On invariant probability measures of regime-switching diffusion processes with singular drifts

J. Math. Anal. Appl. 478 (2019) 655–688

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On invariant probability measures of regime-switching diffusion processes with singular drifts Shao-Qin Zhang School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 27 November 2018 Available online 30 May 2019 Submitted by U. Stadtmueller Keywords: Regime-switching diffusions Singular drift Invariant probability measure Integrability conditions Weakly coupled elliptic system

Under integrability conditions of the coefficients, the existence, uniqueness and regularity estimates are derived for the invariant probability measure of regimeswitching diffusion processes. Moreover, we study the L1 -uniqueness of the semigroup generated by regime-switching diffusion processes. As an application, the L1 -uniqueness of the extension of weakly coupled elliptic systems is obtained. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Let S = {1, 2, 3, · · · , N } where N ∈ N ∪ {∞}, and let (Ω, F , P ) be a complete probability space. A regime-switching diffusion process (RSDP for short) is a two components process {(Xt , Λt )}t≥0 on Rd × S described by dXt = bΛt (Xt )dt +

√

2σΛt (Xt )dWt ,

(1.1)

and  P (Λt+Δt = j | Λt = i, (Xs , Λs ), s ≤ t) =

qij (Xt )Δt + o(Δt),

i = j,

1 + qii (Xt )Δt + o(Δt),

i = j,

(1.2)

where {Wt }t≥0 is a Brownian motion on Rd w.r.t. a right continuous completed reference {Ft }t≥0 and b : Rd × S → Rd , σ : Rd × S → Rd ⊗ Rd , qij : Rd → R, E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.05.049 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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are measurable functions such that 

qij (x) ≥ 0, i = j,

qij (x) ≤ −qii (x).

j=i

The matrix Q(x) = (qij (x))1≤i,j≤N is called Q-matrix, see [5]. If Q(x) is independent of x so that Λt is independent of {Wt }t≥0 , {(Xt , Λt )}t≥0 is called a state-independent RSDP, otherwise called a state-dependence RSDP. Let qi (x) =



qij (x), i ∈ S.

j=i

If Q(x) is conservative, i.e. −qii (x) = qi (x), x ∈ Rd , then as in [17, Chapter II-2.1] or [11,23,16], we can represent {(Xt , Λt )}t≥0 in the form of a system of stochastic differential equations (SDEs for short) driven by {Wt } and a Poisson random measure. Precisely, for each x ∈ Rd , {Γij (x) : i, j ∈ S} is a family of disjoint intervals on [0, ∞) constructed as follows Γ12 (x) = [0, q12 (x)), Γ13 (x) = [q12 (x), q12 (x) + q13 (x)), · · · Γ21 (x) = [q1 (x), q1 (x) + q21 (x)), Γ23 (x) = [q1 (x) + q21 (x), q1 (x) + q21 (x) + q23 (x)), · · · Γ31 (x) = [q1 (x) + q2 (x), q1 (x) + q2 (x) + q13 (x)), · · · ··· We set Γij (x) = ∅ if either i = j or qij (x) = 0. Define a function h : Rd × S × [0, ∞) → R as follows h(x, i, z) =



 (j − i)1Γij (x) (z) =

j∈S

j − i, if z ∈ Γij (x), 0,

otherwise.

(1.3)

  Let dz be the Lebesgue measure restricted on (0, ∞), B(0,∞) . According to [13, Theorem I.8.1, Theo  rem I.9.1], there exist an adapted Poisson point process {pt }t≥0 on (0, ∞), B(0,∞) and a Poisson random   measure N (dz, dt) with intensity dzdt on (0, ∞) × (0, ∞), B(0,∞)×(0,∞) such that N (U, (0, t])(ω) = #{s ∈ Dp(ω) | 0 < s ≤ t, ps (ω) ∈ U }, U ∈ B(0,∞) , where Dp(ω) is the domain of the point function p(ω), which is a countable subset of (0, ∞). Moreover, by [13, Theorem II.6.3], the Poisson point process p with non-random intensity is independent of {Wt }t≥0 . Consider the following SDE: ⎧ √ ⎨dXt = bΛ (Xt )dt + 2σΛ (Xt )dWt , t t (1.4) ⎩dΛ = ∞ h(X , Λ , z)N (dz, dt). t t− t− 0 The solutions to (1.4) will satisfy (1.2) under some continuity assumption on elements of the matrix Q, see [16,22] for instance. The first component Xt can be viewed as a hybrid process from the diffusion processes {Xti | i ∈ S} solving the SDEs dXti = bi (Xti )dt +

√

2σi (Xti )dWt , i ∈ S.

(1.5)

In [22], we have studied the existence, uniqueness and strong Feller property of (1.4), where drifts bi , i = 1, · · · , N can be singular. In this paper, we study the invariant probability measures of (1.4) with non-regular drifts.

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Recently, in term of a nice reference probability measure, [20,21] introduce integrability conditions to investigate the invariant probability measures for SDEs with singular or path dependent drifts. We shall extract similar results in [20,21] for the semigroup corresponding to (1.4) when N < ∞. Since (1.4) is a hybrid system of {Xti | i ∈ S} via the Q-process Λt , it is natural that the desired integrability condition is ¯ in Theorem 1.1 below or Example 1.4. hybrid as well, cf. (1.10) and the semipositiveness of −(K + Q) For each k ∈ S, we assume σk ∈ C 2 (Rd → Rd ⊗ Rd ). We denote ak = σk σk∗ = (aij k )1≤i,j≤d and

d ij d d div(ak ) = i,j=1 ∂i ak ej , where {ej }j=1 is the canonical orthonormal basis of R and ∂j is the directional derivative along ej . Let V ∈ C 2 (Rd ) such that μV (dx) = eV (x) dx is a probability measure. For each k ∈ S, we consider the reference diffusion process generated by L0k where L0k =

d  i,j=1



d  

aij k ∂i ∂j + 

d  ∂ V ∂ + ∂i aij aij j i k k ∂j

i,j=1

i,j=1

= tr ak ∇ + (ak ∇V ) · ∇ + div (ak ) · ∇   ≡ tr ak ∇2 + Zk0 · ∇. 2

It is well known that L0k is symmetric in L2 (μV ). Writing bk = Zk0 + by 0 LZ k := Lk +

√

√

2σk Zk , we see that Xtk is generated

2 (σk Zk ) · ∇.

(1.6)

In this way, Xtk is regarded as a singular perturbation of the reference diffusion process. Let Pt be the semigroup associated with (1.4). Then the generator of Pt , denoted by L, is of the following form (Lf )(x, k) = LZ k fk (x) +

N 

qkj (x)fj (x), f ∈ Cc2 (Rd × S),

j=1

where fi (x) ≡ f (x, i), i ∈ S, x ∈ Rd . The operator L is called weakly coupled elliptic system, see [6]. (Rd ) the Sobolev space which is the completion of Cc2 (Rd ) under the norm Denote by Hσ1,2 j ⎛ ⎝



⎞ 12  ∗  |σj (x)∇u(x)|2 + |u(x)|2 μV (dx)⎠ , u ∈ Cc2 (Rd )

Rd

and by Wσ1,2 (Rd ) the Sobolev space defined as follows j ⎧ ⎫ ⎨    ⎬  1,2 d d  ∗ 2 2 |σ μ Wσ1,2 (R ) = u ∈ W (R ) (x)∇u(x)| + |u(x)| (dx) < ∞ .  V j loc j ⎩ ⎭ Rd

We assume that the coefficients of L satisfy the following assumptions. (H1) For each k ∈ S, σk ∈ C 2 (Rd → Rd ⊗ Rd ) with λk > 0 such that ak (x) ≡ (σσ ∗ ) (x) ≥ λk , x ∈ Rd ; V ∈ C 2 (Rd ) such that eV dx is a probability measure and Hσ1,2 (Rd ) = Wσ1,2 (Rd ). k k

(1.7)

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(H2) For x ∈ Rd , Q(x) = (qkj (x))1≤k,j≤N is a conservative Q-matrix with CQ :=

qk (x) < ∞.

sup

(1.8)

x∈Rd , k∈S

The matrix Q is fully coupled on Rd , i.e. the set S can not be split into two disjoint non-empty S1 and S2 such that a.e. on Rd ,

ql1 l2 (·) = 0,

l 1 ∈ S1 , l 2 ∈ S2 .

(H3) There are γ1 , · · · , γN ∈ (0, ∞) and β1 , . . . , βN ∈ [0, ∞) such that μV (f 2 log f 2 ) ≤ γk μV (|σk∗ ∇f |2 ) + βk , f ∈ Hσ1,2 (Rd ), μV (f 2 ) = 1, k ∈ S. k Remark 1.1. The condition (1.7) implies that for all i ∈ S, L0i generates a unique non-explosive Markov semigroup and has a unique self-adjoint extension in L2 (μV ) such that the associated semigroup is Markovian, see [10, Corollary 3.3.1]. Let π = (πk )k∈S be a probability measure on S with πk > 0, k ∈ S, and let E 0 (f, g) =

N  k=1

 πk

σk∗ ∇fk (x), σk∗ ∇gk (x)μV (dx), fk , gk ∈ Hσ1,2 (Rd ). k

Rd

Then E 0 is a Dirichlet form with D (E 0 ) = {f = (f1 , · · · , fN ) | fi ∈ Hσ1,2 (Rd ), i ∈ S, E 0 (f, f ) < ∞}. i

If N < ∞, then (H3) yields that the following defective log-Sobolev inequality holds, i.e. for fk ∈ Hσ1,2 (Rd ), k N  k ∈ S, with πk μV (fk2 ) = 1, we have k=1 N 

πk μV (fk2 log fk2 ) ≤ (max γk )E 0 (f, f ) + k

k=1

+

N 

βk πk μV (fk2 )

k=1

N 

πk μV (fk2 ) log μV (fk2 )

k=1

≤ γ E 0 (f, f ) + β,

(1.9)

with γ = max γi , 1≤i≤N

β = max (βk − log πk ) . k

Remark 1.2. The full coupling condition on Q in (H2) had been used in [18] to prove the positivity of solutions to elliptic systems. It was proved in [7, Proposition 4.1.] that Q is fully coupled on a domain D of Rd if and only if Q is irreducible on D: for any distinct k, l ∈ S, there exist k0 , k1 , · · · , kr in S with ki = ki+1 , k0 = k and kr = l such that {x ∈ D| qki ,ki+1 (x) = 0} has positive Lebesgue measure for i = 0, 1, · · · , r − 1. If N < ∞, then Q is fully coupled on Rd if and only if there exists a bounded domain D such that Q is fully coupled on D. In fact, it is clear that if Q is fully coupled on a bounded domain D, then Q is full coupled on Rd . On the other hand, if Q is fully coupled on Rd , then Q is irreducible on Rd . Denote

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Dkkli = {x ∈ D| qki ,ki+1 (x) = 0} and Ikl = {k0 , · · · , kr−1 } in the above definition of the irreducibility. Since N < ∞, the set {Dkkli | ki ∈ Ikl , k, l ∈ S, k = l} is finite. Thus, there exists n > 0 such that for all distinct k, l ∈ S, ki ∈ Ikl , {x | |x| < n} ∩Dkkli has positive Lebesgue measure. Hence, Q is irreducible on {x | |x| < n}. Before our integrability condition, we also need the following definition. Definition 1.1. A matrix A ∈ Rn ⊗ Rn is called semipositive if there exists a positive v ∈ Rn such that Av is positive, equivalently, there exists a positive v ∈ Rn such that Av ≥ 1, where 1 = (1, · · · , 1). The invariant probability measure of Pt is a measure on Rd × S. For a measure μ on Rd × S, we define μk (dx) = μ(dx × {k}). Our first main result reads as follows. ¯ = (¯ Theorem 1.1. Assume N < ∞, and (H1)-(H3). Let Q qij )1≤i,j≤N be a matrix with q¯ij = supx∈Rd qij (x). We suppose in addition that there exist positive constants w1 , w2 , · · · , wN such that sup μV (ewk |Zk | ) < ∞, 2

(1.10)

1≤k≤N

¯ is a semipositive matrix, where and −(K + Q)  K = Diag

2 1 2 1 2 1 − , − ,··· , − w1 γ1 w2 γ2 wN γN

 .

