 0, Z | | | t | k k ∞ 0 and P-a.s.

t t t L (1.3) Xt = X0 + Z0(Xs)ds + σ(Xs)Z(Xs, Xs )ds + σ(Xs)dWs, t 0. Z0 Z0 Z0 ≥

d (2) For any µ0 P(R ), ((X˜t)t 0, (W˜ t)t 0) is called a weak solution to (1.1) starting ∈ ≥ ≥ at µ0, if (W˜ t)t 0 is an n-dimensional Brownian motion under a complete ﬁltration ≥ ˜ ˜ ˜ probability space (Ω˜, F , Ft t 0, P˜), (X˜t)t 0 is a continuous Ft-adapted process on d { } ≥ ≥ R with L ˜ P˜ = µ and X˜ F˜ , and (1.2)-(1.3) hold for (X,˜ W,˜ P˜, E˜) replacing X0 | 0 0 ∈ 0 P (X, W, P, E).

(3) We call (1.1) weakly well-posed for an initial distribution µ0, if it has a weak solution starting at µ0 and any weak solution with the same initial distribution is equal in law.

For the well-posedness of distribution dependent SDEs with singular drifts, one can refer to [5, 6, 10, 11, 12, 21, 22, 23, 25, 27, 35] and references within. The remaining part of the paper is organized as follows: In Section 2, we investigate the weak well-posedness of (1.1) under integrable condition. In Section 3, the regularity of invariant probability measure for McKean-Vlasov SDEs with integrable drift is presented. The existence and uniqueness of invariant probability measure are stated in Section 4.

2 Weak Well-posedness

For any µ, ν P(Rd), the total variation distance between µ and ν is deﬁned as ∈

µ ν T V = 2 sup µ(A) ν(A) = sup µ(f) ν(f) . k − k A B(Rd) | − | f ∞ 1 | − | ∈ k k ≤ To obtain the weak well-posedness of (1.1), we make the following assumptions, see [31] for more details.

3 (A) The reference SDE

(2.1) dXt = Z0(Xt)dt + σ(Xt)dWt

is strongly well-posed and has a unique invariant probability measure µ0.

(Ab) There exist constants ε> 0,Kb > 0 such that

0 ε Z( ,µ0) 2 (2.2) µ (e | · | ) < , ∞ and

(2.3) Z(x, γ) Z(x, γ˜) K γ γ˜ , x Rd, γ, γ˜ P(Rd). | − | ≤ bk − kT V ∈ ∈ Let π be the projection map from C([0, ); Rd) to Rd, i.e. t ∞ π (w)= w , w C([0, ); Rd). t t ∈ ∞ For any γ P(Rd), we will prove that (1.1) has a unique weak solution with initial ∈ distribution γ and the distribution of the solution Pγ satisfying

t γ d γ 1 2 (2.4) P w C([0, ); R ), Z(ws, P πs− ) ds< , t 0 =1. ∈ ∞ Z0 | ◦ | ∞ ≥ Firstly, repeating the proof of [31, Theorem 2.1], we can easily extend it to the time dependent case below. Consider

(2.5) dX = Z (X )+ σ(X )Z˜ (X ) dt + σ(X )dW , t { 0 t t t t } t t here Z˜ : [0, ) Rd Rn is measurable. ∞ × → 0 Theorem 2.1. Assume (A) and that the semigroup Pt associated to (2.1) satisﬁes the Harnack inequality, i.e. there exists p> 1 such that

(2.6) (P 0 f )p(z) (P 0 f p)(¯z)eΦp(t,z,z¯), f B (Rd), z, z¯ Rd,t> 0 t | | ≤ t | | ∈ b ∈ with

t 1 0 Φp(s,z, ) p d (2.7) µ (e− · ) − ds< , t> 0, z R . Z ∞ ∈ 0 In addition, if there exists a constant ε> 0 such that

ε Z˜ 2 e | | ∞ 1 0 < , t> 0, k kL ([0,t];L (µ )) ∞ d ˜ γ then for any γ P(R ), (2.5) has a unique weak solution Xt with initial distribution γ and satisfying (2.4)∈ .

