On invariant probability measures of regime-switching diffusion processes with singular drifts
aa r X i v : . [ m a t h . P R ] N ov On invariant probability measures ofregime-switching diffusion processes withsingular drifts
Shao-Qin Zhang
School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, ChinaEmail: [email protected]
November 29, 2018
Abstract
We introduce integrability conditions involving a nice reference probabilitymeasure and the Q -matrix to study the existence of the invariant probabilitymeasures of regime-switching diffusion processes. Regularities and the uniquenessof the invariant probability density w.r.t the nice reference probability measureare also considered. Moreover, we study the L -uniqueness of the semigroupgenerated by regime-switching diffusion processes. Since the generator of theregime-switching diffusion process is a weakly coupled elliptic system, the L -uniqueness of the extension of weakly coupled elliptic systems is obtained. AMS Subject Classification (2010): 60J60, 47D07Keywords: Regime-switching diffusions, singular drift, invariant probability mea-sure, integrability conditions, weakly coupled elliptic system
Let S = { , , , · · · , N } , and let (Ω , F , P ) be a completed probability space. Aregime-switching diffusion process is a two components process { ( X t , Λ t ) } t ≥ de-scribed by d X t = b Λ t ( X t )d t + √ σ Λ t ( X t )d W t , (1.1) equ_b1 and P (Λ t +∆ t = j | Λ t = i, ( X s , Λ s ) , s ≤ t ) = (cid:26) q ij ( X t )∆ t + o (∆ t ) , i = j, q ii ( X t )∆ t + o (∆ t ) , i = j, (1.2) equ_b2 where { W t } t ≥ is a Brownian motion on R d w.r.t a right continuous completed refer-ence { F t } t ≥ and b : R d × S → R d , σ : R d × S → R d ⊗ R d , q ij : R d → R , Shao-Qin Zhang are measurable functions and q ij ( x ) ≥ , i = j, X j = i q ij ( x ) ≤ − q ii ( x ) . The matrix Q ( x ) = ( q ij ( x )) ≤ i,j ≤ N is called Q -matrix, see [6]. If Q ( x ) is independentof x and Λ t is independent of { W t } t ≥ , { ( X t , Λ t ) } t ≥ is called a state-independentRSDP, otherwise called a state-dependence RSDP. Let q i ( x ) = X j = i q ij ( x ) , i ∈ S . If Q ( x ) is conservative, i.e. − q ii ( x ) = q i ( x ), x ∈ R d , then as in [19, Chapter II-2.1]or [11, 23, 18], we can represent { ( X t , Λ t ) } t ≥ in the form of a system of stochasticdifferential equations(SDEs for short) driven by { W t } and a Poisson random measure.Precisely, for each x ∈ R d , { Γ ij ( x ) : i, j ∈ S } is a family of disjoint intervals on [0 , ∞ )constructed as followsΓ ( x ) = [0 , q ( x )) , Γ ( x ) = [ q ( x ) , q ( x ) + q ( x )) , · · · Γ ( x ) = [ q ( x ) , q ( x ) + q ( x )) , Γ ( x ) = [ q ( x ) + q ( x ) , q ( x ) + q ( x ) + q ( x )) , · · · Γ ( x ) = [ q ( x ) + q ( x ) , q ( x ) + q ( x ) + q ( x )) , · · ·· · · We set Γ ii ( x ) = ∅ and Γ ij ( x ) = ∅ if q ij ( x ) = 0 for i = j . Define a function h : R d × S × [0 , ∞ ) → R as follows h ( x, i, z ) = X j ∈ S ( j − i ) Γ ij ( x ) ( z ) = (cid:26) j − i, if z ∈ Γ ij ( x ) , , otherwise . Let N (d z, d t ) be a Poisson random measure with intensity d z d t and independent ofthe Brownian motion { W t } t ≥ . Then we turn to consider the following equation ( d X t = b Λ t ( X t )d t + √ σ Λ t ( X t )d W t , dΛ t = R ∞ h ( X t − , Λ t − , z ) N (d z, d t ) . (1.3) main_equ The first component X t can be viewed as a hybrid process from the diffusion X it ineach state in S : d X it = b i ( X it )d t + √ σ i ( X it )d W t , i ∈ S . (1.4) Xi In [25], we have study the existence, uniqueness and strong Feller property of (1.3),where drifts b i , i = 1 , · · · , N can be singular. In this paper, we shall study theexistence and uniqueness of invariant probability measures of (1.3) with non-regulardrifts.Recently, in term of a nice reference probability measure, [21, 22] introduce in-tegrability conditions to ensure the existence of invariant probability measures forSDEs with singular or path dependent drifts. We shall extract similar results in[21, 22] for the semigroup corresponding to (1.3). However, for this type of semi-groups generated by (1.3), two obstacles have to be concern about. One is that, aspointed out by [17], to have an invariant probability measure, it is not necessary for nvariant measure of Regime-switching processes X it to process an invariant probability measure. The other one is theheavy tail phenomenon found in [10]. Hence the integrability of invariant measuresof (1.3) may be worse than the invariant probability measures of some X it . So newintegrability conditions (see (1.10), (1.11) and Remark 1.4) are needed to study toinvariant probability measure of (1.3).Let a k ( x ) = σ k ( x ) σ ∗ k ( x ) = ( a i,jk ( x )) ≤ i,j ≤ d and div( a j ( x )) l = P dk =1 ∂ k a klj ( x ). Let V ∈ C ( R d ) such that µ V (d x ) = e V ( x ) d x is a probability measure. In this paper, weconsider the drifts that process the following form b k ( x ) ≡ Z k ( x ) + √ σ k ( x ) Z ( x ) ≡ a k ( x ) ∇ V ( x ) + div( a k ( x )) + √ σ k ( x ) Z k ( x ) . Then σ k Z k is the singular part of the drift b k . For the diffusion part, we definedifferential operators L k and L Zk , k = 1 , · · · , N as follows( L k f )( x ) = e − V ( x ) div (cid:16) e V ( x ) a k ( x ) ∇ f ( x ) (cid:17) = tr (cid:0) a k ( x ) ∇ f ( x ) (cid:1) + h a k ( x ) ∇ V ( x ) , ∇ f ( x ) i + D div( a k ( x )) , ∇ f ( x ) E ≡ tr (cid:0) a k ( x ) ∇ f ( x ) (cid:1) + D Z k ( x ) , ∇ f ( x ) E , ( L Zk f )( x ) = ( L k f )( x ) + √ h σ k ( x ) Z k ( x ) , ∇ f ( x ) i , f ∈ C c ( R d ) . Let P t be the semigroup associated with (1.3). Then the generator of P t , denoted by L , is of the following form( Lf )( x, k ) = L Zk f k ( x ) + N X j =1 q kj ( x ) f j ( x ) , f ∈ C c ( R d × S ) . The operator L sometimes is called weakly coupled elliptic system, see [7].Let µ π,V be a probability measure on R d × S : µ π,V ( f ) = N X k =1 Z R d f k ( x ) π k e V ( x ) d x. Denote by H , σ j ( R d ) the Sobolev space which is the completion of C c ( R d ) under thenorm (cid:18)Z R d (cid:0) | σ ∗ j ( x ) ∇ u ( x ) | + | u ( x ) | (cid:1) µ V (d x ) (cid:19) , u ∈ C c ( R d )and by W , σ j ( R d ) the Sobolev space defined as follows W , σ j ( R d ) = (cid:26) u ∈ W , loc ( R d ) (cid:12)(cid:12)(cid:12) Z R d (cid:0) | σ ∗ j ( x ) ∇ u ( x ) | + | u ( x ) | (cid:1) µ V (d x ) < ∞ (cid:27) . We assume that the coefficients of L satisfy the following assumptions(H1) V ∈ C ( R d ) such that e V d x is a probability measure, and σ ijk ∈ C ( R d ). Forall k ∈ S , H , σ k ( R d ) = W , σ k ( R d ) , (1.5) H=W and there is λ k > H1 a k ( x ) ≥ λ k , x ∈ R d . (1.6) Shao-Qin Zhang (H2) For x ∈ R d , Q ( x ) = ( q kj ( x )) ≤ k,j ≤ N is a conservative Q -matrix with C Q := sup x ∈ R d , k ∈ S q k ( x ) < ∞ . The matrix Q is fully coupled on R d , i.e. the set S can not be split into twodisjoint non-empty S and S such that q l l ( · ) = 0 , a.e. , l ∈ S , l ∈ S . There is an x ∈ R d such that Q ( x ) being an irreducible Q -Matrix with aninvariant probability measure π = ( π , · · · , π N ). H2 (H3) There are γ , · · · , γ N ∈ (0 , ∞ ) and β , . . . , β N ∈ [0 , ∞ ) such that H3 µ V ( f log f ) ≤ γ k µ V ( | σ ∗ k ∇ f | ) + β k , f ∈ H , σ k ( R d ) , µ V ( f ) = 1 k ∈ S . rem_logSob Remark 1.1.
The condition (H1) implies that for all i ∈ S , L i generates a uniquenon-explosive Markov semigroup.Let E ( f, g ) = N X k =1 π k Z R d h σ ∗ k ∇ f k ( x ) , σ ∗ k ∇ g k ( x ) i µ V (d x ) , f k , g k ∈ H , σ k ( R d ) . Then E is a Dirichlet form with D ( E ) = { f = ( f , · · · , f N ) | f i ∈ H , σ i ( R d ) , i ∈ S } .The condition (H3) yields that the following defective log-Sobolev inequality holds. For f ∈ H , σ ( R d ) , · · · , f N ∈ H , σ N ( R d ) , with N X k =1 π k µ V ( f k ) = 1 , µ π,V ( f log f ) = N X k =1 π k µ V ( f k log f k ) (1.7) log_N ≤ (max k γ k ) E ( f, f ) + N X k =1 β k π k µ V ( f k )+ N X k =1 π k µ V ( f k ) log µ V ( f k ) ≤ γ E ( f, f ) + β, (1.8) with Remark 1.2.