Then Pt has a unique invariant probability measure μ such that μk  μV for all k ∈ S. Moreover, each √ dμk of the densities ρk := dμ has a continuous, positive version such that ρk ∈ Hσ1,2 (Rd ) and for all p > 0, k V 1,p ρk ∈ Wloc (Rd ).

¯ will be used in Lemma 3.5 to derive Remark 1.3. The inequality (1.10) and the semipositivity of −(K + Q) the entropy estimate on the density of the invariant measure. This estimate is crucial to the proof of Theorem 1.1. A matrix is called semipositive if and only if it is a nonsingular M-matrix, which is also equivalent to that all of the principal minors of the matrix are positive, see [15, Proposition 2.4] or [2] for instance. Hence, we ¯ is semipositive by calculating the principal minors of −(K + Q). ¯ can verify that −(K + Q) Theorem 1.1 implies that the class N 

N N N      ρk μV (dx)δk  ρk ≥ 0, μV (ρk ) = 1, μV (ρk Pt f (·, k)) = μV (ρk fk )

k=1

k=1

k=1

for t ≥ 0, f ∈ B (R × S) d

k=1



contains only one element. According to [20, Theorem 2.1], (H1) and (1.10) yields that for each i, the SDE (1.5) with generator LZ i has a non-explosive strong solution. By (1.3), N < ∞ and (1.8), we have  x∈Rd ,i∈S

supp(h(x, i, ·)) ⊂





x∈Rd ,i∈S

j=i

Γij (x) ⊂ [0, N CQ ].

Then as showed in [8,22] that (1.4) has a non-explosive strong solution. √ Remark 1.4. Comparing with [20, (2.3)], there is extra 2 in (1.6), so w2i refers to the constant λ in [20, 2 Theorem 2.3]. It follows from [20, Theorem 2.3] that if μV (ewi |Zi | ) < ∞ for w2i > γ4i , i.e. wi > γ2i , then LZ i

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has an invariant measure. However, some wi could be small such that wi < γ2i in (1.10), see the following concrete example. So not all diffusion processes Xti satisfy the integrability conditions in [20, Theorem 2.2] and [21, Theorem 1.1(2) or Theorem 5.2]. To illustrate that Theorem 1.1 can be applied to the regime-switching diffusion process that not all the diffusions in all the environments has an invariant probability measure, we present the following example. In this example, the diffusion process in the state i = 2 is singular and the Lyapunov condition does not hold, so it cannot be covered by [15]. Example 1.2. Let N = 2, d = 1, b1 (x) = −x, b2 (x) = −x +

√

 2 δx +

∞  

1 log |x| − n

n=0

 θ21

 1(n,n+1] (|x|) ,

where θ1 ∈ (0, 1) and δ is a positive undetermined constant, σ1 (x) = σ2 (x) = 1, and  Q(x) =



−a b

a −b

 +

−a(x) a(x) b(x) −b(x)



1+a+b with a > 0, b > 0, and |a(x)| ≤ θ2 a, |b(x)| ≤ θ2 b for some θ2 ∈ [0, a+b+4ab ). Then for δ satisfies

 δ<

1 + (1 − θ2 )(a + b) − 4θ2 ab , 2(1 + (1 − θ2 )a)

(1.11)

all the conditions of Theorem 1.1 hold. Moreover, if 1 + 2b + a a + 2b + 8ab

θ2 <

(1.12)

√ holds in addition, then there is δ > 0 such that 2δ − 1 ≥ 0. With this δ, the diffusion process in the state i = 2 is not recurrent, consequently, it does not have an invariant probability measure. Proof. Let V (x) = c − 12 x2 for some c > 0. Then Z1 = 0 and Z2 = δx +

∞ 

 log

n=0

1 |x| − n

 θ21

1(n,n+1] (|x|),

and it is clear that (H1)-(H3) hold with γ1 = γ2 = 2, β1 = β2 = 0. 2 Next, we shall check our integrability conditions. For all w1 > 0, μV (ew1 |Z1 | ) < ∞. Since θ 1 ∞  1 w2 (1 + )  log w2 |Z2 | ≤ w2 ( + 1)δ x + 1(n,n+1] (|x|) |x| − n n=0 2

2 2

≤ w2 ( + 1)δ 2 x2 −

∞  

log

 |x| − n 1(n,n+1] (|x|) + C(w2 , , θ1 )

n=0

for any > 0 and some constant C(w2 , , θ1 ) > 0, we have μV (ew2 |Z2 | ) ≤ eC(w2 ,,θ1 ) 2

∞ 



n=0 n<|x|≤n+1

1  2 2 1  e− 2 −w2 (+1)δ x dx. |x| − n

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

Then, for0 < w2 < 2δ12 , there is >0 such that ¯ = −(1 − θ2 )a (1 + θ2 )a . Then Let Q (1 + θ2 )b −(1 − θ2 )b  ¯ = −(K + Q)

1 2

661

− w2 ( + 1)δ 2 > 0, which implies that μV (ew2 |Z2 | ) < ∞.

(1 − θ2 )a + 1 − −(1 + θ2 )b

2

1 w1

−(1 + θ2 )a (1 − θ2 )b + 1 −

 1 w2

.

¯ is semipositive if and only if all of the principal minors of −(K + Q) ¯ are By [15, Proposition 2.4], −(K + Q) positive, i.e. 

1 + (1 − θ2 )a − w11 > 0 1 + (1 − θ2 )a − w11 1 + (1 − θ2 )b −

1 w2



− (1 + θ2 )2 ab > 0.

(1.13)

If δ and θ2 satisfy (1.11), then there is w2 > 0 such that 1 + (1 − θ2 )b −

1 + (1 − θ2 )(a + b) − 4θab 1 (1 + θ2 )2 ab = > > 2δ 2 1 + (1 − θ2 )a 1 + (1 − θ)a w2

which implies that there is w1 > 0 such that 1 + (1 − θ2 )b −

1 (1 + θ2 )2 ab > w2 1 + (1 − θ2 )a −

1 w1

.

Thus (1.13) and (1.10) holds for some positive w1 , w2 . Moreover, if (1.12) holds in addition, then 1 1 + (1 − θ2 )(a + b) − 4θ2 ab > , 1 + (1 − θ2 )a 2 which implies that there is δ such that

√ 2δ − 1 ≥ 0. 2

If Pt has an invariant probability measure μ, then Pt can be extended to a contraction semigroup in L1 (Rd ×S, μ). Consequently, the weakly coupled elliptic system (L, Cc2 (Rd ×S)) has an extension in L1 (Rd × S, μ) which generates a C0 -semigroup. For the uniqueness of the extension in L1 (Rd × S, μ), we present the following result, where μ is introduced in Theorem 1.1. Theorem 1.3. Under the assumption of Theorem 1.1, if in addition there exists > 0 such that max μV (e||σi || ) < ∞, 2

1≤i≤N

then there is only one extension of (L, Cc2 (Rd × S)) that generates a C0 -semigroup in L1 (Rd × S, μ). The rest of the paper is organized as follows. In Section 2, we shall study an elliptic system corresponding to invariant measures, called the weak coupled system of measures of (1.4), where some local a priori estimates of measures will be presented. In Section 3, the entropy estimate of the density of invariant probability measure will be concern about. In the final section, we shall prove our main results. To make the article more concise, we shall use the following notations in the rest parts of the paper. q (N1) For all q ∈ [1, ∞], q ∗ := 1[1
662

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

(N2) For a Borel measurable function f on Rd × S, despite f (x, k), we also denote by fk (x) the value of f 

1/2 N 2 at (x, k). |f |(x) := f (x) . k=1 k (N3) For a measure μ on Rd × S, we define μk (dx) = μ(dx × {k}). 2. Regularity of weak coupled system of measures In this section, we allow N = ∞. Let μ be an invariant probability measure of Pt on B (Rd × S). Then for a smooth function f on Rd × S with compact support (there is M ≥ 0 such that f (x, i) = 0 if |x| ≥ M and i ≥ M ), Itô’s formula yields that t Pt f (x, k) = f (x, k) +

Ps Lf (x, k)ds. 0

Since μ is an invariant measure, t μ(f ) = μ(Pt f ) = μ(f ) +

t μ(Ps Lf )ds = μ(f ) +

0

μ(Lf )ds. 0

Then t 0=

μ(Lf )ds, t > 0 0

and so μ(Lf ) = 0. Hence, to study the invariant measure of Pt , we shall start from the equation μ(Lf ) = 0. Since μ is a measure on Rd × S, we can view μ as a system of measures (μi )i∈S , where μi is defined as in (N3). Thus the equation μ(Lf ) = 0 is a system of elliptic equations for measures, called weakly coupled elliptic system of measures. Elliptic and parabolic equations for measures have been intensively studied, and results on equations for measures are important to the study of invariant probability measures, see [3,4]. Here, we shall extend some results in [3] to the case of weakly coupled elliptic system of measures. Lemma 2.1. Assume that (H1) holds and N ∈ N ∪ {∞}. Let p ∈ (d, ∞) ∩ [2, ∞) and μ be a probability measure on Rd × S such that μ(Lf ) = 0, f ∈ Cc2 (Rd × S). Suppose in addition that for all j ∈ S, Zj ∈ Lploc (Rd ) ∩ Lloc (Rd , dμj ), and qj (x) is locally bounded and there exists δ(j) ∈ N such that qkj (x) = 0, |k − j| > δ(j). 1,p Then for each j ∈ S, there is a continuous function ρˆj ∈ Wloc (Rd ) such that μj (dx) = ρˆj (x)dx.

Proof. For i ∈ S, let f (x, k) = f (x)1[k=i] + 01[k=i] , and ν(dx) = −

 k=i

qki (x)μk (dx).

(2.1)

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Then 





 LZ i f (x) − qi (x)f (x) μi (dx) =

Rd

f (x)ν(dx).

(2.2)

Rd ∗

According to [3, Corollary 2.3], μi has a density ρˆi ∈ Ldloc (Rd ) and ||ρi ||Ld∗ (O) is bounded by a constant depending on ||Zi ||L(O,dμi ) for any bounded domain O. Then 



LZ i f (x)

⎛





f (x) ⎝−

− qi (x)f (x) μi (dx) =

Rd



⎞ qki (x)ˆ ρk (x)⎠ dx.