4 Proof. Since the proof can be completely the same with that of [31, Theorem 2.1], we omit it here. The main result in this section is the following theorem.

0 Theorem 2.2. Assume (A), (Ab), and that Pt satisﬁes (2.6) and (2.7), then for any γ P(Rd), (1.1) has a unique weak solution with initial distribution γ and satisfying (2.4)∈ .

To prove Theorem 2.2, it is suﬃcient to prove that for any T > 0, (1.1) is weakly well-posed on [0, T ]. So, we ﬁx T > 0 in the following. For any γ P(Rd), µ ∈ ∈ B([0, T ]; P(Rd)), consider

(2.8) dX = Z (X )+ σ(X )Z(X ,µ ) dt + σ(X )dW t { 0 t t t t } t t with initial distribution γ.

Proof of Theorem 2.2. Note that (Ab) implies

ε 2 0 2 2 0 2 2 Z( ,µ·) 0 ε Z( ,µ ) +4εK 0 ε Z( ,µ ) 4εK e 2 | · | ∞ 1 0 µ (e | · | b )= µ (e | · | )e b < , t 0, k kL ([0,t];L (µ )) ≤ ∞ ≥ So, under the assumption of Theorem 2.2, Theorem 2.1 implies that for any γ P(Rd), µ µ ∈ F F µ L µ µ (2.8) has a unique weak solution (Xt , Wt )t [0,T ] on (Ω, , ( t)t 0, P ) with X0 P = γ µ ∈ γ ≥ µ | for some probability measure P . Moreover, we denote Φ (µ) = L µ P . For ν t Xt | ∈ B([0, T ]; P(Rd)), we rewrite (2.8) as

(2.9) dXµ = Z (Xµ)+ σ(Xµ)Z(Xµ, ν ) dt + σ(Xµ)dW˜ , t { 0 t t t t } t t where t ˜ µ µ µ Wt = Wt + ξsds, ξs := Z(Xs ,µs) Z(Xs , νs), s,t [0, T ]. Z0 − ∈

By (Ab), one can see that

1 T 2 1 T 2 2 2 R0 ξs ds R0 K µs νs ds 2TK (2.10) EPµ [e 2 | | ] e 2 b k − kT V e b < . ≤ ≤ ∞ Set t µ 1 t 2 R ξs,dWs R ξs ds R := e− 0 h i− 2 0 | | , t [0, T ]. t ∈ According to the Girsanov theorem, we see that RT is a probability density with respect to µ µ P , and(W˜ t)t [0,T ] is an n-dimensional Brownian motion under the probability Q := RT P . ∈ µ From the weak uniqueness of (2.8) and L µ Q = L µ P = γ, we conclude that X0 | X0 | γ Φ (ν)= L µ Q, t [0, T ]. t Xt | ∈

5 Combining this with (Ab) and applying Pinsker’s inequality [14, 26], we obtain

t γ γ 2 2 Φt (ν) Φt (µ) T V 2EPµ [Rt log Rt]= EQ ξs ds k − k ≤ Z0 | | (2.11) t µ µ 2 = EQ Z(Xs ,µs) Z(Xs , νs) ds. Z0 | − |

This together with (Ab) implies

t γ γ 2 2 2 (2.12) Φt (ν) Φt (µ) T V Kb µs νs T V ds. k − k ≤ Z0 k − k 2 B P d Take λ = Kb and consider the space ET := µ ([0, T ]; (R )) : µ0 = γ equipped with the complete metric { ∈ }

2λt ρ(ν,µ) := sup e− νt µt T V . t [0,T ] k − k ∈ It follows from (2.12) that

t 2λt γ γ 2 2 2λ(t s) 2λs 2 sup e− Φt (ν) Φt (µ) T V sup Kb e− − e− µs νs T V ds t [0,T ] k − k ≤ t [0,T ] Z0 k − k ∈ ∈ t 2λs 2 2 2λ(t s) (2.13) sup e− µs νs T V sup Kb e− − ds ≤ s [0,T ] k − k t [0,T ] Z0 ∈ ∈ 1 2λs 2 sup e− µs νs T V . ≤ 2 s [0,T ] k − k ∈ γ Then Φ is a strictly contractive map on ET , so that the equation