The matrix Q is fully coupled in the sense of [16]. It was proved in[8, Proposition 4.1.] that Q is fully coupled on a domain D of R d is equivalent to Q is irreducible on D : for any distinct k, l ∈ S , there exist k , k , · · · , k r in S with k i = k i +1 , k = k and k r = l such that { x ∈ D | q k,k +1 ( x ) = 0 } has positive Lebesguemeasure for i = 0 , , · · · , r − . thm_eu Theorem 1.1.
Assume (H1)-(H3), and inf x ∈ R d q i ( x ) > , i ∈ S . (1.9) infq nvariant measure of Regime-switching processes Let ¯ Q = (¯ q ij ) ≤ i,j ≤ N be a matrix with ¯ q ij = sup x ∈ R d q ij ( x ) . We suppose in ad-dition that there exist positive vector v = ( v , v , · · · , v N ) and positive constants w , w , · · · , w N such that sup ≤ k ≤ N µ V ( e w k | Z k | ) < ∞ , (1.10) ewZ − ( K + ¯ Q ) v ≥ , (1.11) inequ_MM where K = Diag (cid:18) w − γ , w − γ , · · · , w N − γ N (cid:19) . Then P t has a unique invariant probability measure µ such that µ ≪ µ π,V . Moreover,the density ρ = d µ d µ π,V is positive on R d × S , and for all p > and k ∈ S , ρ k ( · ) ≡ ρ ( · , k ) ∈ W ,ploc ( R d ) , and √ ρ k ∈ H , σ k ( R d ) . Remark 1.3.
A square matrix A is called an M-Matrix if A = sI − B with some B ≥ and s ≥ r ( B ) , where by B ≥ we mean that all elements of B are non-negative, I is the identity matrix, and r ( B ) is the spectral radius of B . When s > r ( B ) , A iscalled a nonsingular M-matrix. The inequality (1.11) is equivalent to that − ( K + ¯ Q ) is a nonsingular M-Matrix, see [17, Proposition 2.4] or [2] for instance.In our theorem, we indeed prove the uniqueness in the sense that the set n ρµ π,V | ρ ≥ , µ π,V ( ρ ) = 1 , µ π,V ( ρP t f ) = µ π,V ( ρf ) for t ≥ , f ∈ B ( R d × S ) o contains only one element.According to [21, Theorem 2.1], (H1) and (1.10) yields that for each i , the SDE (1.4) with generator L Zi has a non-explosive strong solution. Combining with that S is finite and ∪ x ∈ R d ,i ∈ S supp( h ( x, i, · )) is a bounded set of [0 , ∞ ) , it is easy to see, forinstance [5, 25], that (1.3) has a non-explosive strong solution. re_exa Remark 1.4.
In conditions (1.10) and (1.11) , some w i can be small such that w i < γ i , so not all diffusion processes X it satisfy the integrability conditions in [21, Thoerem2.2] and [22, Theorem 1.1(2) or Theorem 5.2]. A concrete example is presented asfollows. To illustrate that Theorem 1.1 can be applied to the regime-switching diffusionprocess that not all the diffusions in all the environments has an invariant probabilitymeasure, we present the following example
Example 1.2.
Let N = 2 , d = 1 , b ( x ) = − x , b ( x ) = − x + √ δx , σ ( x ) = σ ( x ) =1 , and Q ( x ) = (cid:18) − a ab − b (cid:19) + (cid:18) − a ( x ) a ( x ) b ( x ) − b ( x ) (cid:19) with a > , b > , and | a ( x ) | ≤ θa , | b ( x ) | ≤ θb for some θ ∈ [0 , . Then V ( x ) = c − x , γ = γ = 2 , β = β = 0 , Z = 0 and Z = √ δx . Forall w > , µ V ( e w | Z | ) < ∞ . For < w < δ , µ V ( e w | Z | ) < ∞ . Let ¯ Q = (cid:18) − (1 − θ ) a (1 + θ ) a (1 + θ ) b − (1 − θ ) b (cid:19) . Then − ( K + ¯ Q ) = (cid:18) − θ ) a − w − (1 + θ ) a − (1 + θ ) b − θ ) b − w (cid:19) . Shao-Qin Zhang
By [17, Proposition 2.4], − ( K + ¯ Q ) is a nonsingular M-Matrix if and only if ( − θ ) a − w > (cid:16) − θ ) a − w (cid:17) (cid:16) − θ ) b − w (cid:17) − (1 + θ ) ab > . Consequently, if δ and θ satisfy δ < − θ )( a + b ) − θab − θ ) a , then there are w and w such that (1.10) and (1.11) hold. Moreover, if θ < b + aa + 2 b + 8 ab , then there are δ , w and w such that √ δ − ≥ and (1.10) and (1.11) hold. Thatmeans for the state i = 2 , the diffusion of this state is not recurrent, then it does nothave an invariant probability measure. If P t has an invariant probability measure µ , then P t can be extended to bea semigroup in L ( R d × S , µ ). Consequently, the weakly coupled elliptic system( L, C c ( R d × S )) has one extension in L ( R d × S , µ ) which generates a semigroup.For the uniqueness of the extension in L ( R d × S , µ ), we present the following result,where µ is introduced in Theorem 1.1 un_semigroup Theorem 1.3.
Under the assumption of Theorem 1.1, if in addition there exists ǫ > such that max ≤ i ≤ N µ V ( e ǫ || σ i || ) < ∞ , then there is only one extension of ( L, C c ( R d × S )) that generates a C -semigroup in L ( R d × S , µ ) . The rest of the paper is organized as follows. In Section 2, we shall study aelliptic system corresponding to invariant measures, called the weak coupled systemof measures of (1.3), where some local a priori estimates of measures will be presented.In Section 3, the entropy estimate of the density of invariant probability measureswill be concern about. In the last section, we shall prove our main results.To make the article more concise, we shall introduce the following notations inthe rest parts of the paper(N1) For all q ∈ [1 , ∞ ], q ∗ := [1 µ ( Lf ) = 0. Hence, to study the invariant measure of P t , we shall start formthe equation µ ( Lf ) = 0. Since µ is a measure on R d × S , we can view µ as a systemof measures ( µ i ) i ∈ S , where µ i is defined as in (N3). Thus the equation µ ( Lf ) = 0 isa system of elliptic equations for measures, called weakly coupled elliptic system ofmeasures.Elliptic and parabolic equations for measures have been intensively studied, andresults on equations for measures are important to the study of invariant probabilitymeasures, see [3, 4]. Here, we shall extend some results in [3] to the case of weaklycoupled elliptic system of measures. ex_den Lemma 2.1.
Assume that (H1) holds. Let p ∈ ( d, ∞ ) ∩ [2 , ∞ ) and µ be a probabilitymeasure on R d × S such that µ ( Lf ) = 0 , f ∈ C c ( R d × S ) . Suppose in addition that for all j ∈ S , Z j ∈ L ploc ( R d ) ∩ L loc ( R d , d µ j ) , and q j ( x ) islocally bounded and there exists δ ( j ) ∈ N such that q kj ( x ) = 0 , | k − j | > δ ( j ) . (2.2) Then for each j ∈ S , there is a continuous function ˆ ρ j ∈ W ,ploc ( R d ) such that µ j (d x ) =ˆ ρ j ( x )d x .Proof. For i ∈ S , let f ( x, k ) = f ( x ) [ k = i ] + 0 [ k = i ] , and ν (d x ) = − X k = i q ki ( x ) µ k (d x ) . Then Z R d (cid:0) L Zi f ( x ) − q i ( x ) f ( x ) (cid:1) µ i (d x ) = Z R d f ( x ) ν (d x ) . (2.3) equ_Li Shao-Qin Zhang
According to [3, Corollary 2.3], µ i has a density ˆ ρ i ∈ L d ∗ loc ( R d ) and || ρ i || L d ∗ ( O ) isbounded by a constant depending on || Z i || L ( O , d µ i ) . Then Z R d (cid:0) L Zi f ( x ) − q i ( x ) f ( x ) (cid:1) µ i (d x ) = Z R d f ( x ) − X k = i q ki ( x )ˆ ρ k ( x ) d x. (2.4) equ_rhi Let ˜ Z jk ( x ) = Z ,jk ( x ) + d X l =1 σ jlk ( x ) Z lk ( x ) , x ∈ R d , k ∈ S , ≤ j ≤ d. For all f ∈ C c ( R d × S ). By µ ( Lf ) = 0 and the definition of ˆ ρ k , we obtain that N X k =1 Z R d d X i,j =1 a ijk ( x ) ∂ i ∂ j f k ( x ) + d X j =1 ˜ Z jk ∂ j f k ( x ) + N X j =1 q kj ( x ) f j ( x ) ˆ ρ k ( x )d x = 0 . (2.5) equ_pde Fix r ∈ ( p ∗ , d ∗ ]. For all q ≥ r ∗ pp − r ∗ , we have q ∗ p + q ∗ r ≤ , q ∗ < r, which implies that ( | ˜ Z k | + 1)ˆ ρ k ∈ L q ∗ loc ( R d ) for all k if ˆ ρ k ∈ L rloc ( R d ). Since f ∈ C c ( R d × S ), there are m ∈ N such that f k = 0, k > m and a bounded domain O ⊂ R d such that S k ≤ m supp f k ⊂ O . Let β m ( x ) = X j ≤ m max | k − j |≤ δ ( j ) | q kj | ( x ) , M = max j ≤ m ( j + δ ( j )) . Then (2.5) implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 Z O d X i,j =1 a ijk ( x ) ∂ i ∂ j f k ( x )ˆ ρ k ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N X k =1 Z O | ˜ Z k ||∇ f k | ˆ ρ k ( x )d x + Z O | f | ( x ) β m ( x ) M X k =1 ˆ ρ k ( x ) ! d x ≤ M X k =1 Z O ( | ˜ Z k | ˆ ρ k ) q ∗ ( x )d x ! q ∗ N X k =1 Z O |∇ f k | q ( x )d x ! q (2.6) grad_Sum_N + Z O β q ∗ m ( x ) M X k =1 ˆ ρ k ( x ) ! q ∗ d x q ∗ (cid:18)Z O | f | q ( x )d x (cid:19) q ≤ C N X k =1 Z O |∇ f k | q ( x )d x ! q , nvariant measure of Regime-switching processes C depends on O , β m , M , || ˜ Z k || L p ( O ) , || ˆ ρ k || L r ( O ) , λ k , k = 1 , · · · , M .Fix some k . Let f k = f k if k = k and f k = 0 otherwise. Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z O d X i,j =1 a ijk ( x ) ∂ i ∂ j f k ( x )ˆ ρ k ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18)Z O |∇ f k | q ( x )d x (cid:19) q . (2.7) grad_sum Next we can adapt a procedure introduced in [3]. Let x ∈ R d , R >