(2.3)

k=i

Rd

Let Z˜kj (x) = Zk0,j (x) +

d √  2 σkjl (x)Zkl (x), x ∈ Rd , k ∈ S, 1 ≤ j ≤ d. l=1

For all f ∈ Cc2 (Rd × S). By μ(Lf ) = 0 and the definition of ρˆk , we obtain that N   k=1 d R

⎡ ⎣

d 

aij k (x)∂i ∂j fk (x)

+

i,j=1

d 

Z˜kj ∂j fk (x)

+

j=1

N 

⎤ qkj (x)fj (x)⎦ ρˆk (x)dx = 0.

(2.4)

j=1

Since p > d, we have p∗ < d∗ . Then for any r ∈ (p∗ , d∗ ] and q ≥ q∗ q∗ + ≤ 1, p r

r∗ p p−r ∗ ,

we have

q ∗ < r,

∗

which implies that (|Z˜k | + 1)ˆ ρk ∈ Lqloc (Rd ) for all k if ρˆk ∈ Lrloc (Rd ). Since f ∈ Cc2 (Rd × S), there are m ∈ N # such that fk = 0, k > m and a bounded domain O ⊂ Rd such that k≤m suppfk ⊂ O. Let βm (x) =

 j≤m

max

|k−j|≤δ(j)

|qkj |2 (x),

M = max(j + δ(j)). j≤m

Then (2.4) implies that  ⎤  ⎡ N   d    ij  ⎦ ⎣ ak (x)∂i ∂j fk (x)ˆ ρk (x) dx  k=1  i,j=1 O

≤

N   k=1 O

⎛

≤⎝

|f |(x)βm (x)

∗

(|Z˜k |ˆ ρk )q (x)dx⎠



 ρˆk (x) dx

⎝

N  

⎞ q1 |∇fk |q (x)dx⎠

(2.5)

k=1 O q∗ 2

βm (x) O

M 

k=1

⎞ q1∗ ⎛

k=1 O

+⎣

1 2

O

M  

⎡



 |Z˜k ||∇fk |ˆ ρk (x)dx +



M 

k=1

q∗ ρˆk (x)

⎤ q1∗ ⎛ dx⎦

⎝

 O

⎞ q1 |f |q (x)dx⎠

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⎛ ≤C⎝

N  

⎞ q1 |∇fk |q (x)dx⎠ ,

k=1 O

where the last inequality we have used the Poincaré inequality (see [1, Theorem 6.30]), and the constant C depends on O, βm , M , ||Z˜k ||Lp (O) , ||ˆ ρk ||Lr (O) , λk , k = 1, · · · , M . Fix some k0 . Let fk = fk0 if k = k0 and fk = 0 otherwise. Then  ⎡ ⎛ ⎤  ⎞ q1    d     ⎣ aij ρk0 (x)⎦ dx ≤ C ⎝ |∇fk0 |q (x)dx⎠ . k0 (x)∂i ∂j fk0 (x)ˆ    i,j=1 O

(2.6)

O

Next, we shall adapt a procedure introduced in [3]. Let x0 ∈ Rd , R > 0, and BR (x0 ) be the open ball with center x0 and radius R. Let η ∈ Cc∞ (BR (x0 )). Then  ⎤    ⎡ d    ij   ⎣ ak0 (x)∂i ∂j fk0 (x)(η ρˆk0 )(x)⎦ dx     BR (x0 ) i,j=1  ⎤    ⎡ d    ij   ⎣ ≤ ak0 (x)∂i ∂j (ηfk0 )(x)ˆ ρk0 (x)⎦ dx   BR (x0 ) i,j=1   ⎤    ⎡ d    ij   ⎣ + ak0 (x)∂i ∂j η(x)(fk0 (x)ˆ ρk0 )(x)⎦ dx   BR (x0 ) i,j=1    ⎤    ⎡ d      ⎦ ⎣ + 2 aij (x)∂ η∂ f (x)ˆ ρ (x) dx  i j k0 k0 k0   BR (x0 ) i,j=1  ⎛ ⎜ ≤C⎝



(2.7)

⎞ q1 ⎟ |∇(ηfk0 )|q (x) + |fk0 |q (x)dx⎠

BR (x0 )

⎛ ⎜ ≤C⎝



⎞ q1 ⎟ |∇fk0 |q (x)dx⎠ ,

BR (x0 )

the constant C depends on BR , βm , M , η, ||Z˜k ||Lp (BR ) , ||ˆ ρk ||Lr (BR ) , λk , k = 1, · · · , M . By (2.7) and [3, ∗ pr Theorem 2.7], for small enough R (independent of x0 ), η ρˆk0 ∈ W01,q (BR ) with 1 ≤ q ∗ ≤ p+r . The point pr x0 is arbitrary. According to Sobolev embedding theorem, if p+r ≥ d, then ρˆk0 is local bounded, and if pr 1 ˆk0 ∈ Lrloc (Rd ) with p+r < d, then ρ r1 =

prd prd pr = d − p+r pd − r(p − d)

which implies that pd pd r1 = > > 1. r pd − r(p − d) pd − p∗ (p − d)

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Starting from r0 = r ∈ (p∗ , d∗ ], and taking into account that x0 is arbitrary and R is independent of x0 , 1,

prn

we can get a sequence {rn } by this procedure such that ρˆk0 ∈ Wlocp+rn (Rd ) until

prn p+rn

≥ d. Indeed, we can

1,l obtain that ρˆk0 ∈ Wloc (Rd ) with some l > d. Then ρˆk0 has a continuous and local bounded version due to Sobolev embedding theorem. From (2.5), (2.6), |Z˜k | ∈ Lploc (Rd ) and that k0 is arbitrary, we obtain for f ∈ Cc2 (Rd × S)

 ⎛ ⎤  ⎞ p1∗ ⎡ N   d N      ∗  ⎣ aij ρk (x)⎦ dx ≤ C ⎝ |∇fk |p (x)dx⎠ k (x)∂i ∂j fk (x)ˆ  k=1  i,j=1 k=1 O

O

with constant C depends on O, βm , M , ||Z˜k ||Lp (O) , ||ˆ ρk ||Lr (O) , λk , k = 1, · · · , M . According to [3, Theo1,p (Rd ), and ||ˆ ρk ||H 1,2 (O) bounded by a constant depends on p, d, O, rem 2.7], we also obtain that ρˆk ∈ Wloc ||Z˜k ||Lp (O) , ||ˆ ρk ||Ld∗ (O) , λk , ak , M , βm . 2 Next, we shall prove the following weak Harnack inequality, which is crucial to prove that ρk := e−V ρˆk is positive. The proof mainly follows the line of [12, Theorem 4.15] indeed. Lemma 2.2. Under the assumption of Lemma 2.1, fixing k ∈ S, for all 0 < r < R < ∞, it holds that ⎛ ess inf ρk ≥ C ⎝



Br

⎞ q1 ρqk dx⎠ , 0 < q <

BR

d , d−2

where C depends on σk , eV , ||qk ||L p2 (B ) , ||Zk ||Lp (BR ) , d, p, r, R, q. R

1,p Proof. By Lemma 2.1, ρk ∈ Wloc (Rd ) is locally bounded. (2.3) and ρˆk = ρk eV imply that

 (LZ k

− qk )f (x)ρk e

Rd

V (x)

dx = −

 

Rd

qjk (x)ρj (x)f (x)eV dx, f ∈ Cc2 (Rd ).

j=k

Then by the integration by part formula, we have 

σk∗ ∇f (x), σk∗ ∇ρk (x)eV (x) dx +

Rd

 qk (x)f (x)ρk (x)eV (x) dx

Rd

  √  = 2 σk Zk , ∇f (x)ρk (x)eV dx + f (x) qjk (x)ρj (x)eV dx Rd

Rd

(2.8)

j=k

√  ≥ 2 σk Zk , ∇f (x)ρk (x)eV dx, f ≥ 0, f ∈ Cc2 (Rd ). Rd ∗

1,p 1,p Since ρk ∈ Wloc (Rd ), Zk ∈ Lploc (Rd ) and (2.1), (2.8) holds for f ∈ Wloc (Rd ) with compact support. Since p ≥ 2, p ≥ p∗ . Then we can replace f by some cutoff function related to ρk . The remainder of the proof will start from this inequality. Fix any bounded open domain O ⊂ Rd . Let ψ ∈ Cc2 (Rd ) with suppψ ⊂ O, and let δ > 0, β > 0 and f = ψ 2 (ρk + δ)−(β+1) . Then

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

666



σk∗ ∇ψ, σk∗ ∇ρk ψ V (x) e dx − (β + 1) (ρk + δ)β+1

2 Rd



|σk∗ ∇ρk |2 ψ 2 V (x) e dx (ρk + δ)β+2

Rd



√  σk Zk , ∇ψψρk V (x) qk (x)ψ 2 (x)ρk V (x) e dx + 2 2 e dx (ρk + δ)β+1 (ρk + δ)β+1

≥− Rd

Rd

√  σk Zk , ∇ρk ψ 2 ρk V (x) − (β + 1) 2 e dx. (ρk + δ)β+1 Rd

Thus 4(β + 1) β2

β

|σk∗ ∇(ρk + δ)− 2 |2 ψ 2 eV (x) dx

Rd



4 β

≤



β

β

σk∗ ∇ψ, σk∗ ∇(ρk + δ)− 2 ψ(ρk + δ)− 2 eV (x) dx +

Rd

qk (x)ψ 2 V (x) e dx (ρk + δ)β

Rd

√  |Zk ||σk∗ ∇ψ||ψ| V (x) +2 2 e dx (ρk + δ)β √



Rd

2 2(β + 1) + β



β

|Zk |ψ 2 |σk∗ ∇(ρk + δ)− 2 | β

(ρk + δ) 2

Rd

eV (x) dx.

β

Let w = (ρk + δ)− 2 . Then from the inequality above, we can obtain that 

|σk∗ ∇w|2 ψ 2 eV (x) dx

Rd

≤

β 4(β + 1)



|σk∗ ∇ψ||σk∗ ∇w|ψweV (x) dx +

β2 4(β + 1)

Rd

+√

2

β 2(β + 1)



β |Zk ||σk∗ ∇ψ|ψw2 eV (x) dx + √ 2

Rd



 qk (x)ψ 2 w2 eV dx Rd

|Zk ||σk∗ ∇w|ψ 2 weV (x) dx.

Rd

Thus 

|σk∗ ∇w|2 ψ 2 eV dx ≤ (1 + β)

Rd

 + 2β



|σk∗ ∇ψ|2 w2 eV dx + 2β

Rd

 qk (x)ψ 2 w2 eV (x) dx

Rd

|Zk |2 ψ 2 w2 eV dx.