(2.14) Φγ (µ)= µ , t [0, T ] t t ∈ has a unique solution µ E . The proof is completed. ∈ T 3 Regularity of Invariant Probability Measure

In this section, we consider the regularity of invariant probability measure of (1.1) and a general result for the regularity will be presented. Assume that µZ is an invariant probability measure of (1.1) with density ρ with respect to µ0, i.e. µZ = ρµ0. It is clear that µZ is also an invariant probability measure of the following distribution independent SDE:

(3.1) dX = Z (X )+ σ(X )Z(X ,µZ) dt + σ(X )dW . t { 0 t t t } t t

6 Theorem 3.1. Assume (A), (2.3) and

0 2 0 2p 0 (3.2) Pt0 L (µ ) L 0 (µ ) < k k → ∞ 0 λ Z( ,µ0) 2 for some t0 > 0 and p0 > 1. If in addition, µ (e | · | ) < for some λ > κ0 := t0(3p0 1) ∞ − , then it holds 2(p0 1) −

0 2 2 λλ˜ 0 λ Z( ,µ ) +4Kb ˜ 0 | · | λ−λ ˜ 2 0 2p 0 t0(3p0 1) log µ (e )+4λp0 log Pt0 L (µ ) L 0 (µ ) µ0(ρ log ρ) inf − k k → . ≤ λ˜ (κ0,λ) 2λ˜(p 1) t (3p 1) ∈ 0 − − 0 0 − Proof. For any λ˜ (0,λ), it follows from (2.3) that ∈ Z 2 0 Z 0 2 0 λ˜ Z( ,µ ) 0 λ˜( Z( ,µ ) +K µ µ T V ) µ (e | · | ) µ (e | · | bk − k ) ≤ Z 0 0 2 Kbkµ −µ kT V 0 2 Z 0 2 λ˜ Z( ,µ ) +2λ˜ √λ λ˜ Z( ,µ ) +λK˜ µ µ T V (3.3) = µ0(e | · | √λ−λ˜ − | · | b k − k | ) ˜ 0 2 2 λλ Z 0 2 0 λ Z( ,µ ) K ˜ µ µ T V µ (e | · | )e b λ−λ k − k < . ≤ ∞

t0(3p0 1) λ > κ − λ˜ κ ,λ Since 0 = 2(p0 1) , using (3.3) for ( 0 ) and [31, Theorem 3.1, Theorem 4.1], − ∈ we conclude that µZ is the unique invariant probability measure of (3.1). Furthermore, [31, Theorem 4.1] holds with λ replaced by λ˜ (κ ,λ), i.e. ∈ 0 0 λ˜ Z( ,µZ ) 2 0 ˜ 2 0 2p 0 t0(3p0 1) log µ (e | · | )+4λp0 log Pt0 L (µ ) L 0 (µ ) (3.4) µ0(ρ log ρ) − k k → . ≤ 2λ˜(p 1) t (3p 1) 0 − − 0 0 − Substituting (3.3) into (3.4), we complete the proof. To obtain the Sobolev estimate for ρ by log-Sobolev’s inequality of the reference SDE (2.1), let d 1 Z = ∂ (σσ∗) (σσ∗) ∂ V e 0 2 { j ij − ij j } i i,jX=1 for some V C2(Rd). Deﬁne ∈ 0 d E (f,g)= µ ( σ∗ f, σ∗ g ), f,g C∞(R ). 0 h ∇ ∇ i ∈ 0 1,2 0 d Let Hσ (µ ) be the completion of C0∞(R ) under the norm

1 0 2 2 2 E1(f, f) := µ ( f + σ∗ f ) . p { | | | ∇ | } E 1,2 0 2 0 Then ( 0,Hσ (µ )) is a symmetric Dirichlet form on L (µ ). Moreover, we shall introduce the condition (H) in [31]:

7 0 V (H) Assume that µ (dx) = e− dx is a probability measure. There exists k 2 such that σ Ck(Rd, Rd Rn) and vector ﬁelds ≥ ∈ ⊗ d U = σ ∂ , i =1, , n i ji j ··· Xj=1 satisfy the H¨ormander condition up to the k-th order of Lie brackets. Moreover, 1 H1,2(µ ) with E (1, 1) = 0, and defective log-Sobolev inequality ∈ σ 0 0 0 2 2 0 2 d 0 2 (3.5) µ (f log f ) κµ ( σ∗ f )+ β, f C∞(R ),µ (f )=1 ≤ | ∇ | ∈ 0 holds for some κ> 0 and β 0. ≥ One can refer to [1, 2, 3, 13, 17, 32] for more results on the log-Sobolev inequality. 0 λ Z( ,µ0) 2 Theorem 3.2. Assume (H), (2.3) and µ (e | · | ) < for some λ > κ. Then ρ has p ∞ √ a strictly positive continuous version satisfying log ρ, ρ 2 H1,2(µ0) for p (1, λ ) and ∈ σ ∈ √κ 0 2 0 0 2 µ ( σ∗ log ρ ) 4µ (( Z( ,µ ) +2K ) ) | ∇ | ≤ | · | b 0 2 2 λλ˜ 0 2 1 0 λ Z( ,µ ) +4K ˜ µ ( σ∗ √ρ ) inf (log µ (e | · | b λ−λ )+ β), | ∇ | ≤ λ˜ (κ,λ) λ˜ κ ∈ − ˜ p 0 2 2 λλ λ Z( ,µ ) +4K C ˜ 0 2 2 0 b λ−λ˜ p,λ µ ( σ∗ ρ ) inf Cp,λ˜(µ (e | · | )) | ∇ | ≤ λ˜ (p2κ,λ) ∈ 2 for some function Cp, :(p κ, λ) [0, ). · → ∞ Proof. By (3.3) for λ˜ (κ, λ) and [31, Theorem 5.1(2), Theorem 5.2], ρ has a strictly ∈ positive continuous version. So, it remains to prove the three estimates above. By [31, Theorem 5.1(2), Theorem 5.2], we arrive at log ρ H1,2(µ0) with ∈ σ 0 2 0 Z 2 0 0 2 µ ( σ∗ log ρ ) 4µ ( Z( ,µ ) ) 4µ (( Z( ,µ ) +2K ) ), | ∇ | ≤ | · | ≤ | · | b and for any λ˜ (κ, λ), ∈ ˜ 2 0 2 2 λλ 0 2 1 0 λ˜ Z( ,µZ ) 1 0 λ Z( ,µ ) +4K ˜ µ ( σ∗ √ρ ) (log µ (e | · | )+ β) (log µ (e | · | b λ−λ )+ β), | ∇ | ≤ λ˜ κ ≤ λ˜ κ − − here we used (3.3) in the last step. √ √λ˜ Next, for any p (1, λ ) and λ˜ (p2κ, λ), we have p (1, ). Again by (3.3) for ∈ √κ ∈ ∈ √κ ˜ 2 λ (p κ, λ) and [31, Theorem 5.1(2), Theorem 5.2], there exists a constant Cλ,p˜ 1 such that∈ ≥ p 2 0 2 0 λ˜ Z( ,µZ ) C ˜ (3.6) µ ( σ∗ ρ 2 ) C ˜(µ (e | · | )) p,λ . | ∇ | ≤ p,λ So, this together with the third inequality in (3.6) yields

˜ p 0 2 2 λλ 0 2 0 λ Z( ,µ ) +4Kb ˜ C ˜ µ ( σ∗ ρ 2 ) C ˜(µ (e | · | λ−λ )) p,λ . | ∇ | ≤ p,λ

