aa r X i v : . [ m a t h . P R ] N ov Chapter 6 in “Probability Approximations and Beyond”, Lecture Notes inStatistics 205, 2012
Basic Estimates of Stability Rate forOne-dimensional Diffusions
Mu-Fa Chen(Beijing Normal University, Beijing 100875, China)
Abstract
In the context of one-dimensional diffusions, we present basic esti-mates (having the same lower and upper bounds with a factor of 4 only)for four Poincar´e-type (or Hardy-type) inequalities. The derivation oftwo estimates have been open problems for quite some time. The boundsprovide exponentially ergodic or decay rates. We refine the bounds andillustrate them with typical examples.
An earlier topic on which Louis Chen has studied is about the Poincar´e-typeinequalities (see [1, 2], for instance). We now use this good chance to introduc-tion in Section 2 some recent progress on the topic, especially for one-dimen-sional diffusions (elliptic operators). The basic estimates of exponentially er-godic (or decay) rate and the principal eigenvalue in different cases are pre-sented. Here the term “basic” means that upper and lower bounds are givenby an isoperimetric constant up to a factor four. As a consequence, the criteriafor the positivity of the rate and the eigenvalue are obtained. The proof of themain result is sketched in Section 3. The materials given in Sections 4, 5, andAppendix are new. In particular, the basic estimates are refined in Section 4and the results are illustrated through examples in Section 5. The coincidenceof the exponentially decay rate and the corresponding principal eigenvalue isproven in Appendix for a large class of symmetric Markov processes.
Mathematics Subject Classifications . 60J60, 34L15, 26D10.
Key words and phases . First nontrivial eigenvalue, Hardy’s inequality, one-dimensionaldiffusion, principal eigenvalue, Poincar´e-type inequality, stability rate.In the published version as Chapter 6 in the book, Theorem 2.1, Proposition 3.2, Corol-lary 4.3 and Example 5.1 here are relabeled as Theorem 6.1, Proposition 6.1, Corollary 6.1and Example 6.1, respectively. Similarly, formulas (1)–(36) here are relabeled as (6.1)–(6.36).
Mu-Fa Chen
Let us recall two types of exponential convergence often studied for Markovprocesses. Let P t ( x, · ) be a transition probability on a measurable state space( E, E ) with stationary distribution π . Then the process is called exponentiallyergodic if there exists a constant ε > c ( x ) such that k P t ( x, · ) − π k Var c ( x ) e − εt , t > , x ∈ E. (1)Denote by ε max be the maximal rate ε . For convenience, in what follows, weallow ε max = 0. Next, let L ( π ) be the real L ( π )-space with inner product( · , · ) and norm k · k respectively, and denote by { P t } t > the semigroup of theprocess. Then the process is called to have L -exponential convergence if thereexists some η ( >
0) such that k P t f − π ( f ) k k f − π ( f ) k e − ηt , t > , f ∈ L ( π ) , (2)where π ( f ) = R E f d π . It is known that η max is described by λ : λ = inf { ( f, − Lf ) : f ∈ D ( L ) , π ( f ) = 0 , k f k = 1 } , (3)where L is the generator with domain D ( L ) of the semigroup in L ( π ). Eventhough the topologies for these two types of exponential convergence are ratherdifferent, but we do have the following result. Theorem 2.1 ([3, 6])
For a reversible Markov process with symmetric measure π , if with respect to π , the transition probability has a density p t ( x, y ) havingthe property that the diagonal elements p s ( · , · ) ∈ L / ( π ) for some s > , and aset of bounded functions with compact support is dense in L ( π ) , then we have ε max = λ . As an immediate consequence of the theorem, we obtain some criterion for λ > ε max >
0. In our recent study, wego to the opposite direction: estimating ε max in terms of the spectral theory.We are also going to handle with the non-ergodic case in which (2) becomes µ (cid:0) ( P t f ) (cid:1) µ (cid:0) f (cid:1) e − ηt , t > , f ∈ L ( µ ) , (4)where µ is the invariant measure of the process. Then η max becomes λ = inf (cid:8) − µ ( f Lf ) : f ∈ C , µ (cid:0) f (cid:1) = 1 (cid:9) , (5)where C is a suitable core of the generator, the smooth functions with compactsupport for instance in the context of diffusions. However, the totally vari-ational norm in (1) may be meaningless unless the process being explosive.Instead of (1), we consider the following exponential convergence: P t ( x, K ) c ( x, K ) e − εt , t > , x ∈ E, K : compact , (6) asic estimates of exponential rate K , c ( · , K ) is locally µ -integrable. Under some mildcondition, we still have ε max = λ . See Appendix for more details. We now turn to our main object: one-dimensional diffusions. The state spaceis E := ( − M, N ) (
M, N ∞ ). Consider an elliptic operator L = a ( x ) d d x + b ( x ) dd x , where a > E . Then define a function C ( x ) as follows: C ( x ) = Z xθ ba , x ∈ E, where θ ∈ E is a reference point. Here and in what follows, the Lebesguemeasure d x is often omitted. It is convenient for us to define two measures µ and ν as follows. µ (d x ) = e C ( x ) a ( x ) d x, ν (d x ) = e − C ( x ) d x. The first one has different names: speed , or invariant , or symmetrizable mea-sure . The second one is called scale measure . Note that ν is infinite iff theprocess is recurrent. By using these measures, the operator L takes a verycompact form L = dd µ dd ν (cid:16) i.e., Lf ≡ a e − C (cid:0) f ′ e C (cid:1) ′ (cid:17) (7)which goes back to a series of papers by W. Feller, for instance [12].Consider first the special case that M, N < ∞ . Then the ergodic casemeans that the process has reflection boundaries at − M and N . In analyticlanguage, we have Neumann boundaries at − M and N : the eigenfunction g of λ satisfies g ′ ( − M ) = g ′ ( N ) = 0. Otherwise, in the non-ergodic case, oneof the boundaries becomes absorbing. In analytic language, we have Dirichletboundary at − M (say): the eigenfunction g of λ satisfies g ( − M ) = 0. Letus use codes “D” and “N”, respectively, to denote the Dirichlet and Neumannboundaries. The corresponding minimal eigenvalues of − L are listed as follows. • λ NN : Neumann boundaries at − M and N , • λ DD : Dirichlet boundaries at − M and N , • λ DN : Dirichlet at 0 and Neumann at N , • λ ND : Neumann at 0 and Dirichlet at N . Mu-Fa Chen