0, and B R ( x )be the open ball with center x and radius R . Let η ∈ C ∞ c ( B R ( x )). Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B R ( x ) d X i,j =1 a ijk ( x ) ∂ i ∂ j f k ( x )( η ˆ ρ k )( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B R ( x ) d X i,j =1 a ijk ( x ) ∂ i ∂ j ( ηf k )( x )ˆ ρ k ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B R ( x ) d X i,j =1 a ijk ( x ) ∂ i ∂ j η ( x )( f k ( x )ˆ ρ k )( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.8) grad_B_R + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B R ( x ) d X i,j =1 a ijk ( x ) ∂ i η∂ j f k ( x )ˆ ρ k ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z B R ( x ) |∇ ( ηf k ) | q ( x ) + | f k | q ( x )d x ! q ≤ C Z B R ( x ) |∇ f k | q ( x )d x ! q . the constant C depends on B R , β m , M , η , || ˜ Z k || L p ( B R ) , || ˆ ρ k || L r ( B R ) , λ k , k = 1 , · · · , M .By (2.8) and [3, Theorem 2.7], for R small enough(independent of x ), η ˆ ρ k ∈ W ,q ∗ ( B R ) with 1 ≤ q ∗ ≤ prp + r . The point x is arbitrary. According to Sobolevembedding theorem, if prp + r ≥ d , then ˆ ρ k is local bounded, and if prp + r < d , thenˆ ρ k ∈ L r loc ( R d ) with r = prdd − prp + r = prdpd − r ( p − d )which implies that r r = pdpd − r ( p − d ) > pdpd − p ∗ ( p − d ) > . Since p > d yields that p ∗ < d ∗ , starting from r ∈ ( p ∗ , d ∗ ], we can get a sequence { r n } by this procedure such that ˆ ρ k ∈ W , prnp + rn loc ( R d ) until pr n p + r n ≥ d . Indeed, we can obtainthat ˆ ρ k ∈ W ,lloc ( R d ) with some l > d . Then ˆ ρ k is continuous and local boundeddue to Sobolev embedding theorem. From (2.6), (2.7), | ˜ Z k | ∈ L ploc ( R d ) and that k isarbitrary, we obtain for f ∈ C c ( R d × S ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 Z O d X i,j =1 a ijk ( x ) ∂ i ∂ j f k ( x )ˆ ρ k ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C N X k =1 Z O |∇ f k | p ∗ ( x )d x ! p ∗ Shao-Qin Zhang with constant C depends on O , β m , M , || ˜ Z k || L p ( O ) , || ˆ ρ k || L r ( O ) , λ k , k = 1 , · · · , M .According to [3, Theorem 2.7], we also obtain that ˆ ρ k ∈ W ,ploc ( R d ), and || ˆ ρ k || H , ( O ) bounded by a constant depends on p , d , O , || ˜ Z k || L p ( O ) , || ˆ ρ k || L d ∗ ( O ) , λ k , a k , M , β m .Next, we shall prove the following weak Harnack inequality, which is crucial toprove that ρ k := e − V π k ˆ ρ k is positive. The proof mainly follows the line of [13, Theorem 4.15] indeed. wHar Lemma 2.2.
Under the assumption of Lemma 2.1, fixing k ∈ S , for all < r < R < ∞ , it holds that ess inf B r ρ k ≥ C (cid:18)Z B R ρ pk d x (cid:19) p , < p < dd − , where C depends on σ k , e V , || q k || L p ( B R ) , || Z k || L p ( B R ) , d , p , r , R .Proof. By Lemma 2.1, ρ k ∈ W ,ploc ( R d ) is locally bounded. (2.4) and ˆ ρ k = π k ρ k e V imply that Z R d ( L Zk − q k ( x )) f ( x ) ρ k e V ( x ) d x = − Z R d X j = k q jk ( x ) ρ j ( x ) f ( x ) e V d x, f ∈ C c ( R d ) . Then by the integration by part formula, we have Z R d h σ ∗ k ∇ f ( x ) , σ ∗ k ∇ ρ k ( x ) i e V ( x ) d x + Z R d q k ( x ) f ( x ) ρ k ( x ) e V ( x ) d x = Z R d h σ k Z k , ∇ f ( x ) i ρ k ( x ) e V d x + Z R d f ( x ) X j = k q jk ( x ) ρ j ( x ) e V d x (2.9) ineq_rhk ≥ Z R d h σ k Z k , ∇ f ( x ) i ρ k ( x ) e V d x, f ≥ , f ∈ C c ( R d ) . The remainder of the proof will start from this inequality.Let δ > , β > f = ψ ( ρ k + δ ) − ( β +1) with ψ ∈ C c ( R d ). Then2 Z R d h σ ∗ k ∇ ψ, σ ∗ k ∇ ρ k i ψ ( ρ k + δ ) β +1 e V ( x ) d x − ( β + 1) Z R d | σ ∗ k ∇ ρ k | ψ ( ρ k + δ ) β +2 e V ( x ) d x ≥ − Z R d q k ( x ) ψ ( x ) ρ k ( ρ k + δ ) β +1 e V ( x ) d x + 2 Z R d h σ k Z k , ∇ ψ i ψρ k ( ρ k + δ ) β +1 e V ( x ) d x − ( β + 1) Z R d h σ k Z k , ∇ ρ k i ψ ρ k ( ρ k + δ ) β +1 e V ( x ) d x. Thus4( β + 1) β Z R d | σ ∗ k ∇ ( ρ k + δ ) − β | ψ e V ( x ) d x ≤ β Z R d h σ ∗ k ∇ ψ, σ ∗ k ∇ ( ρ k + δ ) − β i ψ ( ρ k + δ ) − β e V ( x ) d x + Z R d q k ( x ) ψ ( ρ k + δ ) β e V ( x ) d x nvariant measure of Regime-switching processes
11+ 2 Z R d | Z k || σ ∗ k ∇ ψ || ψ | ( ρ k + δ ) β e V ( x ) d x + 2( β + 1) β Z R d | Z k | ψ | σ ∗ k ∇ ( ρ k + δ ) − β | ( ρ k + δ ) β e V ( x ) d x. Let w = ( ρ k + δ ) − β . Then from the inequality above, we can obtain that Z R d | σ ∗ k ∇ w | ψ e V ( x ) d x ≤ β β + 1) Z R d | σ ∗ k ∇ ψ || σ ∗ k ∇ w | ψwe V ( x ) d x + β β + 1) Z R d q k ( x ) ψ w e V d x + β β + 1) Z R d | Z k || σ ∗ k ∇ ψ | ψw e V ( x ) d x + β Z R d | Z k || σ ∗ k ∇ w | ψ we V ( x ) d x. Thus Z R d | σ ∗ k ∇ w | ψ e V d x ≤ (1 + β ) Z R d | σ ∗ k ∇ ψ | w e V d x + 2 β Z R d q k ( x ) ψ w e V ( x ) d x + β Z R d | Z k | ψ w e V d x. Suppose supp ψ ⊂ O . Letting osc O e V = sup O e V inf O e V and Λ O = sup O || σ ∗ k || , we obtainthat Z R d |∇ ( wψ ) | d x ≤ C ( λ k , osc O e V , Λ O )(1 + β ) Z R d (cid:0) |∇ ψ | + q k ( x ) ψ + | Z k | ψ (cid:1) w d x. (2.10) ineq_wps By H¨older inequality, Z R d ( q k ( x ) + | Z k | ) ψ w d x ≤ (cid:18)Z O ( q k ( x ) + | Z k | ) p d x (cid:19) p (cid:18)Z R d ( ψw ) pp − d x (cid:19) p − p . Since p > d , H¨older inequality and Sobolev inequality imply that for all ǫ > || ψw || L pp − ( O ) ≤ ǫ || ψw || L dd − ( O ) + C ( d, p ) ǫ − dp − d || ψw || L ( O ) ≤ ǫC ( d, O ) ||∇ ( ψw ) || L ( O ) + C ( d, p ) ǫ − dp − d || ψw || L ( O ) . Substituting this into (2.10) and choosing some small ǫ , we obtain that Z R d |∇ ( wψ ) | d x ≤ C (1 + β ) α Z R d (cid:0) |∇ ψ | + ψ (cid:1) w d x with some α > C depending on λ k , osc O e V , Λ O , || q k || L p ( O ) , || Z k || L p ( O ) , p, d . Bythe Sobolev embedding theorem, (cid:18)Z R d ( ψw ) γ d x (cid:19) γ ≤ C (1 + β ) α Z R d (cid:0) |∇ ψ | + ψ (cid:1) w d x Shao-Qin Zhang where γ = dd − for d ≥ γ > d = 2 ,
1. Choosing ψ such that B r ≤ ψ ≤ B r and |∇ ψ | ≤ r − r , where B r and B r are some balls with radius r and r ,which are subsets of O . We arrive at Z B r w γ d x ! γ ≤ C (1 + β ) α ( r − r ) Z B r w d x. (2.11)Thus Z B r ( ρ k + δ ) − βγ d x ! βγ ≤ (cid:18) C (1 + β ) α ( r − r ) (cid:19) β Z B r ( ρ k + δ ) − β d x ! β . (2.12)Hence, we can get the inverse H¨older inequality for ( ρ k + δ ) − by iteration(see [13]for instance) || ( ρ k + δ ) − || L p ( B r ) ≤ C || ( ρ k + δ ) − || L p ( B r ) (2.13) inver_H1 with any p > p > B r ⊂ B R , where the constant C depends on λ k , osc B r e V ,Λ B r , || q k || L p ( B r ) , || Z k || L p ( B r ) , d, p, r , r . Moreover, letting p → ∞ , we obtainess inf B r ρ k + δ ≥ C || ( ρ k + δ ) − || − L p ( B r ) . (2.14) inf Next, we shall prove that for all 0 < q < dd − and R >
0, there is