8 4 Existence and Uniqueness of Invariant Probability Measure

Consider the stationary nonlinear Fokker-Planck-Kolmogorov equation: L (4.1) µ∗µ =0,

here,

1 2 d d L f = Z , f + σZ( ,µ), f + Tr(σσ∗ f), f C∞(R ),µ P(R ), µ h 0 ∇ i h · ∇ i 2 ∇ ∈ 0 ∈ and d d (L ∗γ)(f)= γ(L f), f C∞(R ),µ,γ P(R ). µ µ ∈ 0 ∈ When (1.1) is weakly well-posed, the invariant probability measure of (1.1) is a solution to (4.1) in the sense d µ(L f)=0, f C∞(R ). µ ∈ 0 Note that 0 ε Z( ,µ) 2 0 ε Z( ,µ0) 2 ε Z( ,µ) Z( ,µ0) 2 d µ (e 2 | · | ) µ (e | · | e | · − · | ), µ P(R ). ≤ ∈ Then according to [31, Theorem 5.2], under (H)and (A ) with ε> 2κ, for any µ P(Rd), b ∈ there exists a unique probability measure Γ(µ) such that L (4.2) µ∗(Γ(µ))=0.

Therefore, Γ construct a map from P(Rd) to P(Rd). Next, we prove that Γ has a unique fixed point. We first give a result characterizing the Fisher information of Γ(µ) and Γ(ν) for µ, ν P(Rd). ∈ d By [31, Theorem 5.2], under (H) and (Ab) with ε > 2κ, for any µ P(R ), Γ(µ) 0 ∈ dΓ(µ) has a strictly positive and continuous density with respect to µ . We denote ρµ = dµ0 dΓ(ν) P Rd and ρν,µ = dΓ(µ) ,µ,ν ( ). Then it is clear that ρν = ρν,µρµ. Moreover, log ρµ, p ∈ 2 1,2 0 √ε ρµ H (µ ) for p (1, ). Next, we prove a useful result on the information entropy ∈ σ ∈ √2κ between Γ(µ) and Γ(ν). The proof is modified from [8, Theorem 2.1], where σσ∗ is assumed to be invertible.

Theorem 4.1. Assume (H) and (Ab) with ε> 2κ. If

σσ (x) Z (x)+ σZ(x, µ) ∗ 0 µ x < , (4.3) | |2 + | | Γ( )(d ) Z d (1 + x ) 1+ x ∞ R | | | | then it holds 2 σ∗ ρ | ∇ ν,µ| dΓ(µ) Z( , ν) Z( ,µ) 2dΓ(ν). ZRd ρν,µ ≤ ZRd | · − · |

9 L 0 1 2 L 0 D L 0 Proof. Let := Z0, + 2 Tr(σσ∗ ). Then under (H), ( , ( )) is a symmetric operator in L2(µ0).h It follows∇i from integration∇ by parts formula that

V (x) σZ( ,µ), f ge− dx ZRd h · ∇ i 0 d = f[ div(σZ( ,µ)g)+ σZ( ,µ), V g]µ (dx), f,g C0∞(R ). ZRd − · h · ∇ i ∈

2 0 So, we conclude that the adjoint operator of Lµ on L (µ ) is deﬁned as

0 d L ∗g = L g div(σZ( ,µ)g)+ σZ( ,µ), V g, g C∞(R ). µ − · h · ∇ i ∈ 0 d 2 For any u, v C∞(R ) and f C (R), we have ∈ 0 ∈ 0 L ∗[uv]= L [uv] div(σZ( ,µ)[uv]) + σZ( ,µ), V [uv] µ − · h · ∇ i 1 2 2 = v Z , u + u Z , v + Tr(σσ∗(v u + u v +2 u v)) h 0 ∇ i h 0 ∇ i 2 ∇ ∇ ∇ ⊗∇ u σZ( ,µ), v v σZ( ,µ), u div(σZ( ,µ))[uv]+ σZ( ,µ), V [uv] − h · ∇ i − h · ∇ i − · h · ∇ i = uL ∗v + vL ∗u + Tr σσ∗ u, v + div(σZ( ,µ))[uv] σZ( ,µ), V [uv] µ µ h ∇ ∇ i · −h · ∇ i and

0 L ∗[f(u)] = L [f(u)] div(σZ( ,µ)[f(u)]) + σZ( ,µ), V [f(u)] µ − · h · ∇ i 1 2 = f ′(u) Z , u + Tr(σσ∗(f ′′(u) u u + f ′(u) u)) h 0 ∇ i 2 ∇ ⊗∇ ∇ (4.4) f(u)div(σZ( ,µ)) f ′(u) σZ( ,µ), u + σZ( ,µ), V f(u) − · − h · ∇ i h · ∇ i 1 = f ′(u)L ∗u + f ′′(u) σσ∗ u, u µ 2 h ∇ ∇ i +(uf ′(u) f(u))div(σZ( ,µ))+ σZ( ,µ), V (f(u) uf ′(u)). − · h · ∇ i −