We call them the first non-trivial or the principal eigenvalue . In the lasttwo cases, setting M = 0 is for convenience in comparison with other resultsto be discussed later but it is not necessary. Certainly, this classificationis still meaningful if M or N is infinite. For instance, in the ergodic case,the process will certainly come back from any starting point and so one mayimagine the boundaries ±∞ as reflecting. In other words, the probabilisticinterpretation remains the same when M , N = ∞ . However, the analyticNeumann condition that lim x →±∞ g ′ ( x ) = 0 for the eigenfunction g of λ NN may be lost (cf. the first example given in Section 5). More seriously, thespectrum of the operator may be continuous for unbounded intervals. Thisis the reason why we need the L -spectral theory. In the Dirichlet case, theanalytic condition that lim x →±∞ g ( x ) = 0 can be implied by the definitiongiven below, once the process goes ±∞ exponentially fast. Now, for general M, N ∞ , let D ( f ) = Z N − M f ′ e C , M, N ∞ , f ∈ A ( − M, N ) , A ( − M, N ) = the set of absolutely continuous functions on ( − M, N ) , A ( − M, N ) = { f ∈ A ( − M, N ) : f has a compact support } . From now on, the inner product ( · , · ) and the norm k · k are taken with respectto µ (instead of π ). Then the principal eigenvalues are defined as follows. λ DD = inf { D ( f ) : f ∈ A ( − M, N ) , k f k = 1 } , (8) λ ND = inf { D ( f ) : f ∈ A (0 , N ) , f ( N − ) = 0 if N < ∞ , k f k = 1 } , (9) λ NN = inf { D ( f ) : f ∈ A ( − M, N ) , µ ( f ) = 0 , k f k = 1 } , (10) λ DN = inf { D ( f ) : f ∈ A (0 , N ) , f (0+) = 0 , k f k = 1 } . (11)Certainly, the above classification is closely related to the measures µ and ν .For instance, in the DN- and NN-cases, one requires that µ (0 , N ) < ∞ and µ ( − M, N ) < ∞ , respectively. Otherwise, one gets a trivial result as can beseen from Theorem 2.2 below.To state the main result of the paper, we need some assumptions. Inthe NN-case (i.e., the ergodic one), we technically assume that a and b arecontinuous on ( − M, N ). For λ DN and λ NN , we allow the process to be explosivesince the maximal domain is adopted in definition of λ DN and λ NN . But for λ ND and λ DD , we are working for the minimal process (using the minimaldomain) only, assuming that µ and ν are locally finite. Theorem 2.2 (Basic estimates [9])
Under the assumptions just mentioned,corresponding to each -case, we have (cid:0) κ (cid:1) − / λ = ε max (cid:0) κ (cid:1) − , (12) asic estimates of exponential rate where (cid:0) κ NN (cid:1) − = inf x
Hardy’s inequality k f k pL p ( µ ) A p Z N − M | f ′ | p e C , f (0) = 0 , p > A p denotes the optimal constant in the inequality. Certainly, A = (cid:0) λ DN (cid:1) − . This was initialed, for the specific operator L = x d / d x , by G.H.Hardy [16] in 1920, motivated from a theorem of Hilbert on double series. Towhich, several famous mathematicians (H. Weyl, F.W. Wiener, I. Schur, etal.) were involved. After a half-century, the basic estimates in the DN-casewere finally obtained by several mathematicians, for instance B. Muckenhoupt(1972). The reason should be now clear why (15) can be so famous in thehistory. The estimate of λ ND was given in Maz’ya (1985). In the DD-case, theproblem was begun by P. Gurka [14] around 1989 and then improved in thebook by Opic and Kufner (1990) with a factor ≈
22. In terms of a splittingtechnique, the NN-case can be reduced to the Muckenhoupt’s estimate witha factor 8, as shown by Miclo (1999) in the context of birth–death processes.A better estimate can be done in terms of variational formulas given in [4;Theorem 3.3]. It is surprising that in the more complicated DD- and NN-cases,by adding one more parameter only, we can still obtain a compact expression(13) and (14). Note that these two formulas have the following advantage: theleft- and the right-hand parts are symmetric; the cases having finite or infiniteintervals are unified together without using the splitting technique. asic estimates of exponential rate Here is a quick overview of our motivation and application of the study on thistopic. Consider the ϕ -model on the d -dimensional lattice Z d . At each site i ,there is a one-dimensional diffusion with operator L i = d / d x i − u ′ ( x i )d / d x i ,where u i ( x i ) = x i − βx i having a parameter β >
0. Between the nearestneighbors i and j in Z d , there is an interaction. That is, we have an interactionpotential H ( x ) = − J P h ij i x i x j with parameter J >
0. For each finite box Λ(denoted by Λ ⋐ Z d ) and ω ∈ R Z d , let H ω Λ denote the conditional Hamiltonian(which acts on those x : x k = ω k for all k / ∈ Λ). Then, we have a local operator L ω Λ = X i ∈ Λ (cid:2) ∂ ii − ∂ i ( u + H ω Λ ) ∂ i (cid:3) . We proved that the first non-trivial eigenvalue λ β (cid:0) Λ , ω (cid:1) (as well as the log-arithmic Sobolev constant σ β (cid:0) Λ , ω (cid:1) which is not touched here) of L ω Λ is ap-proximately exp[ − β / − dJ uniformly with respect to the boxes Λ and theboundaries ω . The leading rate β / Theorem 2.3 ([8])
For the ϕ -model given above, we have inf Λ ⋐ Z d inf ω ∈ R Z d λ β (cid:0) Λ , ω (cid:1) ≈ inf Λ ⋐ Z d inf ω ∈ R Z d σ β (cid:0) Λ , ω (cid:1) ≈ exp (cid:2) − β / − c log β (cid:3) − dJ, where c ∈ [1 , . See Figure 1. β r = 2 dJ λ β , σ β :exp (cid:2) − β / − c log β (cid:3) − rc = c ( β ) ∈ [1 , Figure 1
Phase transition of the ϕ modelThe figure says that in the gray region, the system has a positive principaleigenvalue and so is ergodic; but in the region which is a little away above the Mu-Fa Chen curve, the eigenvalue vanishes. The picture exhibits a phase transition. Thekey to prove Theorem 2.2 is a deep understanding about the one-dimensionalcase. Having one-dimensional result at hand, as far as we know, there are atleast three different ways to go to the higher or even infinite dimensions: theconditional technique used in [8]; the coupling method explained in [6; Chapter2]; and some suitable comparison which is often used in studying the stabilityrate of interacting particle systems. This explains our original motivation andshows the value of a sharp estimate for the leading eigenvalue in dimensionone. The application of the present result to this model should be clear now.