Rd

Let oscO eV =

supO eV inf O eV

and ΛO = supO ||σk∗ ||2 . Then we get that

 |∇(wψ)|2 dx Rd



≤ C(λk , oscO eV , ΛO )(1 + β) Rd



 |∇ψ|2 + qk (x)ψ 2 + |Zk |2 ψ 2 w2 dx

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688



≤ C(λk , oscO e , ΛO )(1 + β) V

667

|∇ψ|2 w2 dx

Rd

⎛



+⎝

⎞ p2 ⎛ (qk (x) + |Zk |2 ) 2 dx⎠ ⎝ p

O



⎞ p−2 p (ψw) p−2 dx⎠ 2p

 .

(2.9)

Rd

Since p > d, Hölder inequality and Sobolev inequality imply that for all > 0 ||ψw||

2p L p−2

(O)

≤ ||ψw||

+ C(d, p) − p−d ||ψw||L2 (O) d

2d

L d−1 (O)

≤ C(d, O)||∇(ψw)||L2 (O) + C(d, p) − p−d ||ψw||L2 (O) . d

Substituting this into (2.9) and choosing ⎛ = √

1 ⎝ 2C(d, O)



⎞− p1



(qk (x) + |Zk |2 ) dx⎠ p 2

− 12

C(λk , oscO eV , ΛO )(1 + β)

,

O

we obtain that  |∇(wψ)| dx ≤ C(1 + β) 2

Rd



p p−d

  |∇ψ|2 + ψ 2 w2 dx

Rd

with some C depending on λk , oscO eV , ΛO , ||qk ||L p2 (O) , ||Zk ||Lp (O) , p, d. By the Sobolev embedding theorem, ⎛ ⎝



⎞ γ1



(ψw) dx⎠ ≤ C(1 + β) p−d p

2γ

Rd



 |∇ψ|2 + ψ 2 w2 dx

Rd

d where γ = d−2 for d ≥ 3 and γ > 2 for d = 2, 1. Choosing ψ such that 1Br1 ≤ ψ ≤ 1Br2 and |∇ψ| ≤ where Br1 and Br2 are some balls in O with radius r1 and r2 , we arrive at

⎛ ⎜ ⎝



Br1

⎞ γ1

p

C(1 + β) p−d ⎟ w2γ dx⎠ ≤ (r2 − r1 )2

2 r2 −r1 ,

 w2 dx. Br2

Thus ⎛ ⎜ ⎝



Br1

1 ⎞ βγ

⎟ (ρk + δ)−βγ dx⎠

 ≤

p p−d

C(1 + β) (r2 − r1 )2

 β1

⎛ ⎜ ⎝



⎞ β1 ⎟ (ρk + δ)−β dx⎠ ,

(2.10)

Br2

where C depending on λk , oscO eV , ΛO , ||qk ||L p2 (O) , ||Zk ||Lp (O) , p, d. Hence, by using (2.10), we can get the inverse Hölder inequality for (ρk + δ)−1 by iteration (see [12] for instance)

||(ρk + δ)−1 ||Lp1 (Br1 ) ≤ C||(ρk + δ)−1 ||Lp2 (Br2 )

(2.11)

with any p1 > p2 > 0 and Br1 ⊂ Br2 , where the constant C depends on λk , oscBr2 eV , ΛBr2 , ||qk ||L p2 (B ) , ||Zk ||Lp (Br2 ) , d, p, r1 , r2 , p1 , p2 . For any positive r, R with r < R and β > 0, we set O = BR and r2

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

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i i define ri = R − j=1 R−r 2j and pi = βγ for any i ∈ N with r0 = R and p0 = β. Then by iteration again, we get from (2.10) that

||(ρk + δ)−1 ||Lpk (Brk ) ≤

k & i=1





p

C(1 + βγ i−1 ) p−d (ri−1 − ri )2

1 βγ i−1

||(ρk + δ)−1 ||Lβ (BR )

(2.12)

where C depending on λk , oscBR eV , ΛBR , ||qk ||L p2 (B ) , ||Zk ||Lp (BR ) , p, d. Because R

∞ & i=1



1  i−1 p C(1 + βγ i−1 ) p−d βγ (ri−1 − ri )2 ∞ p ' C  p−d log(1 + βγ i−1 ) + (i + 1) log 2 + log R−r = exp βγ i−1 i=1

< ∞, we get from (2.12) by letting i → ∞ that ess inf ρk + δ ≥ C||(ρk + δ)−1 ||−1 Lβ (BR )

(2.13)

Br

with some C depending on λk , oscBR eV , ΛBR , ||qk ||L p2 (B ) , ||Zk ||Lp (BR ) , p, d, β. R

Next, we shall prove that for all 0 < q < 

d d−2

and R > 0, there is C > 0 such that

(ρk + δ)−q dx

BR

 (ρk + δ)q dx ≤ C.

(2.14)

BR

Let f = (ρk + δ)−1 ψ 2 in (2.8) and w = log(rhk + δ). Then 

|σk∗ ∇w|2 ψ 2 eV dx ≤

Rd



|σk∗ ∇ψ||σk∗ ∇ψ|ψeV dx +

√  2 |Zk ||∇w|ψ 2 eV dx

Rd

Rd

 √  + 2 |Zk ||∇ψ|ψeV dx + qk (x)ψ 2 dx. Rd

Rd

Hence 

|σk∗ ∇w|2 ψ 2 eV dx ≤ 4

Rd



|σk∗ ∇ψ|2 ψeV dx + 8

Rd

 (|Zk |2 + qk (x))ψ 2 eV dx.

Rd

Choosing ψ such that 1Br ≤ ψ ≤ 1B2r and |∇ψ| ≤ 2r , we obtain that there is a constant C > 0 depending on λk , ΛB2r , B2r such that  Br

Since

⎛ |∇w|2 dx ≤ C ⎝rd−2 +



B2r





⎞

|Zk |2 + qk (x) dx⎠ .

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688





⎛

  |Zk |2 + qk (x) dx ≤ |Zk |2 + qk (x) 

d

L 2 (B2r )

B2r

=2

d−2 d−2

r



⎝

  |Zk |2 + qk (x)

669

⎞ d−2 d dx⎠

B2r d

L 2 (B2r )

,

we have that  |∇w|2 dx ≤ Crd−2 Br

with C depends on λk , ΛB2r , ||qk ||

d

L 2 (B2r )

, ||Zk ||Ld (B2r ) , d. As in [12, Theorem 4.14], the Poincaré inequality

on B2r and John-Nirenberg Lemma implies that there exist p0 > 0 and C > 0 depending on d, r such that 

¯ ep0 |w−w| dx ≤ C

Br

where w ¯=

1 |Br |

Br

wdx. Hence  (ρk + δ) Br

−p0

 dx

 p0

(ρk + δ) dx =

Br

e

−p0 w

Br

 dx Br

⎛



≤⎝

ep0 w dx ⎞2

¯ ep0 |w−w| dx⎠

(2.15)

Br

≤C . 2

To prove (2.14), let f = ψ 2 (ρk + δ)−β , β ∈ (0, 1). Then, as above, we can prove the following inverse Hölder inequality ||ρk + δ||Lp1 (Br1 ) ≤ C||ρk + δ||Lp2 (Br2 ) , 0 < r1 < r2 , 0 < p2 < p1 < where the constant C depends on λk , oscBr2 eV , ΛBr2 , ||qk ||L p2 (B

r2 )

d d−2

, ||Zk ||Lp (Br2 ) , d, p, r1 , r2 . Combining

this with (2.15) and (2.11), (2.14) is proved. d Finally, (2.13) and (2.14) yield that for any q ∈ (0, d−2 ) ess inf ρk + δ ≥ C||(ρk + δ)−1 ||−1 Lq (Br Br1

2)

 −1 ≥ C ||(ρk + δ)−1 ||Lq (Br2 ) ||(ρk + δ)||Lq (Br2 ) ||(ρk + δ)||Lq (Br2 ) ≥ C||(ρk + δ)||Lq (Br2 ) .

Letting δ → 0+ , we obtain the weak Harnack inequality. 2 Corollary 2.3. Assume that Q is fully coupled and the hypotheses of Lemma 2.1 hold, then ρk > 0 for all k ∈ S. Proof. The weakly Harnack inequality implies that if μk is a positive measure, then ρˆk is positive on Rd . Let S1 be the set of all j such that ρj ≡ 0, and S2 = S − S1 . Then ρˆk > 0 for all k ∈ S2 . Since μ is a

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

670

probability measure, there exist k ∈ S such that μk is a positive measure, which implies that S2 is not empty. If S1 is non-empty, then it follows from (2.3) that 

qki ρˆk =



qki ρˆk = 0, a.e.,

i ∈ S1 .

k=i

k=i,k∈S2

Since for each k ∈ S2 , ρˆk > 0, we have i ∈ S1 , k ∈ S2 .

qki = 0, a.e., It contradicts that Q is fully coupled on Rd .

2

3. Entropy estimates of ρ In this section, we shall study the entropy estimate of the density ρ = (ρ1 , · · · , ρN ), and assume that N < ∞ throughout this section. We shall first prove a Girsanov’s theorem, which will be used to prove the existence of the invariant measure. Let (Xt , Λt ) be the solution of the following equation (1)

dXt = bΛt (Xt )dt +

√

2gΛt (Xt )dWt , X0 = ξ

(3.1)

∞ dΛt =

h(Xt , Λt− , z)N (dz, dt), Λ0 = λ,

(3.2)

0

and let t ˜ t = Wt − W

(2)

bΛs (Xs )ds, 0

 t (  ) 1 t (2) (2) |bΛs (Xs )|2 ds . Rt = exp bΛs (Xs ), dWs − 2 0

0

Then we have that following Girsanov’s theorem. Lemma 3.1. Suppose that (3.1)-(3.2) has a pathwise unique non-explosive solution, and {Rt }t∈[0,T ] is a martingale and ⎛ T ⎞  (2) P ⎝ |gΛs (Xs )b (Xs )|2 ds < ∞⎠ = 1. Λs

0

˜ t }t∈[0,T ] is a Brownian process and {pt }t≥0 is a Poisson point process with intensity Then, under RT P , {W dzdt such that they are mutually independent. Consequently, the following equation has a weak solution (1)

dXt = bΛt (Xt )dt +

√

2gΛt (Xt )dWt +

√

∞ dΛt =

h(Xt , Λt− , z)N (dz, dt), Λ0 = λ. 0

(2)

2gΛt (Xt )bΛt (Xt )dt, X0 = ξ

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671

˜ t is a Brownian motion due to Girsanov’s theorem. If we can prove Proof. {Rt }t∈[0,T ] is a martingale, W that the compensator of N (dz, dt) under RT P is dzdt, which is a non-random measure, then according to ˜ t }t≥0 under RT P . [13, Theorem I.6.3], {pt }t≥0 is an adapted Poisson point process and independent of {W Let K be a measurable subset of (0, ∞) with its Lebesgue measure |K| < ∞, N (K, t) := N (K, (0, t]), t ∈ [0, T ]. Then  + *  E RT (N (K, t) − |K|t) Fs  +  +  + * * *    = −E |K|tRT Fs + E N (K, s)RT Fs + E (N (K, t) − N (K, s)) RT Fs  + *  = (N (K, s) − |K|t) Rs + E (N (K, t) − N (K, s)) Rt Fs  + *  = (N (K, s) − |K|t) Rs + Rs E (N (K, t) − N (K, s)) Rs,t Fs , where Rs,t = exp

 t (  ) 1 t (2) (2) |bΛr (Xr )|2 dr . bΛr (Xr ), dWr − 2 s

s

  Let G = Fs ∨ σ N (U, r)  r ≤ t, U ∈ B ((0, ∞)) . Then 

 + * * * + +  E (N (K, t) − N (K, s)) Rs,t Fs = E (N (K, t) − N (K, s)) E Rs,t |G Fs . Let Π[s,t] be the totality of point functions on [s, t] for s ≤ t, and let Ds,t (S) be all the cadlag function on [s, t] with value in S. Since the equation has a unique non-explosive strong solution, there exists F : Rd × S × C([s, t], Rd ) × Π[s,t] → C([s, t], Rd ) × Ds,t (S) such that (Xr , Λr ) = F (Xs , Λs , W[s,t] , p[s,t] )r , r ∈ [s, t]. Since {Wr − Ws }r≥s is independent of G while (Xs , Λs , p[s,t] ) is G -measurable, we have  + *  t ( )  E Rs,t G = E exp b(2) (F (Xs , Λs , W[s,t] , p[s,t] )r ), dWr *

s

−

1 2

t

 +  |b(2) (F (Xs , Λs , W[s,t] , p[s,t] )r )|2 dr G

s

 t ( = E exp

(b(2) (F (x, k, W[s,t] , p[s,t] )r ), dWr

)

s

−

1 2

t

  |b(2) (F (x, k, W[s,t] , p[s,t] )r )|2 dr 

s

Combining with ERs,t

≤ 1, a.s.  +  *  = E E Rs,t G = 1, we have that  + *  E Rs,t G = 1,

a.s.