From the deﬁnition of ρµ and ρν, we arrive at L L µ∗ρµ = ν∗ρν =0 in the sense that

L 0 L 0 d µ(φ)ρµdµ = ν (φ)ρνdµ =0, φ C0∞(R ). ZRd ZRd ∈

2 So, for f C ((0, )) with f ′′ 0, we have ∈ ∞ ≥

L ∗[f(ρ )ρ ]= ρ L ∗f(ρ )+ f ′(ρ )Tr σσ∗ ρ , ρ µ ν,µ µ µ µ ν,µ ν,µ h ∇ ν,µ ∇ µi (4.5) + div(σZ( ,µ))[f(ρ )ρ ] σZ( ,µ), V [f(ρ )ρ ]. · ν,µ µ −h · ∇ i ν,µ µ

10 Replacing u by ρν,µ and multiplying ρµ on both sides of (4.4), we get from (4.5) that L µ∗[f(ρν,µ)ρµ] 1 = ρ f ′′(ρ ) σσ∗ ρ , ρ µ 2 ν,µ h ∇ ν,µ ∇ ν,µi

+ f ′(ρ ) ρ L ∗ρ + ρ ρ div(σZ( ,µ)) ν,µ µ µ ν,µ µ ν,µ ·

+ Tr σσ∗ ρ , ρ σZ( ,µ), V [ρ ρ ] . h ∇ ν,µ ∇ µi−h · ∇ i µ ν,µ Noting that

L ∗[ρ ρ ]=(L ∗ L ∗)ρ = div(σ(Z( , ν) Z( ,µ))ρ )+ σ(Z( ,µ) Z( , ν)), V ρ , µ µ ν,µ µ − ν ν · − · ν h · − · ∇ i ν we know

L ∗[ρ ρ ]= ρ L ∗ρ + Tr σσ∗ ρ , ρ µ µ ν,µ µ µ ν,µ h ∇ µ ∇ ν,µi + div(σZ( ,µ))[ρ ρ ] σZ( ,µ), V [ρ ρ ] · µ ν,µ −h · ∇ i µ ν,µ = div(σ(Z( , ν) Z( ,µ))ρ )+ σ(Z( ,µ) Z( , ν)), V ρ . · − · ν h · − · ∇ i ν This implies that 1 ρ f ′′(ρ ) σσ∗ ρ , ρ µ 2 ν,µ h ∇ ν,µ ∇ ν,µi = L ∗[f(ρ )ρ ] f ′(ρ ) [div(σ(Z( , ν) Z( ,µ))ρ )+ σ(Z( ,µ) Z( , ν)), V ρ ] . µ ν,µ µ − ν,µ · − · ν h · − · ∇ i ν d Let Φ = σ(Z( , ν) Z( ,µ)). It yields that for any ψ C∞(R ), · − · ∈ 0 1 0 ρµ f ′′(ρν,µ) σσ∗ ρν,µ, ρν,µ ψµ (dx) ZRd 2 h ∇ ∇ i 0 0 = [f(ρν,µ)ρµ]Lµψµ (dx) f ′(ρν,µ)div(Φρν )ψµ (dx) ZRd − ZRd 0 (4.6) + Φ, V f ′(ρν,µ)ρν ψµ (dx) ZRd h ∇ i 0 = [f(ρν,µ)ρµ]Lµψµ (dx) ZRd 0 0 + Φ, ρν,µ ρνf ′′(ρν,µ)ψµ (dx)+ Φ, ψ f ′(ρν,µ)ρνµ (dx). ZRd h ∇ i ZRd h ∇ i Similarly to [7, Proof of Theorem 1] or [8, Proof of Theorem 2.1] , there exists a d L sequence of ψN N 1 C0∞(R ) such that ψN 1, ψN 0 and µψN 0 as N { } ≥ ∈ → |∇ | → → goes to inﬁnity. Therefore, replacing ψ with ψN in (4.6) and letting N , (4.3) implies that →∞