C > Z B R ( ρ k + δ ) − q d x Z B R ( ρ k + δ ) q d x ≤ C. (2.15) q-q Let f = ( ρ k + δ ) − ψ in (2.9) and w = log( rh k + δ ). Then Z R d | σ ∗ k ∇ w | ψ e V d x ≤ Z R d | σ ∗ k ∇ ψ || σ ∗ k ∇ ψ | ψe V d x + Z R d | Z k ||∇ w | ψ e V d x + Z R d | Z k ||∇ ψ | ψe V d x + Z R d q k ( x ) ψ d x. Hence Z R d | σ ∗ k ∇ w | ψ e V d x ≤ Z R d | σ ∗ k ∇ ψ | e V d x + 4 Z R d ( | Z k | + q k ( x )) ψ e V d x. Choosing ψ such that B r ≤ ψ ≤ B r ≤ O and |∇ ψ | ≤ r , we obtain that there is aconstant C > λ k , Λ O , O such that Z B r |∇ w | d x ≤ C (cid:18) r d − + Z B r (cid:0) | Z k | + q k ( x ) (cid:1) d x (cid:19) . Since Z B r (cid:0) | Z k | + q k ( x ) (cid:1) d x ≤ (cid:12)(cid:12)(cid:12)(cid:12) | Z k | + q k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) L d ( B r ) (cid:18)Z B r d x (cid:19) d − d = 2 d − r d − (cid:12)(cid:12)(cid:12)(cid:12) | Z k | + q k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) L d ( B r ) , nvariant measure of Regime-switching processes Z B r |∇ w | d x ≤ Cr d − with C depends on λ k , Λ O , O , || q k || L d ( B r ) , || Z k || L d ( B r ) , d . As in [13, Theorem 4.14],the Poincar´e inequality on B r and John-Nirenberg Lemma imply that there exist p > C > d , r such that Z B r e p | w − ¯ w | d x ≤ C where ¯ w = | B r | R B r w d x . Hence Z B r ( ρ k + δ ) − p d x Z B r ( ρ k + δ ) p d x = Z B r e − p w d x Z B r e p w d x ≤ (cid:18)Z B r e p | w − ¯ w | d x (cid:19) (2.16) p0-p0 ≤ C . To prove (2.15), let f = ψ ( ρ k + δ ) − β , β ∈ (0 , || ρ k + δ || L p ( B r ) ≤ C || ρ k + δ || L p ( B r ) , < r < r , < p < p < dd − C depends on λ k , osc B r e V , Λ B r , || q k || L p ( B r ) , || Z k || L p ( B r ) , d , p , r , r . Combing this with (2.16) and (2.13), (2.15) is proved.Lastly, (2.14) and (2.15) yield thatess inf B r ρ k + δ ≥ C || ( ρ k + δ ) − || − L p ( B r ) ≥ C (cid:16) || ( ρ k + δ ) − || L q ( B r ) || ( ρ k + δ ) || L q ( B r ) (cid:17) − || ( ρ k + δ ) || L q ( B r ) ≥ C || ( ρ k + δ ) || L q ( B r ) . Letting δ → + , we obtain the weak Harnack inequality. rem-add Remark 2.1. If Q is fully coupled and µ is a probability measure, then it followsfrom the weakly Harnack inequality that ρ k > for all k ∈ S . Indeed, the weaklyHarnack inequality implies that if µ k is a positive measure, then ρ k is positive on R d .Since µ is a probability measure, there exist k ∈ S such that ρ k π k e V ( x ) d x is a positivemeasure. Let S be the set of all µ j such that µ j is zero, and S = S − S . If S isnon-empty, then it follows from (2.4) that q l l = 0 , a.e. , l ∈ S , l ∈ S . It contradicts that Q is fully coupled on R d . Shao-Qin Zhang ρ In this section, we shall study the the entropy estimate of the density ρ . Firstly, weshall prove a Girsanov’s theorem, which will be used to prove the existence of theinvariant measure. Let ( X t , Λ t ) be the solution of the following equationd X t = b (1)Λ t ( X t )d t + √ g Λ t ( X t )d W t , X = ξ (3.1) BasicEqu1 dΛ t = Z ∞ h ( X t , Λ t − , z ) N (d z, d t ) , Λ = λ. (3.2) BasicEqu2
Let ˜ W t = W t − √ Z t b (2)Λ s ( X s )d s (3.3) R t = exp n √ Z t D b (2)Λ s ( X s ) , d W s E − Z t | b (2)Λ s ( X s ) | d s o . (3.4)Let { p t } t ≥ be the Poisson point process corresponding to the random measure N (d z, d t ), i.e. p is a Poisson point process which takes value in (0 , ∞ ) and is in-dependent of the Brownian motion { W t } t ≥ such that N ( U, (0 , t ])( ω ) = { s ∈ D p ( ω ) | s ≤ t, p s ( ω ) ∈ U } , where D p ( ω ) is the domain of the point function p ( ω ). GirTh
Lemma 3.1.
Suppose that (3.1) - (3.2) has a pathwise unique non-explosive solution,and { R t } t ∈ [0 ,T ] is a martingale and P (cid:18)Z T | g Λ s ( X s ) b (2)Λ s ( X s ) | d s < ∞ (cid:19) = 1 . Then, under R T P , { ˜ W t } t ∈ [0 ,T ] is a Brownian process and { p t } t ≥ is a Poisson pointprocess such that they are mutually independent. Consequently, the following equationhas a weak solution d X t = b (1)Λ t ( X t )d t + √ g Λ t ( X t )d W t + √ g Λ t ( X t ) b (2)Λ t ( X t )d t, X = ξ dΛ t = Z ∞ h ( X t , Λ t − , z ) N (d z, d t ) , Λ = λ. Proof. { R t } t ∈ [0 ,T ] is a martingale, ˜ W t is a Brownian motion due to Girsanov’s theo-rem. According to [14, Theorem I.6.3], we only need to prove that the compensatorof N (d z, d t ) under R T P is d z d t .Let K be a measurable subset of R + with its Lebesgue measure | K | < ∞ , N ( K, t ) := N ( K, (0 , t ]) , t ∈ [0 , T ]. Then E h R T ( N ( K, t ) − | K | t ) (cid:12)(cid:12)(cid:12) F s i = − E h | K | tR T (cid:12)(cid:12)(cid:12) F s i + E h N ( K, s ) R T (cid:12)(cid:12)(cid:12) F s i + E h ( N ( K, t ) − N ( K, s )) R T (cid:12)(cid:12)(cid:12) F s i = ( N ( K, s ) − | K | t ) R s + E h ( N ( K, t ) − N ( K, s )) R t (cid:12)(cid:12)(cid:12) F s i nvariant measure of Regime-switching processes
15= ( N ( K, s ) − | K | t ) R s + R s E h ( N ( K, t ) − N ( K, s )) R s,t (cid:12)(cid:12)(cid:12) F s i , where R s,t = exp n Z ts D b (2)Λ r ( X r ) , d W r E − √ Z ts | b (2)Λ r ( X r ) | d r o . Let G = F s ∨ σ (cid:16) N ( U, r ) (cid:12)(cid:12)(cid:12) r ≤ t, U ∈ B ( R + ) (cid:17) . Then E h ( N ( K, t ) − N ( K, s )) R s,t (cid:12)(cid:12)(cid:12) F s i = E h ( N ( K, t ) − N ( K, s )) E h R s,t | G i F s i . Let Π [ s,t ] be the totality of point functions on [ s, t [ for s ≤ t , and let D s,t ( S ) be all thecadlag function on [ s, t ] with value in S . Since the equation has a unique non-explosivestrong solution, there exists F : R d × S × C ([ s, t ] , R d ) × Π [ s,t ] → C ([ s, t ] , R d ) × D s,t ( S )such that ( X r , Λ r ) = F ( X s , Λ s , W [ s,t ] , p [ s,t ] ) r , r ∈ [ s, t ] . Thus E h R s,t (cid:12)(cid:12)(cid:12) G i = E h exp n Z ts D b (2) ( F ( X s , Λ s , W [ s,t ] , p [ s,t ] ) r ) , d W r E − √ Z ts | b (2) ( F ( X s , Λ s , W [ s,t ] , p [ s,t ] ) r ) | d r o(cid:12)(cid:12)(cid:12) G i = E exp n Z t − s D ( b (2) ( F ( x, k, θ s ( W [0 ,t − s ] ) + x, p [ s,t ] ) r ) , d W u E − √ Z t − s | b (2) ( F ( x, k, θ s ( W [0 ,t − s ] ) + x, p [ s,t ] ) r ) | d u o(cid:12)(cid:12)(cid:12) ( x,k,p )=( X s , Λ s , p ) ≤ , P ( ·| G )-a.s.Combining with E R s,t = E (cid:16) E h R s,t (cid:12)(cid:12)(cid:12) G i(cid:17) = 1, we have that E h R s,t (cid:12)(cid:12)(cid:12) G i = 1 , P ( ·| G )-a.s.Since F s ⊂ G , we obtain that E h ( N ( K, t ) − N ( K, s )) E h R s,t | G i(cid:12)(cid:12)(cid:12) F s i = E h N ( K, t ) − N ( K, s ) (cid:12)(cid:12)(cid:12) F s i = | K | ( t − s ) . Then E h R T (cid:16) N ( K, t ) − | K | t (cid:17)(cid:12)(cid:12)(cid:12) F s i = R s h N ( K, s ) − | K | s i . Thus for A ∈ F s E R T n ( N ( K, t ) − | K | t ) A o = E n R T ( N ( K, t ) − | K | t ) A o = E n R s h N ( K, s ) − | K | s i A o = E n R T h N ( K, s ) − | K | s i A o = E R T nh N ( K, s ) − | K | s i A o . Shao-Qin Zhang
Therefore E R T h N ( K, t ) − | K | t (cid:12)(cid:12)(cid:12) F s i = N ( K, s ) − | K | s, which implies that the compensator of N (d z, d t ) under R T P is d z d t .Let P t be the Markov semigroup generated by the Dirichlet form E . Then L isthe generator of P t , and µ π,V is the invariant probability measure of P t . Moreover,for f ∈ L ( R d × S , µ π,V ),( P t ) f ( x, i ) = P it f i ( x ) , x ∈ R d , i ∈ S , where P it is the semigroup generated by L i , and P t can be extended to a contrastive C -semigroup in L p ( R d × S , µ π,V ), p ∈ [1 , ∞ ]. By (H2), N X i =1 π i Z R d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 q ij ( x ) f j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p µ V (d x ) ≤ sup x ∈ R d N X i =1 π i N X j =1 | q ij | pp − ( x ) π p j p − Z R d N X j =1 π j f pj ( x ) µ V (d x ) . Thus Q is a bounded operator in L p ( R d × S , µ π,V ) for all p ∈ [1 , ∞ ]. Let L Q be definedas follows ( L Q f )( x, i ) = ( L i f i )( x ) + N X j =1 q ij ( x ) f j ( x ) , f ∈ C c ( R d × S ) . Then L Q generates a unique contractive C -semigroup in L p ( R d × S , µ π,V ), say P Qt .By (H1), (H2), and for all f ∈ C c ( R d × S ) we have X i π i X j q ij ( f + j ∧ f i − f + i ∧ X i π i X j = i q ij h ( f + j ∧ f − i i − X i π i X j = i q ij ( f + j ∧ f i − + + X i π i q i ( f + i ∧ f i − + = X i π i X j = i q ij h ( f + j ∧ f − i + ( f + i ∧ f i − + i − X i π i X j = i q ij ( f + j ∧ f i − + ≥ X i π i X j = i q ij (1 − f + j ∧ f i − + ≥ . According to the proof of [7, Theorem 3.1], P Qt is also a Markov semigroup. Moreover,we have nvariant measure of Regime-switching processes hyper Lemma 3.2.