(x,k,p)=(Xs ,Λs ,p)

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

672

Since Fs ⊂ G , we obtain that  + * * + + *   E (N (K, t) − N (K, s)) E Rs,t |G Fs = E N (K, t) − N (K, s)Fs = |K|(t − s). Then *   + * +  E RT N (K, t) − |K|t Fs = Rs N (K, s) − |K|s . Thus for A ∈ Fs 

   (N (K, t) − |K|t) 1A = E RT (N (K, t) − |K|t) 1A  * +   * +  = E Rs N (K, s) − |K|s 1A = E RT N (K, s) − |K|s 1A * +  = ERT N (K, s) − |K|s 1A .

E RT

Therefore  + *  ERT N (K, t) − |K|t Fs = N (K, s) − |K|s, which implies that the compensator of N (dz, dt) under RT P is dzdt.

2

  By (H1), for each i ∈ S, L0i , Cc2 (Rd ) has a unique self-adjoint extension in L2 (μV ) such that the semigroup generated by the extension is Markovian. We shall denote by L0i this extended operator, and denote by Pti the associated symmetric Markov semigroup. Let π = (πi )i ∈ S with πi > 0 for all i ∈ S, and let Pt0 be the Markov semigroup generated by the Dirichlet form E 0 (see Remark 1.1 for the definition of E 0 ) and L0 the generator of Pt0 . Let μπ,V be a probability measure on Rd × S defined as follows μπ,V =

N 

πk μV (dx)δk .

k=1

Since μV is the invariant probability measure of Pti , μπ,V is the invariant probability measure of Pt0 .   Moreover, E 0 can be identified with the quadratic form generated by the operator Diag L01 , · · · , L0N in the ,N weight Hilbert space i=1 L2 (μV ) with the following inner produce (f, g) →

N  i=1

πi μV (fi gi ), f, g ∈

N &

L2 (μV ).

i=1

Then for f ∈ L2 (Rd × S, μπ,V ), (Pt0 )f (x, i) = Pti fi (x), x ∈ Rd , i ∈ S. Moreover, Pt0 can be extended to a contrastive C0 -semigroup in Lp (Rd × S, μπ,V ), p ∈ [1, ∞]. By (H2), ⎛  p ⎛ ⎞p−1 ⎞  p  N   N N N   ⎜ ⎝ |qij | p−1 (x) ⎠ ⎟  πi  qij (x)fj (x) μV (dx) ≤ sup ⎝ πi πj fjp (x)μV (dx). ⎠ 1 d p x∈R   πj i=1 j=1 i=1 j=1 d d j=1

N 

R

R

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

673

Thus Q is a bounded operator in Lp (Rd × S, μπ,V ) for all p ∈ [1, ∞]. Let LQ be defined as follows (LQ f )(x, i) = (L0i fi )(x) +

N 

qij (x)fj (x), f ∈ Cc2 (Rd × S).

j=1

Then LQ generates a unique contractive C0 -semigroup in Lp (Rd × S, μπ,V ), say PtQ . By (H1), (H2), and for all f ∈ Cc1 (Rd × S) we have  i

=

πi



 j





i



πi

. qij (fj+ ∧ 1)fi− + (fi+ ∧ 1)(fi − 1)+

j=i

− 



πi

i

πi

i

j=i

πi qi (fi+ ∧ 1)(fi − 1)+

i

i

≥

.   qij (fj+ ∧ 1)fi− − πi qij (fj+ ∧ 1)(fi − 1)+

j=i

+ =



πi

i

qij (fj+ ∧ 1)(fi − fi+ ∧ 1)





qij (fj+ ∧ 1)(fi − 1)+

j=i

qij (1 − fj+ ∧ 1)(fi − 1)+ ≥ 0.

(3.3)

j=i

Then, according to the proof of [6, Theorem 3.1], PtQ is also a Markov semigroup. Moreover, we have the following lemma on PtQ . Lemma 3.2. For t > 0, PtQ is hyperbounded, and 4t

||PtQ ||Lq (μπ,V )→Lq(t) (μπ,V ) < ∞, q > 1, q(t) = 1 + (q − 1)e γ , recalling that γ = max γi . 1≤i≤N

Proof. By [14, Theorem 3.1.1], we have t PtQ f

=

Pt0 f

+

  0 Pt−s QPsQ f ds, f ∈ L∞ (Rd × S, μπ,V ).

0

Then for all p ≥ 1 ⎛ μπ,V (|PtQ f |p ) ≤ 2p−1 ⎝μπ,V (|Pt0 f |p ) + μπ,V

p ⎞⎞ ⎛ t     0 Q  ⎠⎠ ⎝ Pt−s QP f ds s     0

t ≤2

p−1

μπ,V (|Pt0 f |p )

μπ,V (|QPsQ f |p )ds

p−1 p−1

+2

t

0

t ≤2

p−1

μπ,V (|Pt0 f |p )

p−1 p−1

+2

t

μπ,V (|PsQ f |p )ds,

bp 0

(3.4)

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

674

where bp = ||Q||pLp . Gronwall’s inequality yields p−1 p

μπ,V (|PtQ f |p ) ≤ 2p−1 e2

t bp

μπ,V (|Pt0 f |p ).

According to [19, Theorem 5.1.4], (H3) and Remark 1.1 imply that Pt0 is hyperbounded: ||Pt0 ||Lq (μπ,V )→Lq(t) (μπ,V )

0 /  1 1 − , t > 0, q > 1, ≤ exp β q q(t)

4t

where q(t) := 1 + (q − 1)e γ and β = maxk (βk − log πk ). Thus 

μπ,V (|PtQ f |q(t) )

1 q(t)

 ≤2

1 1− q(t)

exp

β(q(t) − q) 2q(t)−1 tq(t) bq(t) + qq(t) q(t)

 1

(μπ,V (f q )) q .

2

Let (Xt , Λt ) be the process generated by LQ , which satisfies the stochastic differential (1.4) with bk (x) = k ∈ S, x ∈ Rd . The following lemma is a similar result of [21, Theorem 4.1] for (1.4).

Zk0 (x),

4

Lemma 3.3. Assume (H1)-(H3). Let p1 = 12 (1 + e γ ) and  c1 = where b1 = ||Q||L1 (μπ,V ) . There is w >

eb1 − 1 b1 3p1 −1 2(p1 −1)

 12

||P1Q ||L2 (μπ,V )→L2p1 (μπ,V ) ,

such that

N    2 2 μπ,V ew|Z| := πi μV ew|Zi | < ∞.

(3.5)

i=1

Then Pt has an invariant measure μ = ρμπ,V such that

μπ,V (ρ log ρ) ≤

 2 (3p1 − 1) log μπ,V ew|Z| + 4wp1 log c1 2λ(p1 − 1) − (3p1 − 1)

.

Proof. We first consider the Feynman-Kac semigroup for any F ∈ Bb (Rd × S): ⎡

⎛ t ⎞⎤  (x,i) (x,i) (PtF f )(x, i) = E ⎣f (Xt , Λt ) exp ⎝ F (Xs(x,i) , Λ(x,i) )ds⎠⎦ , f ∈ Bb (Rd × S). s 0

Let p = 1 +

p1 −1 2p1 .

By Lemma 3.2, ||P1Q ||L2 (μπ,V )→L2p1 (μπ,V ) < ∞. Then

μπ,V (|P1F f |2p ) ≤ μπ,V

⎛⎛ ⎡ ⎛ 1 ⎞⎤⎞2p ⎞  ⎜⎝ ⎣ ⎟ ⎝ E |f (X1 , Λ1 )| exp ⎝ F (Xs , Λs )ds⎠⎦⎠ ⎠ 0

  p1 1 ≤ μπ,V (P1Q f p )2p1

 μπ,V



 Ee

≤ ||P1Q ||2L2 (μπ,V )→L2p1 (μπ,V ) μπ,V (f 2p )

p p−1

⎧ 1 ⎨ ⎩

0

1 0

F (Xs ,Λs )ds



p

1 2(p−1)p p −1

' p1p−1 1

1

μπ,V Ee p−1 F (Xs ,Λs )



⎫ p1p−1 ⎬ 1 ds ⎭

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

= ||P1Q ||2L2 (μπ,V )→L2p1 (μπ,V ) μπ,V (f 2p )

≤ ||P1Q ||2L2 (μπ,V )→L2p1 (μπ,V ) μπ,V (f 2p )

⎧ 1 ⎨ ⎩

p

μπ,V PsQ e p−1 F



0

⎧ 1 ⎨ ⎩





p

eb1 s μπ,V e p−1 F

0

⎫ p1p−1 ⎬ 1 ds ⎭



⎫ p1p−1 ⎬ 1 . ds ⎭

Thus p1 −1  p p2p1 −1p  3p1 −1 3p 1  1  1 −1 1 ||P1F ||L2p (μπ,V ) ≤ c1p μπ,V e p−1 F = c1p μπ,V e p1 −1 F ,

 where c1 =

eb1 −1 b1

Ee

n 0

12

μ

||P1Q ||L2 (μπ,V )→L2p1 (μπ,V ) . By the Markov property, we have that μ

F (Xs π,V ,Λs π,V )ds

1 −1)  3p1 −1 n(p n  2p1 p = μπ,V (PnF 1) ≤ ||P1F ||nL2p (μπ,V ) ≤ c1p μπ,V e p1 −1 F .