1 0 0 ρµ f ′′(ρν,µ) σσ∗ ρν,µ, ρν,µ µ (dx)= Φ, ρν,µ ρν,µρµf ′′(ρν,µ)µ (dx). ZRd 2 h ∇ ∇ i ZRd h ∇ i

11 Recalling Φ = σ(Z( , ν) Z( ,µ)) and combing the elemental inequality · − ·

1 2 1 2 2 Φ, ρ ρ σ∗ ρ + Z( , ν) Z( ,µ) ρ , h ∇ ν,µi ν,µ ≤ 2| ∇ ν,µ| 2| · − · | ν,µ we have

2 0 2 2 0 f ′′(ρν,µ) σ∗ ρν,µ ρµµ (dx) Z( , ν) Z( ,µ) ρν,µf ′′(ρν,µ)ρµµ (dx). ZRd | ∇ | ≤ ZRd | · − · |

1 Finally, taking f(x)= x log x and noting f ′′(x)= x , we completed the proof. Theorem 4.2. Assume (H) and (A ) with ε> 2κ. If in addition, for any µ P(Rd), b ∈ 2 2 2 1,2 0 2 (4.7) Γ(µ) f Γ(µ) κ˜Γ(µ)( σ∗ f ), f H (µ ), Γ(µ)(f )=1 k − kT V ≤ | ∇ | ∈ σ κ˜ L with 2 Kb < 1, then µ∗µ =0 has a unique solution. q Proof. Taking f = √ρν,µ in (4.7), we have

2 κ˜ 2 κ˜ 2 2 Γ(µ) Γ(ν) T V Z(x, ν) Z(x, µ) dΓ(ν) Kb µ ν T V . k − k ≤ 2 ZRd | − | ≤ 2 k − k

κ˜ When 2 Kb < 1, we ﬁnish the proof due to the ﬁxed point theorem. q 4.1 Some Other Examples When Z has a special form, we can also derive the existence and uniqueness of invariant probability measure. The following three examples are presented to illustrate this case.

Example 4.3. Let Z(x, µ)= F (x, µ)+ F¯(x) and σ = √2Id d. Assume (H) and that ∇ ∇ log 2 × there exist constants ε> 0,C > 0 and δ (0, ) such that ∈ 2 0 F ( ,µ0)+F¯ 0 ε F ( ,µ0)+ F¯ 2 µ (e · )+ µ (e |∇ · ∇ | ) < , ∞ F (x, µ) F (x, ν) C µ ν , µ,ν P(Rd), x Rd, |∇ −∇ | ≤ k − kT V ∈ ∈ and F (x, µ) F (x, ν) δ µ ν , µ,ν P(Rd), x Rd. | − | ≤ k − kT V ∈ ∈ Then (1.1) has a unique invariant probability measure.

Proof. It is clear that dΓ(µ) eF ( ,µ)+F¯ = · . 0 0 F ( ,µ)+F¯ dµ µ (e · )

12 By Taylor’s expansion, we arrive at

∞ F (x, µ) F (x, ν) k eF (x,µ) eF (x,ν) eF (x,ν) | − | − ≤ k! Xk=1

∞ δk µ ν k eF (x,ν) k − kT V ≤ k! Xk=1 ∞ δk2k 1 eF (x,ν) µ ν − ≤ k − kT V k! Xk=1 e2δ 1 = eF (x,ν) µ ν − . k − kT V 2 As a result, it holds Γ(µ) Γ(ν) k − kT V eF ( ,µ)+F¯ eF ( ,ν)+F¯ = · · µ0(dx) 0 F ( ,µ)+F¯ 0 F ( ,ν)+F¯ ZRd µ (e · ) − µ (e · )

F ( ,µ)+F¯ 0 F ( ,ν)+F¯ F ( ,ν)+ F¯ 0 F ( ,µ)+F¯ e · µ (e · ) e · µ (e · ) = µ0(dx) 0 F ( ,µ)+F¯− 0 F ( ,ν)+F¯ ZRd µ (e · )µ (e · )