The hardest part of Theorem 2.2 is the assertion for λ NN . Here we sketch itsproof. Meanwhile, the proof for λ DD is also sketched. The proof for the firstassertion consists mainly of three steps by using three methods: the couplingmethod, the dual method, and the capacitary method. The next result was proved by using the coupling technique.
Theorem 3.1 (Chen and Wang (1997))
For the operator L on (0 , ∞ ) withreflection at , we have λ = λ NN > sup f ∈ F inf x> (cid:20) − b ′ − af ′′ + ( a ′ + b ) f ′ f (cid:21) ( x ) , (17) F = (cid:8) f ∈ C (0 , ∞ ) : f (0) = 0 , f | (0 , ∞ ) > (cid:9) . (18) Actually, the equality sign holds once the eigenfunction ot λ belongs to C . We now rewrite the above formula in terms of an operator, Schr¨odingeroperator L S . λ = sup f ∈ F inf x> (cid:20) − b ′ − af ′′ + ( a ′ + b ) f ′ f (cid:21) ( x ) (19)= sup f ∈ F inf x> (cid:18) − L S ff (cid:19) ( x ) =: λ S , (20) L S = a ( x ) d d x + (cid:0) a ′ ( x ) + b ( x ) (cid:1) dd x + b ′ ( x ) . (21)The original condition π ( f ) = 0 in the definition of λ NN means that f hasto change its sign. Note that f is regarded as a mimic of the eigenfunction g . The difficulty is that we do not know where the zero-point of g is located.In the new formula (20), the zero-point of f ∈ F is fixed at the boundary 0, asic estimates of exponential rate b ′ ( x ).Since b ′ ( x ) can be positive, the operator L S is Schr¨odinger but may not be anelliptic operator with killing. Up to now, we are still unable to handle withgeneral Schr¨odinger operator (even with killing one), but at the moment, thepotential term is very specific so it gives a hope to go further. To overcome the difficulty just mentioned, the idea is a use of duality. Thedual now we adopted is very simple: just an exchange of the two measures µ and ν . Recall that the original operator is L = dd µ dd ν by (7). Hence the dualoperator takes the following form L ∗ = dd µ ∗ dd ν ∗ = dd ν dd µ , (22) L ∗ = a ( x ) d d x + (cid:0) a ′ ( x ) − b ( x ) (cid:1) dd x , x ∈ (0 , ∞ ) . (23)This dual goes back to Siegmund (1976) and Cox & R¨osler (1983) (in whichthe probabilistic meaning of this duality was explained), as an analog of theduality for birth–death process (cf. [9] for more details and original references).It is now a simple matter to check that the dual operator is a similar transformof the Schr¨odinger one L ∗ = e C L S e − C . (24)This implies that − L S ff = − L ∗ f ∗ f ∗ , where f ∗ := e C f is one-to-one from F into itself. Therefore, we have λ S = sup f ∈ F inf x> − L S ff ( x ) = sup f ∗ ∈ F inf x> − L ∗ f ∗ f ∗ ( x ) = λ ∗ DD , where the last equality is the so-called Barta’s equality .we have thus obtained the following identity.