Next, let ⎧ t ⎨

μ

Rt π,V = exp

⎩

Z(Xsμπ,V , Λμs π,V ), dWs  −

0

1 2

t 0

⎫ ⎬ |Z|2 (Xsμπ,V , Λμs π,V )ds . ⎭

Then, similar to [21, Theorem 4.1],

ERnμπ,V log Rnμπ,V

⎛ ⎞ n 1 ⎝ μπ,V = E Rn |Z(Xs , Λs )|2 ds⎠ 2 0

≤ ERnμπ,V log Rnμπ,V

* 1 n 2 + + log E e 2 0 |Z| (Xs ,Λs ) ds , ∈ (0, 1).

Thus ERnμπ,V log Rnμπ,V

1 2 1 −1   3p1 −1 p2p 2 n 1 |Z| log c1 μπ,V e 2(p1 −1) ≤ , ∈ (0, 1). p(1 − )

By Lemma 3.1, 1 νn (f ) := n

n μπ,V 0

1 (Ps f ) ds = n

n ERnμπ,V f (Xsμπ,V , Λμs π,V )ds 0

n μπ,V μπ,V 1 1 ≤ ERnμπ,V log Rnμπ,V + log Ee 0 f (Xs ,Λs )ds n n 1 2 1 −1   3p1 −1 p2p 2 1 |Z| log c1 μπ,V e 2(p1 −1) ≤ p(1 − ) 1 2 1 −1   3p1 −1 p2p 1 1 f + log c1 μπ,V e p1 −1 , f ≥ 0, f ∈ Bb (Rd × S). p

By these estimates above, following the proof of [21, Theorem 4.1], we can prove the theorem. 2

675

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

676

The inequality (3.5) implies that for each i ∈ S, Zi has nice integrability w.r.t. μV indeed. However, it is not the desired hybrid integrability condition. In the rest part of this section, we shall give a weaker condition to get a priori estimate to the entropy of the density of the invariant probability measure. Define x log x = 0 if x = 0. We have the following a priori estimate. Lemma 3.4. Assume (H1)-(H3) and that for all i ∈ S, Zi is bounded with compact support. Then Pt has a unique invariant probability measure μ such that μ = ρμπ,V . Moreover, the density ρ = (ρ1 , · · · , ρN ) is √ positive and ρi ∈ Hσ1,2 (Rd ) with i √ 2πi μV (|σi∗ ∇ ρi |2 ) + πi μV (qi ρi log ρi )   ≤ πi μV (|Zi |2 ρi ) + πk μV (qki ρk log(qki ρk )) k=i

 − μV (qki ρk ) log μV (qki ρk ) + μV (qki ρk ) log μV (ρi ) , i ∈ S. (n)

(n)

Proof. Let qki = qki 1[|x|≤n] , Q(n) = (qki )1≤k,i≤N and L[n] = LZ + Q(n) . By Remark 1.2, we can choose large enough n such that Q(n) is also full coupled. Because Zi is bounded with compact support, Lemma 3.3 [n] and Lemma 2.2 can be applied to L[n] and the associated semigroup Pt . Then the semigroup generated by L[n] has an invariant measure with positive density ρ[n] w.r.t. μπ,V such that supn≥1 μπ,V (ρ[n] log ρ[n] ) < ∞, [n]

which implies that ρi → ρi weakly in L1 (μV ). Moreover, ρ is a probability density and also satisfies μπ,V (ρLg) = 0, g ∈ Cc2 (Rd × S). Hence, Lemma 2.2 implies that ρ is positive. In fact, ρμπ,V is also the invariant probability measure of Pt , that is μπ,V (ρf ) = μπ,V (ρPt f ) for all f ∈ Bb (Rd × S). Notice that for f ∈ Bb (Rd × S) [n]

μπ,V (ρ[n] f ) = μπ,V (ρ[n] Pt f ) [n]

= μπ,V (ρ[n] (Pt f − Pt f )) + μπ,V ((ρ[n] − ρ)Pt f ) + μπ,V (ρPt f ), and lim μπ,V ((ρ[n] − ρ)Pt f ) = 0,

lim μπ,V (ρ[n] f ) = μπ,V (ρf ).

n→∞

n→∞

In order to prove μπ,V (ρf ) = μπ,V (ρPt f ), we only need to prove [n]

lim μπ,V (ρ[n] (Pt f − Pt f )) = 0.

n→∞

(3.6)

Since supn≥1 μπ,V (ρ[n] log ρ[n] ) < ∞, which implies that {ρ[n] }n≥1 is uniformly integrable, we have that for any > 0, there is m > 0 such that sup μπ,V (ρ[n] 1[ρ[n] ≥m] ) ≤ .

n≥1

Thus [n]

[n]

μπ,V (ρ[n] |Pt f − Pt f |) ≤ μπ,V (ρ[n] 1[ρ[n]
≤ mμπ,V (|Pt f − Pt f |) + 2 ||f ||∞ . Hence, to prove (3.6), we only need to prove

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

[n]

lim μπ,V (|Pt f − Pt f |) = 0, f ∈ Bb (Rd × S).

n→∞

677

(3.7)

√ Next, we shall prove (3.7). Let Bi = ( 2σi Zi ) · ∇ and B = Diag{B1 , · · · , BN }. Since Zi is bounded with √ compact support, the Multiplier 2σi Zi generates a bounded operator in L2 (μV ). Combining this with that the gradient operator is closed in L2 (μV ), we have that B is a closable operator on L2 (μπ,V ). Moreover, it follows (H1) and the boundedness of Zi that     μV |Bi f |2 ≤ CμV |σi∗ ∇f |2 , i ∈ S,

(3.8)

√ √  √  which implies that D (B) ⊃ D −L0 , where D (B) and D −L0 are the domain of B and −L0 respectively. Then LB = L0 + B is closable, and the closure generates an analytic semigroup by [14, Corollary 3.2.4.]. Since Zi is bounded with compact support, Zi is in the Kato’s class. Combining this with ˜ such ˜ Λ) (1.8) and (3.3), it follows from the proof of [6, Theorem 3.1] that there exists a Hunt process (X, that ˜ t ), f ∈ Bb (Rd × S), ˜t, Λ P˜t f (x, i) = Ef (X

(3.9)

where P˜t is the semigroup generated by LB + Q in L2 (μπ,V ). Since Zi is bounded with compact support and σi is non-degenerate, the diffusion process generated by LZ i is pathwise unique, see for instance [20, Theorem 2.1]. Then the process generated by L is pathwise unique, which together with (3.9) implies Pt f = P˜t f for all f ∈ Bb (Rd × S). It follows from [14, Theorem 3.1.1] that t Pt f = P˜t f =

PtB f

B Pt−s (QPs f ) ds,

+ 0

t

[n]

Pt f = PtB f +

 B Pt−s Q[n] Ps[n] f ds, f ∈ Bb (Rd × S).

0

Then 

μπ,V |Pt f −

[n] Pt f |2



≤Ct

 t μπ,V

 2    (n) Ps f  ds  Q−Q

0

t +

μπ,V

 2     [n] P − P f  ds .  s s

0

By Gronwall’s inequality, we have  [n] lim μπ,V |Pt f − Pt f |2 = 0, f ∈ Bb (Rd × S),

n→∞

(3.10)

which implies (3.7). Since Zi is bounded with compact support, Lemma 2.1 and Corollary 2.3 imply that any invariant probability measure of Pt has positive density. Then the uniqueness follows from [9, Proposition 3.2.7, Proposition 3.2.5]. The equation μπ,V (ρ[n] L[n] g) = 0, g ∈ Cc2 (Rd × S) yields, for all i ∈ S and f ∈ Cc2 (Rd ), √

 2πi μV

[n] σi Zi , ∇f ρi



+ μV

N  k=1

 [n] [n] πk qki f ρk

 [n] = πi μV σi∗ ∇ρi , σi∗ ∇f  .

(3.11)

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

678

Then there is Cn > 0 which is independent of f such that 3      [n] μV σi∗ ∇ρi , σi∗ ∇f   ≤ Cn μV (|σi∗ ∇f |2 ) + μV (f 2 ).

(3.12)

The defective log-Sobolev inequality yields the Poincaré inequality: μV (f 2 ) ≤ CμV (|σi∗ ∇f |2 ) + μV (f )2 , f ∈ Hσ1,2 (Rd ). i So the norm of f , Hσ1,2 (Rd ): i

 μV (|σi∗ ∇f |2 ) + μV (f )2 , which is induced by the following inner product of g1 , g2 on μV (g1 )μV (g2 ) + μV (σi∗ ∇g1 , σi∗ ∇g2 ), g1 , g2 ∈ Hσ1,2 (Rd ) i [n]

is equivalent to the Sobolev norm on Hσ1,2 (Rd ). Hence (3.12) implies that ρi ∈ Hσ1,2 (Rd ). Moreover, (3.11) i i holds for all f ∈ Hσ1,2 (Rd ). i 3 [n] [n] Next, we shall prove that supn≥1 μV (|σi∗ ∇ ρi |2 ) < ∞. Let f = log(ρi + ) with ∈ (0, 1). Then (3.11)

N [n] and j=1 πj μV (ρj ) = 1 imply that  2πi μV ≤

 j=i

≤





3

|σi∗ ∇

[n]

ρi + |2 (n) [n]

(n) [n]

(n)

+ πi μV (qi ρi log(ρi [n]

+ ))

[n]

πj μV (qji ρj log(ρi + )) + πi μV (|Zi |2 ρi )      (n) [n] (n) [n] (n) [n] (n) [n] πj μV qji ρj log qji ρj − μV qji ρj log μV qji ρj

j=i

      (n) [n] [n] [n] + μV qji ρj log μV ρi + + sup |Zi |2 πi μV ρi ≤



x∈Rd

   (n) [n] (n) (n) [n] [n] πj μV qji ρj log qji + μV qji ρj log ρj + e−1

j=i

  CQ 1 + log +1 + sup |Zi |2 πj πi x∈Rd    [n] [n] ≤ CQ log+ CQ + CQ πj μV ρj log ρj + πj (CQ + 1)e−1 j=i



+ CQ log

πi + 1 πi



+ sup |Zi |2 , x∈Rd

and πi μV (qi ρi log(ρi + )) ≥ πi μV (qi ρi log ρi ) ≥ −πi CQ e−1 . (n) [n]

[n]

(n) [n]

[n]

Hence, the monotone convergence theorem implies that there is C > 0 which is independent of i and n such that   2     ∗ 3 [n] 2  ∗ 3 [n]  μV σi ∇ ρi  = lim+ μV σi ∇ ρi +  ≤ C. →0 The defective log-Sobolev inequality implies the existence of a super Poincaré inequality, and hence the essential spectrum of L0i is empty, which implies that Hσ1,2 (Rd ) is compactly embedded into L2 (μV ). Thus, i

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

√

ρi ∈ weakly

Hσ1,2 (Rd ), and i in Hσ1,2 (Rd ). i

[n] ρi

by a subsequence,

 [n] For m > 0, let f = log (ρi ∨ m) ∧

679

3 √ [n] converges to ρi in L (μV ) and ρi convergences to ρi 1

. Then (3.11) yields that

1 m



    ∗ [n] 2 1[ m1 ≤ρ[n] i ≤m] πi μV σi ∇ρi  [n] ρi ⎛ 2 ⎞ 4   1   [n] = 4πi μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠  m =