0 F ( ,ν)+F¯ 0 F ( ,µ)+F¯ F ( ,µ)+F¯ F ( ,ν)+F¯ µ (e · ) µ (e · ) e · e · − µ0(dx)+ − µ0(dx) 0 F ( ,ν)+F¯ 0 F ( ,ν)+F¯ ≤ ZRd µ (e · ) ZRd µ (e · ) µ ν (e2δ 1) ≤k − kT V − By Banach’s ﬁxed point theorem and δ (0, log 2 ), we ﬁnish the proof. ∈ 2 The next example concentrates on the stochastic Hamiltonian system. d Example 4.4. Let Z0(x, y)=(y, x y),x,y R , σ = √2Id d. Consider − − ∈ × dX = Y (4.8) t t dYt = Xt Yt H( , L(X ,Y ))(Xt)dt H¯ (Xt)dt + √2dWt, − − −∇ · t t −∇ here H : Rd P(R2d) R, H¯ : Rd R. Z(x, µ) = H( ,µ)(x) H¯ (x). Then 2 2 0 × x →y → −∇ · −∇ µ (dx, dy) = exp 2 2 dxdy. Assume that there exist constants ε > 0,C > 0 and n− − o δ (0, log 2 ) such that ∈ 2 0 H( ,µ0)+H¯ 0 ε H( ,µ0)+ H¯ 2 µ (e · )+ µ (e |∇ · ∇ | ) < , ∞ H(x, µ) H(x, ν) C µ ν , µ,ν P(R2d), x Rd, |∇ −∇ | ≤ k − kT V ∈ ∈ and H(x, µ) H(x, ν) δ µ ν , µ,ν P(R2d), x Rd. | − | ≤ k − kT V ∈ ∈ Then (1.1) has a unique invariant probability measure.

13 Proof. It is easy to see that dΓ(µ) eH( ,µ)+H¯ = · 0 0 H( ,µ)+H¯ dµ µ (e · ) In fact, the inﬁnitesimal generator of (4.8) is

2 2d Lf(x, y)= y f +( x y H(x, µ) H¯ ) f + f, f C∞(R ), ∇x − − −∇ −∇ ∇y ∇y ∈ 0 which yields

x2 y2 Lf(x, y) exp H¯ (x, µ) H(x) dxdy ZR2 − 2 − 2 − − x2 y2 = xf, y exp H¯ (x, µ) H(x) dxdy − ZRd Rd ∇ ∇ − 2 − 2 − − × x2 y2 + yf, x exp H¯ (x, µ) H(x) dxdy =0. ZRd Rd ∇ ∇ − 2 − 2 − − × Then repeating the proof of Example 4.3, we complete the proof. In the one-dimensional case, since the invariant probability measure has explicit rep- resentation, the drift term Z can be more general.

Example 4.5. Let d = 1, σ = 1, and Z(y,µ) = b(y,µ)+ ¯b(y). Assume that there exist constants ε> 0,c R, C > 0 and δ (0, log 2 ) such that 0 ∈ ∈ 2 · 0 · 0 R b(y,µ )dy+R ¯b(y)dy 0 ε b( ,µ0)+¯b 2 µ (e c0 c0 )+ µ (e | · | ) < , ∞ b(x, µ) b(x, ν) C µ ν , µ,ν P(R), x Rd, | − | ≤ k − kT V ∈ ∈ and x x b(y,µ)dy b(y, ν)dy δ µ ν T V , µ,ν P(R), x R. Z − Z ≤ k − k ∈ ∈ c0 c0

Then (1.1) has a unique invariant probability measure. Proof. It is clear that 0 R · b(y,µ )dy+R · ¯b(y)dy dΓ(µ) e c0 c0 = . 0 R · b(y,µ0)dy+R · ¯b(y)dy dµ µ0(e c0 c0 ) Again the remaining is just repeating the proof of Example 4.3, we obtain the desired result.

Acknowledgement. The authors would like to thank Professor Feng-Yu Wang for help- ful comments.

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