For t > , P Qt is hyperbounded, and || P Qt || L q ( µ π,V ) → L q ( t ) ( µ π,V ) < ∞ , q > , q ( t ) = 1 + ( q − e tγ , recalling that γ = max ≤ i ≤ N γ i .Proof. By [15, Theorem 3.1.1], we have P Qt f = P t f + Z t P t − s (cid:0) QP Qs f (cid:1) d s, f ∈ L ∞ ( R d × S , µ π,V ) . (3.5) Pt_Pt0
Then for all p ≥ µ π,V ( | P Qt f | p ) ≤ p µ π,V ( | P t f | p ) + t p − Z t µ π,V ( | QP Qs f | p )d s ≤ p − µ π,V ( | P t f | p ) + 2 p − t p − Z t µ π,V ( | QP Qs f | p )d s ≤ p − µ π,V ( | P t f | p ) + 2 p − t p − b p Z t µ π,V ( | P Qs f | p )d s, where b p = || Q || pL p . Gronwall’s inequality yields µ π,V ( | P Qt f | p ) ≤ p − e p − t p b p µ π,V ( | P t f | p ) . According to [20, Theorem 5.1.4], (H3) and Remark 1.1 imply that P t is hyper-bounded: || P t || L q ( µ π,V ) → L q ( t ) ( µ π,V ) ≤ exp (cid:20) β (cid:18) q − q ( t ) (cid:19)(cid:21) , t > , q > , where q ( t ) := 1 + ( q − e tγ and β = max k ( β k − log π k ). Thus (cid:16) µ π,V ( | P Qt f | q ( t ) ) (cid:17) q ( t ) ≤ − q ( t ) exp β ( q ( t ) − q ) qq ( t ) + 2 q ( t ) − t q ( t ) b q ( t ) q ( t ) ! ( µ π,V ( f q )) q , and the proof of the lemma is completed.Let ( X t , Λ t ) be the process generated by L Q , which satisfies the stochastic differ-ential (1.3) with b k ( x ) = Z k ( x ), k ∈ S , x ∈ R d . The following lemma is a similarresult of [22, Theorem 4.1] for (1.3). lem_mu Lemma 3.3.
Assume (H1)-(H3). Let p = (1 + e γ ) and c = (cid:18) e b − b (cid:19) || P Q || L ( µ π,V ) → L p ( µ π,V ) , where b = || Q || L ( µ π,V ) . There is w > p − p − such that µ π,V (cid:16) e w | Z | (cid:17) := N X i =1 π i µ V (cid:16) e w | Z i | (cid:17) < ∞ . (3.6) inequ_int Shao-Qin Zhang
Then P t has an invariant measure µ = ρµ π,V such that µ π,V ( ρ log ρ ) ≤ (3 p −
1) log µ π,V (cid:16) e w | σ − Z | (cid:17) + 4 wp log c λ ( p − − (3 p − . Proof.
We first consider the Feynman-Kac semigroup for any F ∈ B b ( R d × S ):( P Ft f )( x, i ) = E (cid:20) f ( X ( x,i ) t , Λ ( x,i ) t ) exp (cid:18)Z t F ( X ( x,i ) s , Λ ( x,i ) s )d s (cid:19)(cid:21) , f ∈ B b ( R d × S ) . Let p = 1 + p − p . By Lemma 3.2, || P Q || L ( µ π,V ) → L p ( µ π,V ) < ∞ . Then µ π,V ( | P F f | p ) ≤ µ π,V (cid:18) E (cid:20) | f ( X , Λ ) | exp (cid:18)Z F ( X s , Λ s d s (cid:19)(cid:21)(cid:19) p ! ≤ (cid:16) µ π,V (cid:16) ( P Q f p ) p (cid:17)(cid:17) p ( µ π,V (cid:16) E e pp − R F ( X s , Λ s )d s (cid:17) p − p p − !) p − p ≤ || P Q || L ( µ π,V ) → L p ( µ π,V ) µ π,V ( f p ) (cid:26)Z µ π,V (cid:16) E e pp − F ( X s , Λ s ) (cid:17) d s (cid:27) p − p = || P Q || L ( µ π,V ) → L p ( µ π,V ) µ π,V ( f p ) (cid:26)Z µ π,V (cid:16) P Qs e pp − F (cid:17) d s (cid:27) p − p ≤ || P Q || L ( µ π,V ) → L p ( µ π,V ) µ π,V ( f p ) (cid:26)Z e b s µ π,V (cid:16) e pp − F (cid:17) d s (cid:27) p − p . Thus || P F || L p ( µ π,V ) ≤ c p (cid:16) µ π,V (cid:16) e pp − F (cid:17)(cid:17) p − p p = c p (cid:18) µ π,V (cid:18) e p − p − F (cid:19)(cid:19) p − p − , where c = (cid:16) e b − b (cid:17) || P Q || L ( µ π,V ) → L p ( µ π,V ) . By the Markov property, we have that E e R n F ( X µπ,Vs , Λ µπ,Vs )d s = µ π,V ( P Fn ) ≤ || P F || nL p ( µ π,V ) ≤ c np (cid:18) µ π,V (cid:18) e p − p − F (cid:19)(cid:19) n ( p − p p . Next, let R µ π,V t = exp (cid:26)Z t h Z ( X µ π,V s , Λ µ π,V s ) , d W s i − Z t | Z | ( X µ π,V s , Λ µ π,V s )d s (cid:27) . Then, similar to [22, Theorem 4.1], E R µ π,V n log R µ π,V n = 12 E (cid:18) R µ π,V n Z n | Z ( X s , Λ s ) | d s (cid:19) ≤ ǫ E R µ π,V n log R µ π,V n + ǫ log E h e ǫ R n | Z | ( X s , Λ s ) d s i , ǫ ∈ (0 , . nvariant measure of Regime-switching processes E R µ π,V n log R µ π,V n ≤ nǫp (1 − ǫ ) log " c (cid:18) µ π,V (cid:18) e p − ǫ ( p − | Z | (cid:19)(cid:19) p − p , ǫ ∈ (0 , . By Lemma 3.1, ν n ( f ) := 1 n Z n µ π,V ( P s f ) d s = 1 n Z n E R µ π,V n f ( X µ π,V s , Λ µ π,V s )d s ≤ n E R µ π,V n log R µ π,V n + 1 n log E e R n f ( X µπ,Vs , Λ µπ,Vs )d s ≤ ǫp (1 − ǫ ) log " c (cid:18) µ π,V (cid:18) e p − ǫ ( p − | Z | (cid:19)(cid:19) p − p + 1 p log " c (cid:18) µ π,V (cid:18) e p − p − f (cid:19)(cid:19) p − p , f ≥ , f ∈ B b ( R d × S ) . By these estimates above, following the proof of [22, Theorem 4.1], we can prove thetheorem.
Remark 3.1.
The inequality (3.6) implies that for each i ∈ S , Z i has nice integrabilityw.r.t µ V indeed. It does not work well for (1.3) . In the rest part of this section, weshall give a weaker condition to get a priori estimate to the entropy of the density ofinvariant probability measures. We define x log x = 0 if x = 0. Then we have ineq_nn_en Lemma 3.4.