√

 [n] 2πi μV Zi , σi∗ ∇ρi 1[ 1 ≤ρ[n] ≤m] m



i

/   0 1 (n) [n] [n] ρk πk μV qki log (ρi ∨ m) ∧ + m k=1     πi  ∗ [n] 2 1[ m1 ≤ρ[n] i ≤m] 2 [n] ≤ πi μV (|Zi | ρi ) + μV σi ∇ρi  [n] 2 ρi     1 (n) [n] [n] πk μV qki ρk log (ρi ∨ m) ∧ + m k=i    1 (n) [n] [n] . − πi μV qi ρi log (ρi ∨ m) ∧ m N 

[n]

Since ρi → ρi in L1 (μV ),

3

[n]

ρi

√ ρi weakly in Hσ1,2 (Rd ) and i

convergences to

    (n) 1  [n] < ∞, sup qi log (ρi ∨ m) ∧ m  n≥1,x∈Rd [n]

we have lim μV (|Zi |2 ρi ) = μV (|Zi |2 ρi ), n→∞

lim

n→∞

=

N 

 πk μV

k=1

N 

πk μV

k=1

  1 (n) [n] [n] qki ρk log (ρi ∨ m) ∧ m

   1 qki ρk log (ρi ∨ m) ∧ , m

and ⎛ ⎛ 2 ⎞ 2 ⎞ 4 4     1 1     [n] lim μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠ ≥ μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠ .   m m n→∞ Combining these with (3.13), we obtain that ⎛ 2 ⎞ 4  0 /   1 1   ⎠ ∗ ⎝ 2πi μV σi ∇ (ρi ∨ m) ∧  + πi μV qi ρi log (ρi ∨ m) ∧  m m ≤ πi μV (|Zi | ρi ) + 2

 k=i

 πk μV



1 qki ρk log (ρi ∨ m) ∧ m



(3.13)

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

680

≤ πi μV (|Zi |2 ρi ) +



 πk μV (qki ρk log (qki ρk ))

k=i

(3.14) 

− μV (qki ρk ) log μV (qki ρk ) + μV (qki ρk ) log μV

(ρi ∨ m) ∧

1 m

 ,

where the Young inequality is used in the last inequality. Since ρi > 0 and ⎛ 2 ⎞ 4   1   μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠  m   2 √ √ 2 = μV σi∗ ∇ ρi ∨ m − μV |σi∗ ∇ ρi | 1[ρi ≤ m1 ] , the monotone convergence theorem yields ⎛ 2 ⎞ 4   1  √ 2  lim μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠ ≥ μV |σi∗ ∇ ρi | .  m m→∞ Combining this with (3.14) and Fatou’s Lemma, we have  √ 2 μV |σi∗ ∇ ρi | + πi μV (qi ρi log ρi ) ⎛ 2 ⎞ 4   1   ≤ 2πi lim μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠   m m→∞ 

0 / 1 + πi lim μV qi ρi log (ρi ∨ m) ∧ m m→∞ ⎛ 2 ⎞ 4    1   ≤ lim 2πi μV ⎝σi∗ ∇ (ρi ∨ m) ∧  ⎠  m m→∞ 

0  / 1 + πi μV qi ρi log (ρi ∨ m) ∧ m   πk μV (qki ρk log (qki ρk )) ≤ πi μV (|Zi |2 ρi ) + k=i

 − μV (qki ρk ) log μV (qki ρk ) + μV (qki ρk ) log μV (ρi ) .

2

The final result of this section is the following lemma on a priori estimate of entropy of ρi for Zi satisfies the integrability condition (1.10), which is crucial to the proof of Theorem 1.1. Lemma 3.5. Let πmin = min πk . Then, under conditions of Lemma 3.4 and Theorem 1.1, we have the 1≤k≤N

following an a priori estimate N  k=1

/ πk μV (ρk log ρk ) ≤ max

1≤k≤N

+ C˜Q

0 −1 vk wk |σk Z k |2 μV (e ) − log πmin wk

N  k=1

vk + 2 max

1≤k≤N

vk βk , γk

(3.15)

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

681

where C˜Q = CQ log+ CQ + 2(CQ + 1)e−1 − 2CQ log πmin . Moreover, there is a constant C depending on 2 πk , CQ , wk , vk and μV (ewk |Zk | ), k ∈ S such that N 

√ πk μV (|σk∗ ∇ ρk |) ≤ C.

(3.16)

k=1

Proof. Let qˆij = inf x∈Rd qij (x). Then Lemma 3.4 and the Young inequality imply   √ 2πk μV |σk∗ ∇ ρk |2 + πk μV (qk ρk log ρk ) πk (μV (ρk log ρk ) − μV (ρk ) log μV (ρk )) ≤ wk  2 πk μV (ρk ) log μV ewk |Zk | + wk   πj q¯jk μV (ρj log ρj ) + μV (ρj ) sup (qjk (x) log qjk (x)) +

(3.17)

x∈Rd

j=k

+ (¯ qjk + qˆjk )e−1 + e−1 − CQ μV (ρj ) log πj . Combining this with (H3) and qk ≥ −¯ qkk , we have 

 2πk 2πk βk − πk q¯kk (μV (ρk log ρk ) − μV (ρk ) log μV (ρk )) − μV (ρk ) γk γk   √ ≤ 2πk μV |σk∗ ∇ ρk |2 + πk q¯k (μV (ρk log ρk ) − μV (ρk ) log μV (ρk )) πk (μV (ρk log ρk ) − μV (ρk ) log μV (ρk )) ≤ wk  2 πk μV (ρk ) log μV ewk |Zk | + wk  πj q¯jk (μV (ρj log ρj ) − μV (ρj ) log μV (ρj )) + j=k

+ CQ log+ CQ + 2(CQ + 1)e−1 − 2CQ log πmin .

(3.18)

Let ψ = (ψj )1≤j≤N = (μV (ρj log ρj ) − μV (ρj ) log μV (ρj ))1≤j≤N ,  2 1 Ψ= μV (ρk ) log μV ewk |Zk | . 2wk ¯ is semipositive, there exists positive v = (v1 , · · · , vN ) such that Since −(K + Q) ¯ ≥ 1. −(K + Q)v Multiplying both sides of (3.18) by vk and summing all k in S, we have ¯ π + Ψ, vπ + C˜Q 0 ≤ ψ, (K + Q)v

N  k=1

where v, uπ =

N i=1

πi vi ui , u, v ∈ RN . Hence

vk ≤ −ψ, 1π + Ψ, vπ + C˜Q

N  k=1

vk ,

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

682 N 

πk μV (ρk log(ρk ))

k=1

≤ Ψ, vπ + C˜Q

N 

vk +

k=1

≤ Ψ, vπ + C˜Q

N 

N 

πk μV (ρk ) log μV (ρk ) + 2 max

1≤k≤N

k=1

vk − log πmin + 2 max

1≤k≤N

k=1

βk vk γk

βk vk , γk

N

in the last inequality, we use k=1 πk μV (ρk ) = 1. Then it is easy to see that the inequality (3.15) holds. Consequently, (3.16) holds by (3.17). 2 4. Proofs of main results 4.1. The proof of Theorem 1.1 (n)

(n)

(n)

(n)

Let Zi = Zi 1[|Zi |≤n] , Li = LZ + Q. Then, according to Lemma 3.4, the Markov semigroup Pt i generated by L(n) has an invariant probability measure, denoted by ρ(n) μπ,V . Moreover, Lemma 3.5 yields that    3 N  (n) 2 (n) (n) ∗ < ∞, sup μπ,V (ρ log ρ ) + μV πi |σi ∇ ρi | n

i=1 (n)

which implies there exists a ρ ∈ L1 (Rd × S, μπ,V ) such that, by a subsequence, ρi → ρi in L1 (μV ) and 3 √ (n) ρi → ρi weakly in Hσ1,2 (Rd ). i Next, we shall prove that ρμπ,V is an invariant probability measure of Pt . Indeed, noticing that μπ,V (ρ(n) f ) = μπ,V (ρ(n) Ptn f ) = μπ,V (ρ(n) (Ptn f − Pt f )) + μπ,V ((ρ(n) − ρ)Pt f ) + μπ,V (ρPt f ), and for f ∈ Bb (Rd × S) lim μπ,V ((ρ(n) − ρ)Pt f ) = 0,

n→∞

lim μπ,V (ρ(n) f ) = μπ,V (ρf ),

n→∞

in order to prove μπ,V (ρf ) = μπ,V (ρPt f ), we only need to prove lim μπ,V (ρn (Ptn f − Pt f )) = 0.

n→∞

(4.1)

By the Markov property, we only need to prove (4.1) holds for some small t. Let (Xt , Λt ) be the process generated by LQ . Then, by Lemma 3.1, μπ,V (|Ptn f − Pt f |) ) * t ( (n)   (n) ZΛs (Xs ),dWs − 12 0t |ZΛs (Xs )|2 ds  0 ≤ μπ,V Ef (Xt , Λt ) e + 6 1 t t5 2  − e 0 ZΛs (Xs ),dWs − 2 0 |ZΛs (Xs )| ds   ≤ ||f ||∞

Ee

2

t 0

7 Z

μπ,V Λs

 8  μ (Xs π,V ),dWs − 0t Z

μπ,V Λs

 1  2 μ (Xs π,V )ds

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

683

⎛  7 8 2 ⎞ 12  t (Z n −Z) μ (Xsμπ,V ),dWs − 1 t |(Z n −Z)|2 μ (Xsμπ,V )ds  π,V π,V  0  2 0 Λs Λs × ⎝E e − 1 ⎠ .   For all η > 0 and measurable function g from Rd × S to Rd so that |g(x, i)| ≤ |Zi (x)|, (x, i) ∈ Rd × S, we have t

μπ,V

t

μπ,V

μπ,V

μπ,V

Ee2η 0 g(Xs ,Λs ),dWs −η 0 |g(Xs ,Λs )|  12 t μπ,V μπ,V μπ,V μπ,V 2 t ≤ Ee4η 0 g(Xs ,Λs ),dWs −8η 0 |g(Xs ,Λs )|  12 μπ,V μπ,V 2 + t 2 × Ee(8η −2η) 0 |g(Xs ,Λs )| ds ⎛ ≤⎝

1 t

⎞ 12

t Ee

μ (8η 2 −2η)+ t|g(Xs π,V

μ ,Λs π,V

)|2

ds⎠

0

⎛ 1 ≤⎝ t

t



μπ,V PsQ e(8η

2

−2η) t|g| +

2



⎞ 12 ds⎠

0

⎛ ≤⎝

1 t

t

⎞ 12 eb1 s ds⎠

3

  μπ,V e(8η2 −2η)+ t|Z|2

0

< ∞,

t<

min1≤k≤N wk . (8η 2 − 2η)+

So, for t small enough, lim μπ,V (|Ptn f − Pt f |) = 0.

n→∞

(4.2)

Since {ρ(n) }n≥1 is uniformly integrable, for > 0, there is m > 0 such that sup μπ,V (ρ(n) 1[ρ(n) ≥m] ) ≤ .

n≥1

Thus μπ,V (ρ(n) |Ptn f − Pt f |) ≤ μπ,V (ρ(n) 1[ρ(n)
n→∞

Finally, we shall prove our claim on the uniqueness. Since ||σi || is local bounded, it follows from the Young inequality that