Proposition 3.2 λ = λ S = λ ∗ DD . Actually, we have a more general conclusion that L S and L ∗ are isospectralfrom L (cid:0) e C d x (cid:1) to L (cid:0) e − C d x (cid:1) . This is because of Z e C f L S g = Z e − C ( e C f ) (cid:0) e C L S e − C (cid:1) ( e C g ) = Z e − C f ∗ L ∗ g ∗ , and L S and L ∗ have a common core. But L on L ( µ ) and its dual L ∗ on L (cid:0) e − C d x (cid:1) are clearly not isospectral.0 Mu-Fa Chen
The rule mentioned in the remark after Theorem 2.2, and used to deduced(14) from (13), comes from this duality. Nevertheless, it remains to compute λ DD for the dual operator. To compute λ DD , we need a general result which comes from a different di-rection to generalize the Hardy-type inequalities. In contract to what we havetalked so far, this time we extend the inequalities to the higher dimensional sit-uation. This leads to a use of the capacity since in the higher dimensions, theboundary may be very complicated. After a great effort by many mathemati-cians (see for instance Maz’ya 1985; Hasson 1979; Vondraˇcek 1996; Fukushima& Uemura 2003; and [7]), we have finally the following result. Theorem 3.3
For a regular transient Dirichlet form ( D, D ( D )) with locallycompact state space ( E, E ) , the optimal constant A B in the Poincar´e-type in-equality (cid:13)(cid:13) f (cid:13)(cid:13) B A B D ( f ) , f ∈ C ∞ K ( E ) satisfies B B A B B B , where k · k B is the norm in a normed linear space B and B B = sup compact K Cap( K ) − k K k B . The space B can be very general, for instance L p ( µ ) ( p >
1) or the Orliczspaces. In the present context, D ( f ) = R N − M f ′ e C , D ( D ) is the closure of C ∞ K ( − M, N ) with respect to the norm k · k D : k f k D = k f k + D ( f ), andCap( K ) = inf (cid:8) D ( f ) : f ∈ C ∞ K ( − M, N ) , f | K > (cid:9) . Note that we have the universal factor 4 here and the isoperimetric constant B B has a very compact form. We now need to compute the capacity only. Theproblem is that the capacity is usually not computable explicitly. For instance,at the moment, I do not know how to compute it for Schr¨odinger operatorseven for the elliptic operators having killings. Very lucky, we are able tocompute the capacity for the one-dimensional elliptic operators. The resulthas a simple expression: B B = sup − M Here is the summery of our proof. First, by a change of the topology, we reducethe study on ε max to λ NN . Then, by coupling, we reduce λ NN to λ S . Next, byduality, we reduce λ S to λ ∗ DD . We use capacitary method to compute λ ∗ DD .Finally, we use duality again to come back to λ NN . Recall that our originalpurpose is using λ = λ NN to study the phase transition, a basic topic in thestudy on interacting particle systems (abbrev. IPS). It is very interesting thatwe now have an opposite interaction. We use the main tools (coupling andduality) developed in the study on IPS to investigate a very classical problemand produce an interesting result. The basic estimates given in Theorem 2.2 can be further improved. For half-line at least, we have actually an approximating procedure for each of theprincipal eigenvalues. Refer to [6, 9] and references therein. Moreover, onemay approach the whole line by half-lines. Here we consider an additionalmethod but concentrate on λ DD and λ NN only. As will be seen soon, theresulting bounds are much more complicated, less simple and less symmetry,than those given in Theorem 2.2.Let us begin with a simper but effective result. Proposition 4.1 We have λ DD (cid:0) ¯ κ DD (cid:1) − (cid:0) κ DD (cid:1) − Mu-Fa Chen and λ NN (cid:0) ¯ κ NN (cid:1) − (cid:0) κ NN (cid:1) − , where (cid:0) ¯ κ DD (cid:1) − = inf x 0, define h − ( z ) = h − f ( z ) = ν (cid:16) µ (cid:0) ( · , θ ) f (cid:1) ( − M, z ) (cid:17) = Z z − M e − C ( x ) d x Z θx e C fa , z θ, (25) h + ( z ) = h + f ( z ) = ν (cid:16) µ (cid:0) ( θ, · ) f (cid:1) ( z, N ) (cid:17) = Z Nz e − C ( x ) d x Z xθ e C fa , z > θ, (26)i.e. (by exchanging the order of the integrals), h − ( z ) = µ (cid:0) f ν ( − M, · ∧ z ) (cid:1) = µ (cid:16) f ν ( − M, · ) ( − M,z ) (cid:17) + µ (cid:16) f ( z,θ ) (cid:17) ν ( − M, z ) , z θ,h + ( z ) = µ (cid:0) f ν ( · ∨ z, N ) (cid:1) = µ (cid:16) f ν ( · , N ) ( z, N ) (cid:17) + µ (cid:16) f ( θ, z ) (cid:17) ν ( z, N ) , z > θ, asic estimates of exponential rate x ∧ y = min { x, y } , x ∨ y = max { x, y } , and θ = θ ( f ) ∈ ( − M, N ) is theunique root of the equation: h − ( θ ) = h + ( θ )provided h ± f < ∞ . Next, define II ± ( f ) = h ± /f . Theorem 4.2 (Variational formula) Let a and b be continuous and a > on ( − M, N ) .