Assume (H1)-(H3) and that for all i ∈ S , Z i is bounded with compactsupport. Then there is positive ρ ∈ L ( R d × S , u π,V ) such that √ ρ i ∈ H , σ i ( R d ) with π i µ V ( | σ ∗ i ∇√ ρ i | ) + π i µ V ( q i ρ i log ρ i ) ≤ π i µ V ( | Z i | ρ i ) + X k = i π k n µ V ( q ki ρ k log( q ki ρ k )) − µ V ( q ki ρ k ) log µ V ( q ki ρ k ) + µ V ( q ki ρ k ) log µ V ( ρ i ) o , i ∈ S . Proof.
Let q ( n ) ki = q ki [ | x |≤ n ] , Q ( n ) = ( q ( n ) ki ) ≤ k,i ≤ N and L [ n ] = L Z + Q ( n ) . Then Lemma3.3 and Lemma 2.2 imply that the semigroup generated by L [ n ] has an invariantmeasure with positive density ρ [ n ] w.r.t µ π,V such that sup n ≥ µ π,V ( ρ [ n ] log ρ [ n ] ) < ∞ ,which implies that ρ [ n ] i → ρ i weakly in L ( µ V ). Moreover, ρ is a probability densityand also satisfies µ V ( ρLg ) = 0, g ∈ C c ( R d × S ). Hence, Lemma 2.2 implies that ρ ispositive.The equation µ π,V ( ρ [ n ] L [ n ] g ) = 0, g ∈ C c ( R d × S ) yields, for all i ∈ S , π i µ V (cid:16) h Z i , ∇ f i ρ [ n ] i (cid:17) + µ V N X k =1 π k q [ n ] ki f ρ [ n ] k ! = π i µ V (cid:16) h σ ∗ i ∇ ρ [ n ] i , σ ∗ i ∇ f i (cid:17) , f ∈ C c ( R d ) . (3.7) equa_muLf Shao-Qin Zhang
Then there is C n > f such that (cid:12)(cid:12)(cid:12) µ V (cid:16) h σ ∗ i ∇ ρ [ n ] i , σ ∗ i ∇ f i (cid:17)(cid:12)(cid:12)(cid:12) ≤ C n q µ V ( | σ ∗ i ∇ f | ) + µ V ( f ) (3.8) rhH12 The defective log-Sobolev inequality yields the Poincar´e inequality: µ V ( f ) ≤ Cµ V ( | σ ∗ i ∇ f | ) + µ V ( f ) , f ∈ H , σ i ( R d ) . So the norm of f , p µ V ( | σ ∗ i ∇ f | ) + µ V ( f ) , which is induced by the following innerproduct of g , g on H , σ i ( R d ): µ V ( g ) µ V ( g ) + µ V ( h σ ∗ i ∇ g , σ ∗ i ∇ g i ) , g , g ∈ H , σ i ( R d )is equivalent to the Sobolev norm on H , σ i ( R d ). Hence (3.8) implies that ρ [ n ] i ∈ H , σ i ( R d ). Moreover, (3.7) holds for all f ∈ H , σ i ( R d ).Next, we shall prove that sup n ≥ µ V ( | σ ∗ i ∇ q ρ [ n ] i | ) < ∞ . Let f = log( ρ [ n ] i + ǫ )with ǫ ∈ (0 , P Nj =1 π j µ V ( ρ [ n ] j ) = 1 imply that2 π i µ V (cid:18) | σ ∗ i ∇ q ρ [ n ] i + ǫ | (cid:19) + π i µ V ( q ( n ) i ρ [ n ] i log( ρ ( n ) i + ǫ )) ≤ X j = i π j µ V ( q ( n ) ji ρ [ n ] j log( ρ [ n ] i + ǫ )) + π i µ V ( | Z i | ρ [ n ] i ) ≤ X j = i π j n µ V (cid:16) q ( n ) ji ρ [ n ] j log (cid:16) q ( n ) ji ρ [ n ] j (cid:17)(cid:17) − µ V (cid:16) q ( n ) ji ρ [ n ] j (cid:17) log µ V (cid:16) q ( n ) ji ρ [ n ] j (cid:17) + µ V (cid:16) q ( n ) ji ρ [ n ] j (cid:17) log µ V (cid:16) ρ [ n ] i + ǫ (cid:17) o + 12 sup x ∈ R d | Z i | ! π i µ V (cid:16) ρ [ n ] i (cid:17) ≤ X j = i π j n µ V (cid:16) q ( n ) ji ρ [ n ] j log q ( n ) ji (cid:17) + µ V (cid:16) q ( n ) ji ρ [ n ] j log ρ [ n ] j (cid:17) + e − + C Q π j log (cid:18) π i + 1 (cid:19) o + 12 sup x ∈ R d | Z i | ≤ X j = i n (cid:0) C Q log + C Q (cid:1) + C Q π j µ V (cid:16) ρ [ n ] j log ρ [ n ] j (cid:17) + π j ( C Q + 1) e − + C Q log (cid:18) π i + 1 π i (cid:19) o + 12 sup x ∈ R d | Z i | , and π i µ V ( q ( n ) i ρ [ n ] i log( ρ [ n ] i + ǫ )) ≥ π i µ V ( q ( n ) i ρ [ n ] i log ρ [ n ] i ) ≥ − π i C Q e − . Hence, the monotone convergence theorem implies that there is
C > i and n such that µ V (cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ q ρ [ n ] i (cid:12)(cid:12)(cid:12)(cid:12) ! = lim ǫ → + µ V (cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ q ρ [ n ] i + ǫ (cid:12)(cid:12)(cid:12)(cid:12) ! ≤ C. Since the defective log-Sobolev inequality implies the existence of a super Poincar´einequality, the essential spectrum of L i is empty, which implies that H , σ i ( R d ) is nvariant measure of Regime-switching processes L ( µ V ). Thus, √ ρ i ∈ H , σ i ( R d ), and by a subsequence, ρ [ n ] i converges to ρ i in L ( µ V ) and q ρ [ n ] i convergences to √ ρ i weakly in H , σ i ( R d ).For m >
0, let f = log (cid:16) ( ρ [ n ] i ∨ m ) ∧ m (cid:17) . Then (3.7) yields that π i µ V (cid:12)(cid:12)(cid:12) σ ∗ i ∇ ρ [ n ] i (cid:12)(cid:12)(cid:12) [ m ≤ ρ [ n ] i ≤ m ] ρ [ n ] i ! = 4 π i µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ [ n ] i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = π i µ V (cid:16) h Z i , σ ∗ i ∇ ρ [ n ] i i [ m ≤ ρ [ n ] i ≤ m ] (cid:17) + N X k =1 π k µ V (cid:18) q ( n ) ki (cid:20) log (cid:18) ( ρ [ n ] i ∨ m ) ∧ m (cid:19)(cid:21) ρ [ n ] k (cid:19) (3.9) inequ_sq ≤ π i µ V ( | Z i | ρ [ n ] i ) + π i µ V (cid:12)(cid:12)(cid:12) σ ∗ i ∇ ρ [ n ] i (cid:12)(cid:12)(cid:12) [ m ≤ ρ [ n ] i ≤ m ] ρ [ n ] i ! + X k = i π k µ V (cid:18) q ( n ) ki ρ [ n ] k log (cid:18) ( ρ [ n ] i ∨ m ) ∧ m (cid:19)(cid:19) − π i µ V (cid:18) q ( n ) i ρ [ n ] i log (cid:18) ( ρ [ n ] i ∨ m ) ∧ m (cid:19)(cid:19) . Since ρ [ n ] i → ρ i in L ( µ V ), q ρ [ n ] i convergences to √ ρ i weakly in H , σ i ( R d ) andsup n ≥ ,x ∈ R d (cid:12)(cid:12)(cid:12)(cid:12) q ( n ) i log (cid:18) ( ρ [ n ] i ∨ m ) ∧ m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , we have lim n →∞ µ V ( | Z i | ρ [ n ] i ) = µ V ( | Z i | ρ i ),lim n →∞ N X k =1 π k µ V (cid:18) q ( n ) ki ρ [ n ] k log (cid:18) ( ρ [ n ] i ∨ m ) ∧ m (cid:19)(cid:19) = N X k =1 π k µ V (cid:18) q ki ρ k log (cid:18) ( ρ i ∨ m ) ∧ m (cid:19)(cid:19) , and lim n →∞ µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ [ n ] i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Hence, combining these with (3.9), we obtain that2 π i µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + π i µ V (cid:18) q i ρ i log (cid:20) ( ρ i ∨ m ) ∧ m (cid:21)(cid:19) Shao-Qin Zhang ≤ π i µ V ( | Z i | ρ i ) + X k = i µ V (cid:18) q ki ρ k log (cid:18) ( ρ i ∨ m ) ∧ m (cid:19)(cid:19) ≤ π i µ V ( | Z i | ρ i ) + X k = i n µ V ( q ki ρ k log ( q ki ρ k )) (3.10) inequ_muQ − µ V ( q ki ρ k ) log µ V ( q ki ρ k ) + µ V ( q ki ρ k ) log µ V (cid:18) ( ρ i ∨ m ) ∧ m (cid:19) o . Since ρ i > µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = µ V (cid:16)(cid:12)(cid:12) σ ∗ i ∇√ ρ i ∨ m (cid:12)(cid:12) (cid:17) − µ V (cid:16) | σ ∗ i ∇√ ρ i | [ ρ i ≤ m ] (cid:17) , by monotone convergence theorem, we havelim m →∞ µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ µ V (cid:16) | σ ∗ i ∇√ ρ i | (cid:17) . Combining this with (3.10) and Fatou’s Lemma, we have µ V (cid:16) | σ ∗ i ∇√ ρ i | (cid:17) + π i µ V ( q i ρ i log ρ i ) ≤ π i lim m →∞ µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + π i lim m →∞ µ V (cid:18) q i ρ i log (cid:20) ( ρ i ∨ m ) ∧ m (cid:21)(cid:19) ≤ lim m →∞ n π i µ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ∗ i ∇ r ( ρ i ∨ m ) ∧ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + π i µ V (cid:18) q i ρ i log (cid:20) ( ρ i ∨ m ) ∧ m (cid:21)(cid:19) o ≤ π i µ V ( | Z i | ρ i ) + X k = i π k n µ V ( q ki ρ k log ( q ki ρ k )) − µ V ( q ki ρ k ) log µ V ( q ki ρ k ) + µ V ( q ki ρ k ) log µ V ( ρ i ) o The last result of this section is the following lemma on a priori estimate of entropyof ρ i for Z i satisfies the integrability condition (1.10), which is crucial to the proof ofTheorem 1.1. entropy Lemma 3.5.