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

684

μV (ρi |σi Zi |1O ) ≤

  supx∈O ||σi ||  μV (ρi log ρi ) − μV (ρi ) log μV (ρi ) + μV (ρi )μV ewi |Zi |2 wi

<∞ for all bounded domain O ⊂ Rd . Then Zi ∈ L1loc (μi ) for all i ∈ S. Consequently, Lemma 2.2 and Corollary 2.3 can be applied to μ. So, for each i ∈ S, ρi > 0. By [9, Proposition 3.2.7, Proposition 3.2.5], there is only one invariant probability measure μ such that μ  μπ,V . 4.2. The proof of Theorem 1.3 Let ˆ0 = L i

  1 div eV ρi ai ∇· , ρi eV

1 Zˆi = Zi − √ σi∗ ∇ log ρi . 2

√ ˆ 0 − ai (∇ log ρi ) · ∇ and LZ = L ˆ 0 + 2σi Zˆi · ∇. Then L0i = L i i i We shall prove first that for each i ∈ S, gi ∈ L∞ so that the following equality holds for all fi ∈ Cc2 (Rd ), i ∈ S 0=

N 

1 πi μV

gi (1 −

ˆ 0 )fi L i

−

√

2σi Zˆi , ∇fi gi −

i=1

N 

2 πi qij fj gi ρi

 ,

(4.3)

i=1

1,2 then gi ∈ Hloc (ρi dμV ). Fixing i ∈ S, and let fi ∈ Cc2 (Rd ) and fk = 0 for k = i. Then N  √   ˆ 0 )fi = 2πi μV σi Zˆi , ∇fi gi ρi + πi μV gi ρi (1 − L πk μV (qki fi gk ρk ) . i

(4.4)

k=1

Let ζ ∈ Cc∞ (Rd ). Then  ˆ 0 )fi πi μV (ζgi )ρi (1 − L i   ˆ 0 )(ζfi ) + 2πi μV (ai ∇ζ, ∇fi ρi ) + πi μV (L ˆ 0 ζ)gi ρi = πi μV gi ρi (1 − L i i N   πk μV (qki ζfi gk ρk ) = πi μV σi Zˆi , ∇(ζfi )gi ρi + k=1

 ˆ 0 ζ)fi gi ρi . + 2πi μV (ai ∇ζ, ∇fi gi ρi ) + πi μV (L i By Lemma 2.1, μV (ewi |Zi | ) < ∞ and ||σi || is local bounded, there is a constant C which is independent on fi such that 2

4       μV σi Zˆi , ∇(ζfi )gi ρi  ≤ ||gi ||∞ μV (ζ + |∇ζ|)2 |Zˆi |2 ρi μV ((fi2 + |σi∗ ∇fi |2 )ρi ) ≤C By Lemma 2.2, we also have

3 μV ((fi2 + |σi∗ ∇fi |2 )ρi ).

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

685

9  ⎛  : N 2 ⎞  N  :    ζ 2 ⎠  :  πk μV (qki ζfi gk ρk ) ≤ ;μV (fi2 ρi ) μV ⎝ πk qki gk ρk    ρi   k=1 k=1 9 ⎛ : 2 ⎞ N  : 2   μV (fi ρi ) : ⎝ ;μV ≤ πk qki gk ρk ζ 2 ⎠ inf x∈ suppζ ρi (x)   k=1 3 ≤ C μV (fi2 ρi ), for some positive constant C which is independent of fi . Similarly, 3  ˆ 0 ζ)gi fi ρi ≤ C μV (f 2 ρi ). 2μV (ai ∇ζ, ∇fi ρi ) + μV (L i i Hence 3      0 ˆ ≤ C )ρ (1 − L )f μV ((fi2 + |σi∗ ∇fi |2 )ρi ), fi ∈ Cc2 (Rd ), (ζg μV i i i i  which implies that ζgi ∈ Hσ1,2 (ρi dμV ), i ,0 where Hσ1,2 (ρi dμV ) is the closure of Cc2 (Rd ) under the following norm i ,0 f→

3

μV ((f 2 + |σi∗ ∇f |2 )ρi ),

f ∈ Cc1 (Rd ).

1,2 Therefore gi ∈ Hloc (ρi dμV ). Next, we shall prove that there is only one extension of (L, Cc2 (Rd × S)) that generates a C0 -semigroup in L1 (Rd × S, μ). To this end, we only need to prove that (1 − L)(Cc2 (Rd × S)) is dense in L1 (Rd × S, μ). Let ζn ∈ Cc∞ (Rd ) with 1Bn (0) ≤ ζn ≤ 1B2n (0) and ||∇ζn ||∞ ≤ n1 . Let gi ∈ L∞ , i ∈ S such that (4.3) holds. Fixing i ∈ S, and let fi = ζn2 gi and fk = 0 for k = i. It follows from (4.3),

  0 = πi μV (ζn2 gi2 ρi ) + πi μV ai ∇(ζn2 gi ), ∇gi ρi N    √ − 2πi μV σi Zˆi , ∇(ζ 2 gi )gi ρi − μV πk qki (ζ 2 gi )gk ρk n

n

k=1

  = πi μV (ζn2 gi2 ) + πi μV (ai ∇(ζn gi ), ∇(ζn gi )ρi ) − πi μV ai ∇ζn , ∇ζn gi2 ρi N    √ 2 2 ˆ − 2πi μV σi Zi , ∇(ζn gi )gi ρi − μV πk qki (ζn gi )gk ρk .

(4.5)

k=1

Let fk (x) = (ζn gi )2 (x)1[k=i] (k), x ∈ Rd , k ∈ S. ˆ 0 f ) = 0 and μ(Lf ) = 0. By μ(L ˆ 0 f ) = μ(Lf ), we have that Then (ζn gi )2 ∈ Hσ1,2 (ρi dμV ). Consequently, μ(L i 0=

√

 2πi μV

σ Zˆi , ∇(ζn2 gi2 )ρi + μV



N 

k=1

 πk qki ζn2 gi2 ρk

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

686

=

√

 2πi μV



√  σ Zˆi , ∇(ζn2 gi )gi ρi + 2πi μV Zˆi , ∇gi gi ζn2 ρi + μV

 √  √ = 2 2πi μV σ Zˆi , ∇gi gi ζn2 ρi + 2πi μV σ Zˆi , ∇ζn2 gi2 ρi + μV



N 

 πk qki ζn2 gi2 ρk

k=1 N 



πk qki ζn2 gi2 ρk

.

k=1

Thus N   1  2 2 2 2 ˆ ˆ πi μV σ Zi , ∇gi gi ζn ρi = −πi μV Zi , ∇ζn ζn gi ρi − μV πk qki ζn gi ρk , 2 k=1    πi μV σi Zˆi , ∇(ζn2 gi )gi ρi = 2πi μV σi Zˆi , ∇ζn ζn gi2 ρi + πi μV σi Zˆi , ∇gi gi ζn2 ρi N   1  2 2 πk qki (ζn gi ) ρk . = πi μV σi Zˆi , ∇ζn ζn gi ρi − μV 2 

k=1

Combining these with (4.5), we arrive at πi μV (ζn2 gi2 ρi ) + πi μV (ai ∇(ζn gi ), ∇(ζn gi )ρi ) N     qki (ζn2 gi )gk ρk = πi μV ai ∇ζn , ∇ζgi2 ρi + μV k=1

 1 + πi μV σi Zˆi , ∇ζn ζn gi2 ρi − μV 2



N 

(4.6)  2

πk qki (ζn gi ) ρk

.

k=1

Since N 

1 πk qki gi2 ρk 2 N

πk qki gi gk ρk −

k=1

k=1

=−

πi 2 1 qi gi ρi + 2 2

N 

1 πk qki gi gk ρk 2 N

πk qki gi (gk − gi )ρk +

k=i

k=i

πi 2 1 qi gi ρi − πk qki (gk − gi )2 ρk 2 2 N

=−

k=i

1 1 πk qki gk (gk − gi )ρk + πk qki gi gk ρk 2 2 N

+

N

k=i

=−

πi 2 1 qi gi ρi − 2 2

k=i

N 

1 πk qki gk2 ρk , 2 N

πk qki (gk − gi )2 ρk +

k=i

k=i

we have that N N   i=1

k=1

N



k=1

1  1  1 πi qi gi2 ρi − πk qki (gk − gi )2 ρk + πk qki gk2 ρk 2 i=1 2 i=1 2 i=1 N

=−

1 πk qki gi gk ρk − πk qki gi2 ρk 2 N

N

k=i

N

N

k=i

S.-Q. Zhang / J. Math. Anal. Appl. 478 (2019) 655–688

1  1 1  πk qki (gk − gi )2 ρk − πi qi gi2 ρi + πk qki gk2 ρk 2 i=1 2 i=1 2 N

=−

N

N

k=i

N

687

N

(4.7)

k=1 i=k

1  πk qki (gk − gi )2 ρk . =− 2 i=1 N

N

k=i

Hence, (4.6), (4.7) and the definition of ζn yield that ⎧ N ⎨ 

⎛ ⎞⎫ N ⎬ 2   2 2  ζ πi μV ζn gi ρi + ai ∇(ζn gi ), ∇(ζn gi )ρi + μV ⎝ n πk qki (gk − gi )2 ρk ⎠ ⎩ ⎭ 2 i=1 k=i

=

N 

N    √  πi μV ai ∇ζn , ∇ζn gi2 ρi + 2 πi μV σi Zˆi , ∇ζn ζn gi2 ρi

i=1

≤

i=1

maxi∈S ||gi ||2∞ n2

N 

√ μV (||ai ||ρi ) +

i=1

N  2 maxi∈S ||gi ||2∞  πi μV |σi Zˆi |ρi . n i=1

For r small enough, we have  μV (||ai ||ρi ) + μV |σi Zˆi |ρi   1 μV ρi (|Zi |2 + 3||σi ||2 ) + μV (||σi || · |σi∗ ∇ρi |) 2      1 √  ≤ μV ρi |Zi |2 + 2μV ρi ||σi ||2 + 2μV |σi∗ ∇ ρi | 2  1 1 √  ≤ μV (ρi log ρi ) + 2μV |σi∗ ∇ ρi | − μV (ρi ) log μV (ρi ) 2r 2r    2 2 1 + log μV er|Zi | + log μV e3r||σi || 2r < ∞, i ∈ S. ≤

(4.8)

Therefore lim

n→∞

N 

πi μV (ζn2 gi2 ) ≤ 0,

i=1

which implies that gi = 0, i ∈ S. Acknowledgment The author would like to thank the editors and referees for their helpful comments. The author was supported by the National Natural Science Foundation of China (Grant No. 11771326). References [1] R.A. Adams, J.J.F. Fournier, Sobolev Spaces, second edition, Academic Press, 2009. [2] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, SIAM Press, Philadelphia, 1994. [3] V.I. Bogachev, N.V. Krylov, M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations 26 (2001) 2037–2080. [4] V.I. Bogachev, N.V. Krylov, M. Röckner, Elliptic and parabolic equations for measures, Russian Math. Surveys 64 (2002) 5–116.

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