(1) Assume that ν ( − M, N ) < ∞ . Using the notation above, we have λ DD = sup f ∈ C + h inf z ∈ ( − M,θ ) II − ( f )( z ) − i ^ h inf z ∈ ( θ,N ) II + ( f )( z ) − i , (27) where C + = { f ∈ C ( − M, N ) : f > on ( − M, N ) } .(2) Assume that µ ( − M, N ) < ∞ . Then (27) holds replacing λ DD by λ NN provided in definition of h ± , µ and ν are exchanged. Proof . By duality, it suffices to prove the first assertion.(a) Without loss of generality, assume that h ± f < ∞ . Otherwise, theassertion is trivial. First, we prove “ > ”. Let h ( z ) = ( h − ( z ) , z θ,h + ( z ) , z > θ, Clearly, h | ( − M,N ) > h ∈ C ( − M, N ) in view of definition of θ . Next,note that h −′ ( x ) = e − C ( x ) Z θx e C a f, h −′′ ( x ) = e − C ( x ) (cid:20) − ba Z θx e C a f − e C a f (cid:21) , x < θ ; h + ′ ( x ) = − e − C ( x ) Z xθ e C a f, h + ′′ ( x ) = e − C ( x ) (cid:20) ba Z xθ e C a f − e C a f (cid:21) , x > θ. Obviously, h ′ ( θ ± 0) = 0. Since a , b and f are continuous and a > − M, N ), we also have h ′′ ( θ +0) = h ′′ ( θ − 0) and so h ∈ C ( − M, N ). Therefore,by Barta’s equality, we have λ DD = sup g ∈ F inf z ∈ ( − M,N ) − Lgg ( z ) > inf z ∈ ( − M,N ) − Lhh ( z )= h inf z ∈ ( − M,θ ) − Lh − h − ( z ) i ^ h inf z ∈ ( θ,N ) − Lh + h + ( z ) i . Mu-Fa Chen Now, by (7), required assertion follows by a simple computation.(b) Next, we show that the equality sign in (27) holds. The assertionbecomes trivial if λ DD = 0. Otherwise, the eigenfunction g of λ DD should beunimodal (which seems known in the Sturm–Liouville theory and is proved inthe discrete context [9; Proposition 7.14]. Actually, the discrete case is evenmore complex since the eigenfunction can be a simple echelon, not necessarilyunimodal). By setting f = g and θ to be the maximum point of g ( g ′ ( θ ) = 0),it follows that II ± ( f ) − ≡ λ DD and hence the equality sign holds. (cid:3) We now introduce a typical application of Theorem 4.2. Fix x < y . Define f x,y ( s ) = s ν ( y, N ) ν ( − M, x ) ν ( − M, s ∧ x ) , s y p ν ( s, N ) , s > y and set κ DD = inf x Under assumptions of Theorem 4.2, we have λ DD > (cid:0) κ DD (cid:1) − and λ NN > (cid:0) κ NN (cid:1) − . We remark that the assumption in part (1) of Theorem 4.2 is necessaryfor DD-case (cf. (13)). Recall that (27) is a complete variational formulafor the lower estimates of λ DD . Starting at f = f used in Corollary 4.3,replacing f and h used in Theorem 4.2 by f n − and f n , respectively, we obtainan approximating procedure from below for λ DD . Dually, we can obtain avariational formula for the upper estimates of λ DD and an approximatingprocedure from above. Here we omit all of the details. The same remark ismeaningful for λ NN , which is especially interesting since here we do not usethe property that µ ( f ) = 0 for the test function f . The new difficulty of (27)is that θ ( f ) may not be computable analytically. This costs a question toprove that κ DD κ DD which should be true in view of our knowledge on thehalf-line, and is illustrated by examples in the next section. It is noticeablethat the method works for the whole line and the use of θ ( f ) is essentiallydifferent from what used in the splitting technique. Finally, we mention thatthe method used here is meaningful for birth–death processes, refer to [9;Lemma 7.12]. asic estimates of exponential rate h ± used in Corollary 4.3 moreexplicitly. Let ν − ( s ) = ν ( − M, s ) and ν + ( s ) = ν ( s, N ) for simplicity. Then f ( s ) = f x,y ( s ) = p ν + ( y ) ν − ( s ) (cid:14)p ν − ( x ) , s x p ν + ( y ) , x s y p ν + ( s ) , s > y, (28)and h − ( z ) = µ (cid:16) f ν − ( − M, z ) (cid:17) + ν − ( z ) µ (cid:0) f ( z, θ ) (cid:1) , z θ, (29) h + ( z ) = µ (cid:16) f ν + ( z, N ) (cid:17) + ν + ( z ) µ (cid:0) f ( θ, z ) (cid:1) , z > θ. (30)We now consider the typical case that θ ∈ [ x, y ]. Then, h − ( θ ) = s ν + ( y ) ν − ( x ) µ (cid:16) ν / − ( − M, x ) (cid:17) + p ν + ( y ) µ (cid:16) ν − ( x, θ ) (cid:17) ,h + ( θ ) = µ (cid:16) ν / ( y, N ) (cid:17) + p ν + ( y ) µ (cid:16) ν + ( θ, y ) (cid:17) . Hence the equation h − ( θ ) = h + ( θ ) becomes1 p ν − ( x ) µ (cid:16) ν / − ( − M, x ) (cid:17) + µ (cid:16) ν − ( x,θ ) (cid:17) = 1 p ν + ( y ) µ (cid:16) ν / ( y, N ) (cid:17) + µ (cid:16) ν + ( θ, y ) (cid:17) , θ ∈ [ x, y ] . (31)Furthermore, by some computations, we obtain the ratio h ± /f x,y as follows.We have for z : z x θ y that II − (cid:0) f x,y (cid:1) ( z ) = 1 p ν − ( z ) µ (cid:16) ν / − ( − M, z ) (cid:17) + p ν − ( z ) µ (cid:16) √ ν − ( z, x ) (cid:17) + p ν − ( z ) ν − ( x ) µ ( x, θ ) , (32)and for z : z > y > θ that II + (cid:0) f x,y (cid:1) ( z ) = 1 p ν + ( z ) µ (cid:16) ν / ( z, N ) (cid:17) + p ν + ( z ) µ (cid:16) √ ν + ( y, z ) (cid:17) + p ν + ( z ) ν + ( y ) µ ( θ, y ) . (33)Note that by (25) and (26), h − is increasing on [ x, θ ] and h + is decreasing on[ θ, y ]. Since f x,y is a constant on [ x, y ], it follows thatmax z ∈ [ x,θ ] h − ( z ) f x,y ( z ) = h − ( θ ) f x,y ( x ) and max z ∈ [ θ,y ] h + ( z ) f x,y ( z ) = h + ( θ ) f x,y ( x ) . Mu-Fa Chen By assumption, h − ( θ ) = h + ( θ ). Hencemax z ∈ [ x,θ ] II − (cid:0) f x,y (cid:1) ( z ) = max z ∈ [ θ,y ] II + (cid:0) f x,y (cid:1) ( z ) = h − ( θ ) f x,y ( x )= 1 p ν − ( x ) µ (cid:16) ν / − ( − M, x ) (cid:17) + µ (cid:16) ν − ( x, θ ) (cid:17) . (34)Thus, for computing κ DD , by (32)–(34), we arrive at h sup z ∈ ( − M, θ ) II − ( f x,y )( z ) i _ h sup z ∈ ( θ,N ) II + ( f x,y )( z ) i = sup z ∈ ( − M, x ) (cid:20) p ν − ( z ) µ (cid:16) ν / − ( − M, z ) (cid:17) + p ν − ( z ) µ (cid:16) √ ν − ( z, x ) (cid:17) + p ν − ( z ) ν − ( x ) µ ( x, θ ) (cid:21)_ (cid:20) p ν − ( x ) µ (cid:16) ν / − ( − M, x ) (cid:17) + µ (cid:16) ν − ( x, θ ) (cid:17)(cid:21)_ sup z ∈ ( y, N ) (cid:20) p ν + ( z ) µ (cid:16) ν / ( z, N ) (cid:17) + p ν + ( z ) µ (cid:16) √ ν + ( y, z ) (cid:17) + p ν + ( z ) ν + ( y ) µ ( θ, y ) (cid:21) . (35)Finally, let ( x ∗ , y ∗ , θ ∗ ) solve equation (31) and two more equations modifiedfrom (35) ignoring its left-hand side and replacing the last two “ ∨ ” with “=”.Then we have κ DD = 1 p ν − ( x ∗ ) µ (cid:16) ν / − ( − M, x ∗ ) (cid:17) + µ (cid:16) ν − ( x ∗ , θ ∗ ) (cid:17) . (36) This section illustrates the application of the basic estimates given in Theorem2.2 and the improvements given in Proposition 4.1 and Corollary 4.3. Example 5.1 (OU-processes) The state space is R and the operator is L = 12 (cid:18) d d x − x dd x (cid:19) . This is a typical example of the use of special functions. It has discrete eigen-values λ n = n with eigenfunctions (Hermite polynomials) g n ( x ) = ( − n e x d n d x n (cid:0) e − x (cid:1) , n > . asic estimates of exponential rate (cid:0) κ DD (cid:1) − = λ = 0, λ NN = λ = 1 with eigenfunction g ( x ) = x .To compute κ NN , noting that the operator, the eigenfunction are all symmetricwith respect to 0 and so does κ NN , one can split the whole line into two parts( −∞ , 0) and (0 , ∞ ) with common Dirichlet boundary at 0. This simplifies thecomputation and leads to (cid:0) κ NN (cid:1) − = (cid:0) κ DN (cid:1) − ≈ . 1. Note that g ′ ( x ) ≡ | x |→∞ (cid:0) e C g ′ (cid:1) ( x ) = 0.For the half-space (0 , ∞ ), as we have just mentioned, λ DN = λ DD = 1with g ( x ) = x , (cid:0) κ DN (cid:1) − = (cid:0) κ DD (cid:1) − ≈ . 1. For λ NN , the symmetry in thewhole line is lost. We have λ NN = 2 with g ( x ) = − x , (cid:0) κ NN (cid:1) − ≈ . x, y ) ≈ (0 . , . x →∞ g ′ ( x ) = ∞ butlim x →∞ (cid:0) e C g ′ (cid:1) ( x ) = 0.To study ¯ κ NN , recall that we can reduce the NN-case to the DD-oneby an exchange of µ and ν . By Proposition 3.1, we have (cid:0) ¯ κ NN (cid:1) − ≈ . (cid:0) κ NN (cid:1) − ≈ . 83 with ( x ∗ , y ∗ , θ ∗ ) ≈ (0 . , . , . x, y ) ≈ (0 . , . κ NN in the last paragraph.The ratio becomes 2 . / . ≈ . < 4. We mention that similar estimatescan also be obtained by using a different approximating procedure in parallelwith [9; Theorem 6.3]. Refer to [5; Footnotes 12 and 14].The following examples are often illustrated in the textbooks on ordinarydifferential equations, see for instance Hartman (1982), § Example 5.2 The equation u ′′ + σ u = 0 ( σ = 0)has the general solution u = c cos( σx ) + c sin( σx ) . From this, it should be clear that for the operator L = d / d x with finite statespace ( α, β ), we have λ DD = (cid:18) πβ − α (cid:19) , g ( x ) = sin (cid:18) π ( x − α ) β − α (cid:19) ; λ NN = (cid:18) πβ − α (cid:19) , g ( x ) = cos (cid:18) π ( x − α ) β − α (cid:19) ; λ DN = (cid:18) π β − α ) (cid:19) , g ( x ) = sin (cid:18) π ( x − α )2( β − α ) (cid:19) ; λ ND = (cid:18) π β − α ) (cid:19) , g ( x ) = cos (cid:18) π ( x − α )2( β − α ) (cid:19) . Mu-Fa Chen The corresponding estimates are as follows. (cid:0) κ DD (cid:1) − = (cid:0) κ NN (cid:1) − = (cid:18) β − α (cid:19) , (cid:0) κ DN (cid:1) − = (cid:0) κ ND (cid:1) − = (cid:18) β − α (cid:19) . Note that by symmetry, the DD- and NN-cases can be split at θ = ( α + β ) / λ DD and λ NN by usingthe known approximating method for λ DN and λ ND (cf. [5; Theorem 1.2]).However, as an illustration of Theorem 4.2 and Corollary 4.3, we now compute¯ κ DD and κ DD .Consider first the simpler interval ( α, β ) = (0 , µ = ν = d x , bysymmetry, one may choose y = 1 − x . Then x < / (cid:0) ¯ κ DD (cid:1) − = inf x ∈ (0 , / x (cid:20) − x + x − Z x z d z + x − Z − x (1 − z ) d z (cid:21) − = inf x ∈ (0 , / x − (2 x ) = 323 (with x = 3 / . To compute κ DD , set again y = 1 − x with x ∈ (0 , / f x,y becomes f x ( s ) = ( √ s ∧ x s − x √ − s