Let π min = min ≤ k ≤ N π k . Then, under the conditions of Lemma 3.4 andTheorem 1.1, we have the following an a priori estimate N X k =1 π k µ V ( ρ k log ρ k ) ≤ max ≤ k ≤ N (cid:20) v k w k µ V ( e w k | σ − k Z k | ) (cid:21) − log π min nvariant measure of Regime-switching processes
23+ ˜ C Q N X k =1 v k + 2 max ≤ k ≤ N v k β k γ k , (3.11) inequ_ent where ˜ C Q = C Q log + C Q + 2( C Q + 1) e − − C Q log π min . Moreover, there is a constant C depending on π k , C Q , w k , v k and µ V ( e w k | Z k | ) , k ∈ S such that N X k =1 π k µ V ( | σ ∗ k ∇√ ρ k | ) ≤ C. (3.12) inequ_nn Proof.
Let ˆ q ij = inf x ∈ R d q ij ( x ). Then by Lemma 3.4,2 π k µ V (cid:0) | σ ∗ k ∇√ ρ k | (cid:1) + π k µ V ( q k ρ k log ρ k ) ≤ π k w k ( µ V ( ρ k log ρ k ) − µ V ( ρ k ) log µ V ( ρ k ))+ π k w k µ V ( ρ k ) log µ V (cid:16) e w k | Z k | (cid:17) (3.13) nnleqent + X j = k π j (cid:16) ¯ q jk µ V ( ρ j log ρ j ) + µ V ( ρ j ) sup x ∈ R d ( q jk ( x ) log q jk ( x ))+ (¯ q jk − ˆ q jk ) e − + e − − C Q µ V ( ρ j ) log π j (cid:17) . Combining with (H3), we have (cid:18) π k γ k + π k ¯ q k (cid:19) ( µ V ( ρ k log ρ k ) − µ V ( ρ k ) log µ V ( ρ k )) − π k β k γ k µ V ( ρ k ) ≤ π k µ V (cid:0) | σ ∗ k ∇√ ρ k | (cid:1) + π k ¯ q k ( µ V ( ρ k log ρ k ) − µ V ( ρ k ) log µ V ( ρ k )) ≤ π k w k ( µ V ( ρ k log ρ k ) − µ V ( ρ k ) log µ V ( ρ k )) (3.14) entropy_1 + π k w k µ V ( ρ k ) log µ V (cid:16) e w k | Z k | (cid:17) + X j = k π j ¯ q jk ( µ V ( ρ j log ρ j ) − µ V ( ρ j ) log µ V ( ρ j ))+ C Q log + C Q + 2( C Q + 1) e − − C Q log π min . Let ψ = ( ψ j ) ≤ j ≤ N = ( µ V ( ρ j log ρ j ) − µ V ( ρ j ) log µ V ( ρ j )) ≤ j ≤ N , Ψ = 12 w k µ V ( ρ k ) log µ V (cid:16) e w k | Z k | (cid:17) . Then (1.11) and (3.14) yields that0 ≤ h ψ, ( K + ¯ Q ) v i π + h Ψ , v i π + ˜ C Q N X k =1 v k ≤ −h ψ, i π + h Ψ , v i π + ˜ C Q N X k =1 v k , where h v, u i π = P Ni =1 π i v i u i , u, v ∈ R N . Hence N X k =1 π k µ V ( ρ k log( ρ k ))4 Shao-Qin Zhang ≤ h Ψ , v i π + ˜ C Q N X k =1 v k + N X k =1 π k µ V ( ρ k ) log µ V ( ρ k ) + 2 max ≤ k ≤ N β k v k γ k ≤ h Ψ , v i π + ˜ C Q N X k =1 v k − log π min + 2 max ≤ k ≤ N β k v k γ k , in the last inequality, we use P Nk =1 π k µ V ( ρ k ) = 1. Then it is easy to see that theinequality (3.11) holds. Consequently, (3.12) holds by (3.13). Let Z ( n ) i = Z i [ | Z i |≤ n ] , lim n →∞ Z ( n ) i = Z i . Then, according to Lemma 3.3, the Markovsemigroup P ( n ) t generated by L ( n ) has an invariant probability measure, denoted by ρ ( n ) µ π,V . Moreover, Lemma 3.5 yields thatsup n µ π,V ( ρ ( n ) log ρ ( n ) ) + N X i =1 µ V (cid:18) π i | σ ∗ i ∇ q ρ ( n ) i | (cid:19)! < ∞ , which implies there exists a ρ ∈ L ( R d × S , µ π,V ) such that, by a subsequence, ρ ( n ) i → ρ i in L ( µ V ) and q ρ ( n ) i → √ ρ i weakly in H , σ i ( R d ). Next, we shall prove that ρµ π,V is an invariant probability measure of P t . Indeed, noticing that µ π,V ( ρ ( n ) f ) = µ π,V ( ρ ( n ) P nt f )= µ π,V ( ρ ( n ) ( P nt f − P t f )) + µ π,V (( ρ ( n ) − ρ ) P t f ) + µ π,V ( ρP t f ) , and for f ∈ B b ( R d × S )lim n →∞ µ π,V (( ρ ( n ) − ρ ) P t f ) = 0 , lim n →∞ µ π,V ( ρ ( n ) f ) = µ π,V ( ρf ) , in order to prove µ π,V ( ρf ) = µ π,V ( ρP t f ), we only need to prove thatlim n →∞ µ π,V ( ρ n ( P nt f − P t f )) = 0 . Let ( X t , Λ t ) be the process generated by L Q . Then, by Lemma 3.1, µ π,V ( | P nt f − P t f | ) ≤ µ π,V (cid:16) E (cid:12)(cid:12)(cid:12) f ( X t , Λ t ) h e R t D Z ( n )Λ s ( X s ) , d W s E − R t | Z ( n )Λ s ( X s ) | d s − e R t h Z Λ s ( X s ) , d W s i− R t | Z Λ s ( X s ) | d s i(cid:12)(cid:12)(cid:12)(cid:17) ≤ || f || ∞ E e R t (cid:28) Z Λ µπ,Vs ( X µπ,Vs ) , d W s (cid:29) − R t (cid:12)(cid:12)(cid:12)(cid:12) Z Λ µπ,Vs ( X µπ,Vs ) (cid:12)(cid:12)(cid:12)(cid:12) d s ! nvariant measure of Regime-switching processes × E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e R t (cid:28) ( Z n − Z ) Λ µπ,Vs ( X µπ,Vs ) , d W s (cid:29) − R t | ( Z n − Z ) | µπ,Vs ( X µπ,Vs )d s − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . For all η > h from R d × S to R d with | h ( x, i ) | ≤ | Z i ( x ) | ,( x, i ) ∈ R d × S , we have E e η R t h h ( X µπ,Vs , Λ µπ,Vs ) , d W s i− η R t | h ( X µπ,Vs , Λ µπ,Vs ) | ≤ (cid:16) E e η R t h h ( X µπ,Vs , Λ µπ,Vs ) , d W s i− η R t | h ( X µπ,Vs , Λ µπ,Vs ) | (cid:17) × (cid:16) E e (8 η − η ) + R t | h ( X µπ,Vs , Λ µπ,Vs ) | d s (cid:17) ≤ (cid:18) t Z t E e (8 η − η ) + t | h ( X µπ,Vs , Λ µπ,Vs ) | d s (cid:19) ≤ (cid:18) t Z t µ π,V (cid:16) P Qs e (8 η − η ) + t | h | (cid:17) d s (cid:19) ≤ (cid:18) t Z t e b s d s (cid:19) q µ π,V (cid:0) e (8 η − η ) + t | Z | (cid:1) < ∞ , t < min ≤ k ≤ N w k (8 η − η ) + . So, for t small enough, lim n →∞ µ π,V ( | P nt f − P t f | ) = 0 . (4.1) limPP Next, since { ρ ( n ) } n ≥ is uniformly integrable, for ǫ >
0, there is m > n ≥ µ π,V ( ρ ( n ) [ ρ ( n ) ≥ m ] ) ≤ ǫ . Thus µ π,V ( ρ ( n ) | P nt f − P t f | ) ≤ µ π,V ( ρ ( n ) [ ρ ( n )
Then L i = ˆ L i − a i ( ∇ log ρ i ) · ∇ and L Zi = ˆ L i + σ i ˆ Z i · ∇ . We shall prove first that for g i ∈ L ∞ , i ∈ S so that the following equality holds for all f i ∈ C c ( R d ) , i ∈ S N X i =1 π i µ V " g i (1 − ˆ L i ) f i − h σ i ˆ Z i , ∇ f i i g i − N X i =1 π i q ij f j g i ρ i ! , (4.2) equ_mu_fg then g i ∈ H , loc ( ρ i d µ V ). Indeed, fixing some i ∈ S , and letting f k = 0, if k = i and f i ∈ C c ( R d ), we have π i µ V (cid:16) g i ρ i (1 − ˆ L i ) f i (cid:17) = π i µ V (cid:16) h σ i ˆ Z i , ∇ f i i g i ρ i (cid:17) + N X k =1 π k µ V ( q ki f i g k ρ k ) . (4.3) equ_mu_fg-1 Let ζ ∈ C ∞ c ( R d ). Then π i µ V (cid:16) ( ζg i ) ρ i (1 − ˆ L i ) f i (cid:17) = π i µ V (cid:16) g i ρ i (1 − ˆ L i )( ζf i ) (cid:17) + 2 π i µ V ( h a i ∇ ζ, ∇ f i i ρ i ) + π i µ V (cid:16) ( ˆ L i ζ ) g i ρ i (cid:17) = π i µ V (cid:16) h σ i ˆ Z i , ∇ ( ζf i ) i g i ρ i (cid:17) + N X k =1 π k µ V ( q ki ζf i g k ρ k )+ 2 π i µ V ( h a i ∇ ζ, ∇ f i i g i ρ i ) + π i µ V (cid:16) ( ˆ L i ζ ) f i g i ρ i (cid:17) . Since Lemma 2.1, µ V ( e w i | Z i | ) < ∞ and that || σ i || is local bounded, there is a constant C which is independent on f i such that (cid:12)(cid:12)(cid:12) µ V (cid:16) h σ i ˆ Z i , ∇ ( ζf i ) i g i ρ i (cid:17)(cid:12)(cid:12)(cid:12) ≤ || g i || ∞ r µ V (cid:16) ( ζ + |∇ ζ | ) | ˆ Z i | ρ i (cid:17) µ V (cid:0) ( f i + | σ ∗ i ∇ f i | ) ρ i (cid:1) ≤ C q µ V (cid:0) ( f i + | σ ∗ i ∇ f i | ) ρ i (cid:1) . By Lemma 2.2, we also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 π k µ V ( q ki ζf i g k ρ k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ vuuut µ V ( f i ρ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ V N X k =1 π k q ki g k ρ k ! ζ ρ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q µ V ( f i ρ i )inf x ∈ supp ζ ρ i ( x ) vuuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ V N X k =1 π k q ki g k ρ k ! ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C q µ V ( f i ρ i ) , for some positive constant C which is independent of f i . Similarly,2 µ V ( h a i ∇ ζ, ∇ f i i ρ i ) + µ V (cid:16) ( ˆ L i ζ ) g i f i ρ i (cid:17) ≤ C q µ V ( f i ρ i ) . Hence (cid:12)(cid:12)(cid:12) µ V (cid:16) ( ζg i ) ρ i (1 − ˆ L i ) f i (cid:17)(cid:12)(cid:12)(cid:12) ≤ C q µ V (( f i + | σ ∗ i ∇ f i | ) ρ i ) , f i ∈ C c ( R d ) , nvariant measure of Regime-switching processes ζg i ∈ H , σ, ( ρ i d µ V ) , where H , σ i , ( ρ i d µ V ) is the complete of C c ( R d ) under the following norm of f q µ V (( f + | σ ∗ i ∇ f | ) ρ i ) . Thus g i ∈ H , loc ( ρ i d µ V ).Next, we shall prove that there is only one extension of ( L, C c ( R d × S )) thatgenerates a C -semigroup in L ( R d × S , µ ). To this end, we only have to prove that(1 − L )( C c ( R d × S )) is dense in L ( R d × S , µ ). Let ζ n ∈ C ∞ c ( R d ) with B n (0) ≤ ζ n ≤ B n (0) and |||∇ ζ n ||| ∞ ≤ n . Let g i ∈ L ∞ , i ∈ S such that (4.2) hold. Fixing i ∈ S ,and letting f k = 0 if k = i ; f i = ζ n g i , we have, following from (4.2),0 = π i µ V ( ζ n g i ρ i ) + π i µ V (cid:0) h a i ∇ ( ζ n g i ) , ∇ g i i ρ i (cid:1) − π i µ V (cid:16) h σ i ˆ Z i , ∇ ( ζ n g i ) i g i ρ i (cid:17) − µ V N X k =1 π k q ki ( ζ n g i ) g k ρ k ! = π i µ V ( ζ n g i ) + π i µ V ( h a i ∇ ( ζ n g i ) , ∇ ( ζ n g i ) i ρ i ) − π i µ V (cid:0) h a i ∇ ζ n , ∇ ζ n i g i ρ i (cid:1) (4.4) equa_zeg − π i µ V (cid:16) h σ i ˆ Z i , ∇ ( ζ n g i ) i g i ρ i (cid:17) − µ V N X k =1 π k q ki ( ζ n g i ) g k ρ k ! . Letting f k ( x ) = ( ζ n g i ) ( x ) [ k = i ] ( k ) , x ∈ R d , k ∈ S , since ( ζ n g i ) ∈ H , σ i ( ρ i d µ V ), we have µ ( ˆ L f ) = 0 and µ ( Lf ) = 0. Then the equality µ ( ˆ L f ) = µ ( Lf ) yields that0 = π i µ V (cid:16) h σ ˆ Z i , ∇ ( ζ n g i ) i ρ i (cid:17) + µ V N X k =1 π k q ki ζ n g i ρ k ! = π i µ V (cid:16) h σ ˆ Z i , ∇ ( ζ n g i ) i g i ρ i (cid:17) + π i µ V (cid:16) h ˆ Z i , ∇ g i i g i ζ n ρ i (cid:17) + µ V N X k =1 π k q ki ζ n g i ρ k ! = 2 π i µ V (cid:16) h σ ˆ Z i , ∇ g i i g i ζ n ρ i (cid:17) + π i µ V (cid:16) h σ ˆ Z i , ∇ ζ n i g i ρ i (cid:17) + µ V N X k =1 π k q ki ζ n g i ρ k ! . Thus π i µ V (cid:16) h σ ˆ Z i , ∇ g i i g i ζ n ρ i (cid:17) = − π i µ V (cid:16) h ˆ Z i , ∇ ζ n i ζ n g i ρ i (cid:17) − µ V N X k =1 π k q ki ζ n g i ρ k ! ,π i µ V (cid:16) h σ i ˆ Z i , ∇ ( ζ n g i ) i g i ρ i (cid:17) = 2 π i µ V (cid:16) h σ i ˆ Z i , ∇ ζ n i ζ n g i ρ i (cid:17) + π i µ V (cid:16) h σ i ˆ Z i , ∇ g i i g i ζ n ρ i (cid:17) = π i µ V (cid:16) h σ i ˆ Z i , ∇ ζ n i ζ n g i ρ i (cid:17) − µ V N X k =1 π k q ki ( ζ n g i ) ρ k ! . Combining with (4.4), we arrive at π i µ V ( ζ n g i ρ i ) + π i µ V ( h a i ∇ ( ζ n g i ) , ∇ ( ζ n g i ) i ρ i )8 Shao-Qin Zhang = π i µ V (cid:0) h a i ∇ ζ n , ∇ ζ i g i ρ i (cid:1) + µ V N X k =1 q ki ( ζ n g i ) g k ρ k ! (4.5) zeqq + π i µ V (cid:16) h σ i ˆ Z i , ∇ ζ n i ζ n g i ρ i (cid:17) − µ V N X k =1 π k q ki ( ζ n g i ) ρ k ! . Since N X k =1 π k q ki g i g k ρ k − N X k =1 π k q ki g i ρ k = − π i q i g i ρ i + 12 N X k = i π k q ki g i ( g k − g i ) ρ k + 12 N X k = i π k q ki g i g k ρ k = − π i q i g i ρ i − N X k = i π k q ki ( g k − g i ) ρ k + 12 N X k = i π k q ki g k ( g k − g i ) ρ k + 12 N X k = i π k q ki g i g k ρ k = − π i q i g i ρ i − N X k = i π k q ki ( g k − g i ) ρ k + 12 N X k = i π k q ki g k ρ k , we have that N X i =1 N X k =1 π k q ki g i g k ρ k − N X k =1 π k q ki g i ρ k ! = − N X i =1 π i q i g i ρ i − N X i =1 N X k = i π k q ki ( g k − g i ) ρ k + 12 N X i =1 N X k = i π k q ki g k ρ k = − N X i =1 N X k = i π k q ki ( g k − g i ) ρ k − N X i =1 π i q i g i ρ i + 12 N X k =1 N X i = k π k q ki g k ρ k (4.6) qqgg = − N X i =1 N X k = i π k q ki ( g k − g i ) ρ k . Hence, (4.5), (4.6) and the definition of ζ n yield that N X i =1 π i µ V (cid:0) ζ n g i ρ i + h a i ∇ ( ζ n g i ) , ∇ ( ζ n g i ) i ρ i (cid:1) + µ V ζ n N X k = i π k q ki ( g k − g i ) ρ k = N X i =1 π i µ V (cid:0) h a i ∇ ζ n , ∇ ζ n i g i ρ i (cid:1) + N X i =1 π i µ V (cid:16) h σ i ˆ Z i , ∇ ζ n i ζ n g i ρ i (cid:17) ≤ max i ∈ S || g i || ∞ n N X i =1 µ V ( || a i || ρ i ) + max i ∈ S || g i || ∞ n N X i =1 π i µ V (cid:16) | σ i ˆ Z i | ρ i (cid:17) . nvariant measure of Regime-switching processes r small enough, we have µ V ( || a i || ρ i ) + µ V (cid:16) | σ i ˆ Z i | ρ i (cid:17) ≤ µ V (cid:0) ρ i ( | Z i | + 3 || σ i || ) (cid:1) + µ V ( || σ i || · | σ ∗ i ∇ ρ i | ) ≤ µ V (cid:0) ρ i | Z i | (cid:1) + 2 µ V (cid:0) ρ i || σ i || (cid:1) + 2 µ V (cid:0) | σ ∗ i ∇√ ρ i | (cid:1) ≤ r µ V ( ρ i log ρ i ) + 2 µ V (cid:0) | σ ∗ i ∇√ ρ i | (cid:1) − r µ V ( ρ i ) log µ V ( ρ i ) (4.7) ZA + 12 r n log µ V (cid:16) e r | Z i | (cid:17) + log µ V (cid:16) e r || σ i || (cid:17) o < ∞ , i ∈ S . 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