s ∈ (1 − x, . By symmetry again, we have θ = 1 / 2. Fix x ∈ (0 , / f x as ( f , f ): f ( s ) = √ s for s ∈ [0 , x ] and f ( s ) = √ x for s ∈ [ x, / ν − ( s ) = s , we have h − = (cid:0) h − ( z ) , h − ( z ) (cid:1) : h − ( z ) = Z z f ( s ) s d s + z (cid:20) Z xz f + Z / x f (cid:21) , z ∈ [0 , x ] h − ( z ) = (cid:20) Z x f ( s ) s d s + Z zx f ( s ) s d s (cid:21) + z Z / z f , z ∈ [ x, / . Hence by (32), we have II − ( f x )( z ) = h − ( z ) f x ( z ) = ((cid:16) − x / + x / (cid:17) √ z − z , z ∈ [0 , x ] , (cid:0) z (1 − z ) − x (cid:1) , z ∈ [ x, / . Define H ( x ) = − x / + 12 x / and γ ( z ) = H ( x ) √ z − z . Then γ ′ ( z ) = H ( x )2 √ z − z, γ ′′ ( z ) = − H ( x )4 z / − < . asic estimates of exponential rate γ achieves its maximum at z ∗ ( x ) = (cid:18) H ( x ) (cid:19) / . Furthermore, γ ( z ∗ ( x )) = H ( x ) (cid:18) H ( x ) (cid:19) / − (cid:18) H ( x ) (cid:19) / = 38 (cid:18) (cid:19) / H ( x ) / . Note that z ∗ ( x ) x iff x > / 14. Besides, on the subinterval [ x, / h − ( z ) /f x ( z ) has maximum 1 / − x / 10 by (34). Solving the equation38 (cid:18) (cid:19) / H ( x ) / = 18 − x , x ∈ (5 / , / , we obtain x ∗ ≈ . x ∈ (5 / , / sup z / h − ( z ) f x ( z ) = γ ( z ∗ ( x ∗ )) ≈ . . From these facts and (36), we conclude that (cid:0) κ DD (cid:1) − ≈ / . ≈ . . By the way, we mention that a similar but simpler study shows thatinf x ∈ (0 , / sup z / h − ( z ) f x ( z ) = 18 . This shows that to get a less sharp lower bound 1 / 8, the computation becomesmuch simpler. It needs to study the extremal case that x = 0 only; thecorresponding test function becomes f x ≡ 1. Return to the original interval( α, β ), by Proposition 4.1 and Corollary 4.3, we obtain8( β − α ) < . β − α ) < λ DD = (cid:18) πβ − α (cid:19) β − α ) = 23 (cid:18) β − α (cid:19) . The ratio becomes (cid:14) . ≈ . 13. The same assertion holds if λ DD isreplaced by λ NN because of the symmetry.It is a good chance to discuss the approximating procedure remarked afterCorollary 4.3. Here we consider the lower estimate only. Replacing f x =( f , f ) by ( h − , h − ), one produces a new ( h − , h − ) and then a new II − ( f )which provides a new lower bound. By using this procedure twice with fixed θ = 1 / x = x ∗ ≈ . . β − α ) , . β − α ) . Mu-Fa Chen Clearly, they are quite close to the exact value of λ DD and λ NN : π ( β − α ) ≈ . β − α ) . Example 5.3 By a substitute u = ze − bx/ , the equation u ′′ + bu ′ + γu = 0 ( b, γ are real constants)is reduced to z ′′ + σ z = 0 (cid:0) σ = γ − b / (cid:1) . From the last example, it follows that the equation has general solutions u = e − bx/ ( c + c x ) if γ = b / c e ξ x + c e ξ x if γ < b / e − bx/ (cid:16) c cos (cid:0) x p γ − b / (cid:1) + c sin (cid:0) x p γ − b / (cid:1)(cid:17) if γ > b / , where ξ , ξ are solution to the equation ξ + b ξ + γ = 0 . Thus, for the operator L = d / d x + b d / d x ( b is a constant) with state space(0 , ∞ ), we have the following principal eigenfunctions • g ( x ) = (2 /b + x ) e − bx/ and g ( x ) = xe − bx/ in ND- and DD-cases, respec-tively, when b > • g ( x ) = xe − bx/ and g ( x ) = (1 + bx/ e − bx/ in DN- and NN-cases,respectively, when b < λ = b / (cid:0) κ (cid:1) − = b . Moreover, (cid:0) ¯ κ DD (cid:1) − , (cid:0) ¯ κ NN (cid:1) − = b / 2. Clearly, the lower es-timate (cid:0) κ (cid:1) − / Example 5.4 (Cauchy–Euler equation) Consider the operator L = x d d x + bx dd x , where b is a constant. By a change of variable x = e y , the equation x u ′′ + bxu ′ + γu = 0 ( b , γ are constants)is reduced to the last example:d u d y + ( b − 1) d u d y + γu = 0 . asic estimates of exponential rate u = x (1 − b ) / ( c + c log x ) if γ = (1 − b ) / c x ξ + c x ξ if γ < (1 − b ) / x (1 − b ) / (cid:16) c cos (cid:0)p γ − (1 − b ) / x (cid:1) + c sin (cid:0)p γ − (1 − b ) / x (cid:1)(cid:17) if γ > (1 − b ) / ξ , ξ are solution to the equation ξ + ( b − ξ + γ = 0 : ξ , ξ = (1 − b ) / ± p (1 − b ) / − γ. Here we have used Euler’s formula: x i √ ξ = e i √ ξ log x = cos (cid:0)p ξ log x (cid:1) + i sin (cid:0)p ξ log x (cid:1) . In particular,(1) when b = 0, we have solutions u = √ x ( c + c log x ) if γ = 1 / c x ξ + c x ξ if γ < / √ x (cid:16) c cos (cid:0)p γ − / x (cid:1) + c sin (cid:0)p γ − / x (cid:1)(cid:17) if γ > / . Now, corresponding to γ = 1 / 4, we have λ DN = 14 , g ( x ) = ( √ x if the state space is (0 , ∞ ) √ x log √ x if the state space is (1 , ∞ ) . The first case is the original Hardy’s inequality. Corresponding to γ = 1 / , ∞ ), we have λ NN = 14 , g ( x ) = √ x (cid:0) log √ x − (cid:1) . Here lim x →∞ (cid:0) e C g ′ (cid:1) ( x ) = lim x →∞ g ′ ( x ) = 0. We have (cid:0) κ DN (cid:1) − , (cid:0) κ NN (cid:1) − = 1, (cid:0) ¯ κ DN (cid:1) − , (cid:0) ¯ κ NN (cid:1) − = 1 / 2, respectively. The lower estimate (cid:0) κ (cid:1) − / b = 1, for finite state space (1 , N ) with Dirichlet boundaries, wehave λ n = (cid:18) nπ log N (cid:19) , g ( x ) = sin (cid:18) nπ log N log x (cid:19) , n > . In particular, λ DD = (cid:18) π log N (cid:19) , g ( x ) = sin (cid:18) π log N log x (cid:19) . Mu-Fa Chen Next, for Neumann boundaries, we have λ NN = (cid:18) π log N (cid:19) , g ( x ) = cos (cid:18) π log N log x (cid:19) . In both cases, we have (cid:0) κ DD (cid:1) − , (cid:0) κ NN (cid:1) − = (cid:0) / log N (cid:1) . Besides, we have λ DN = (cid:18) π N (cid:19) , g ( x ) = sin (cid:18) π N log x (cid:19) ; λ ND = (cid:18) π N (cid:19) , g ( x ) = cos (cid:18) π N log x (cid:19) . In these cases, we have (cid:0) κ DN (cid:1) − , (cid:0) κ ND (cid:1) − = (cid:0) / log N (cid:1) . Note that the presentcase can be reduced to Example 5.2 under the change of variable x = e y , theresults here can be obtained from Example 5.2 replacing ( α − β ) by log N .In view of this, we also have (cid:0) ¯ κ DD (cid:1) − = (cid:0) ¯ κ NN (cid:1) − = 323 log N , (cid:0) κ DD (cid:1) − = (cid:0) κ NN (cid:1) − ≈ . N . The next result is a generalization of [9; Proposition 1.2]. Proposition 6.1 Let P t ( x, · ) be symmetric and have density p t ( x, y ) with re-spect to µ . Suppose that the diagonal elements p s ( · , · ) ∈ L / ( µ ) for some s > and a set K of bounded functions with compact support is dense in L ( µ ) . Then λ = ε max . Proof . The proof is similar to the ergodic case (cf. [6; Section 8.3] and [9;proof of Theorem 7.4]), and is included here for completeness.(a) Certainly, the inner product and norm here are taken with respect to asic estimates of exponential rate µ . First, we have P t ( x, K ) = P s P t − s K ( x )= Z µ (d y ) d P s ( x, · )d µ ( y ) P t − s K ( y ) (since P s ≪ µ )= µ (cid:18) d P s ( x, · )d µ P t − s K (cid:19) = µ (cid:18) K P t − s d P s ( x, · )d µ (cid:19) (by symmetry of P t ) p µ ( K ) (cid:13)(cid:13)(cid:13)(cid:13) P t − s d P s ( x, · )d µ (cid:13)(cid:13)(cid:13)(cid:13) (by Cauchy-Schwarz inequality) p µ ( K ) (cid:13)(cid:13)(cid:13)(cid:13) d P s ( x, · )d µ (cid:13)(cid:13)(cid:13)(cid:13) e − λ ( t − s ) (by L -exponential convergence)= (cid:16)p µ ( K ) p s ( x, x ) e λ s (cid:17) e − λ t (by [6; (8.3)]) . By assumption, the coefficient on the right-hand side is locally µ -integrable.This proves that ε max > λ .(b) Next, for each f ∈ K with k f k = 1, we have k P t f k = ( f, P t f ) (by symmetry of P t ) k f k ∞ Z supp ( f ) µ (d x ) P t | f | ( x ) k f k ∞ Z supp ( f ) µ (d x ) P t ( x, supp ( f )) k f k ∞ Z supp ( f ) µ (d x ) c ( x, supp ( f )) e − ε max t =: C f e − ε max t . The technique used here goes back to Hwang et al. (2005).(c) The constant C f in the last line can be removed. Following Lemma2.2 in Wang (2002), by the spectral representation theorem and the fact that k f k = 1, we have k P t f k = Z ∞ e − λt d( E λ f, f ) > (cid:20) Z ∞ e − λs d( E λ f, f ) (cid:21) t/s (by Jensen’s inequality)= k P s f k t/s , t > s. Note that here the semigroup is allowed to be sub-Markovian. Combining thiswith (b), we have k P s f k C s/tf e − ε max s . Letting t → ∞ , we obtain k P s f k e − ε max s , Mu-Fa Chen first for all f ∈ K and then for all f ∈ L ( µ ) with k f k = 1, because of thedenseness of K in L ( µ ). Therefore, λ > ε max . Combining this with (a), wecomplete the proof. (cid:3) The main result (Theorem 2.2) of this paper is presented in the last section(section 10) of the paper [9], as an analog of birth-death processes. Paper [9],as well as [8] for ϕ -model, is available on arXiv.org. Acknowledgments . Research supported in part by the Creative Research GroupFund of the National Natural Science Foundation of China (No. 10721091), by the“985” project from the Ministry of Education in China. The author has been luckilyinvited by Professor Louis Chen three times with financial support to visit Singapore.Deep appreciation is given to him for his continuous encouragement and friendshipin the past 30 years. Sections 2–4 of the paper are based on the talks presented in“Workshop on Stochastic Differential Equations and Applications” (December, 2009,Shanghai), “Chinese-German Meeting on Stochastic Analysis and Related Fields”(May, 2010, Beijing), and “From Markov Processes to Brownian Motion and Beyond— An International Conference in Memory of Kai-Lai Chung” (June, 2010, Beijing).The author acknowledges the organizers of the conferences: Professors Xue-RongMao; Zhi-Ming Ma and Michael R¨okner; and the Organization Committee headed byZhi-Ming Ma (Elton P. 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