A LARGE DEVIATION PRINCIPLE FOR SYMMETRIC MARKOV PROCESSES NORMALIZED BY FEYNMAN-KAC FUNCTIONALS

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1 A LARGE DEVIATION PRINCIPLE FOR SYMMETRIC MARKOV PROCESSES NORMALIZED BY FEYNMAN-KAC FUNCTIONALS MASAYOSHI TAKEDA AND YOSHIHIRO TAWARA Abstract. We establish a large deviation principle for the occupation distribution of a symmetric Markov process normalized by Feynman-Kac functional. The obtained theorem means a large deviation from a ground state, not from an invariant measure.. Introduction Let M = Ω, t, P x, ζ be an m-symmetric irreducible Markov process on a locally compact separable metric space. Here ζ is the lifetime and m is a positive Radon measure with full support. Let E, DE be the Dirichlet form on L 2 ; m generated by M For the definition, see 2.. We denote by P the set of probability measures with the weak topology, and for a positive Green-tight Kato measure µ Definition 2. define the function I µ on the set P by {. I µ E µ f, f if ν = f m, f DE ν = otherwise, where E µ = E, µ. Given ω Ω with < t < ζω, let L t ω P be the normalized occupation distribution: for a Borel set A of L t ωa = t t A s ωds, where A is the indicator function of the set A. We denote by A µ t the positive continuous additive functional with Revuz measure µ. One of authors proved Donsker- Varadhan type large deviation principle with rate function I µ. Theorem. [24]. Assume that the Markov process M possesses the strong Feller property and the tightness property see III in Section 2. i For each open set G P lim inf t t log E x ii For each closed set K P lim sup t e Aµ t ; Lt G, t < ζ log sup E x e Aµ t ; Lt K, t < ζ t x inf ν G Iµ ν. inf ν K Iµ ν. Varadhan [29] gave an abstract formulation for the large deviation principle. The statement in Theorem. is slightly different from his formulation. In fact, the rate function I µ is not always non-negative because it is defined by the Schrödinger form E µ, not by the Dirichlet form E. Furthermore, Theorem. does not represent a Date: July 3, 2. 2 Mathematics Subject Classification. Primary 6J45, 6J4, 35J; Secondary. Key words and phrases. large deviation principle, Feynman-Kac semigroup, symmetric Markov process, Dirichlet form.

2 2 M. TAKEDA AND Y. TAWARA large deviation from a invariant measure because the Markov process is allowed to be explosive. By this reason, we consider the normalized probability measure Q x,t on P defined by, for a Borel set B P, Q x,t B = E xe Aµ t ; Lt B, t < ζ, E x e Aµ t ; t < ζ and prove that the family of probability measures {Q x,t } t> obeys the large deviation principle as t in the sense of Varadhan s formulation. In other words, {Q x,t } t> satisfies the full large deviation principle with a good rate function in the sense of [, Section 2.]. This is the main theorem of this paper Theorem 4.. The rate function is given by.2 Jν := I µ ν λ 2 µ, ν P. Here λ 2 µ is the bottom of the spectrum of the Schrödinger type operator L + µ, where L is the generator of the Markov process: λ 2 µ = inf {E µ u, u : u DE, u 2 = }. To obtain the main theorem, we need to show that the rate function J is good, that is, enjoys the properties i iv in Lemma 4.. In particular, we must show that J has a unique zero point, that is, the existence of a ground state ϕ of the operator L + µ. In order to show the existence of a ground state, we usually use the L 2 -weak compactness of the set {u DE : E µ u, u l} l R and the lower semi-continuity of the Schrödinger form E µ with respect to the L 2 -weak topology e.g. [7]; however we can not derive these properties from our general setting. Hence we here use the following properties instead, the tightness of the level set {ν P : I µ ν l} and the lower semi-continuity of the function I µ with respect to the weak topology. This is a key to the proof of the goodness of the rate function J. We would like to emphasis that the tightness follows from the condition III and the Green-tightness of µ, and the lower semi-continuity of I µ follows from a variational formula for the Schrödinger form Proposition 2., that is, the identification of the Schrödinger form with the modified I-function defined in 2.8. The latter is an extension of a well-known fact due to Donsker and Varadhan that for a symmetric Markov process, the I-function is identical with the Dirichlet form. On account of Lemma 4., we can regard the main theorem as a large deviation from the ground state of the Schrödinger operator. In [], [25], [28], L p -indepedence of growth bounds of non-local Feynman- Kac semigroups have been considered. In this paper we also deal with non-local Feynman-Kac transforms and extended Theorem. to symmetric Markov processes with non-local Feynman-Kac functional Theorem 2.. The existence of ground states implies the existence of a quasi-stationary distribution, ηb := B ϕ xdmx/ ϕ xdmx. In [6], they prove that if a Markov semigroup is intrinsically ultracontractive, then the measure η is the so-called Yaglom limit and a unique quasi-stationary distribution. In the last section, we will give an extension of this fact to generalized Feynman-Kac semigroups by employing Fukushima s ergodic theorem. 2. Symmetric Markov processes with non-local Feynman-Kac functionals Let be a locally compact separable metric space and B the Borel σ-field. Adjoining an extra point to the measurable set, B, we set = { } and B = B {B { } : B B}. Let M = Ω, t, P x, ζ be a right

3 LARGE DEVIATION PRINCIPLE 3 Markov process on with lifetime ζ := inf{t > : t = }. semigroup and the resolvent by p t fx = E x f t ; t < ζ, R β fx = e βt p t fxdt We define the for a bounded Borel function f on. We assume that the Markov process M is m-symmetric, p t f, g m = f, p t g m, where m is a positive Radon measure with full support. Let E, DE be the Dirichlet form on L 2 ; m generated by M: { } DE = u L 2 ; m : lim t t u p tu, u m < 2. Eu, v = lim t t u p tu, v m. For basic materials on right processes and associated Dirichlet forms quasi-regular Dirichlet forms, we refer to [7], [8]. We impose three assumptions on M. I Irreducibility If a Borel set A is p t -invariant, i.e., p t A fx = A p t fx m-a.e. for any f L 2 ; m B b and t >, then A satisfies either ma = or m \ A =. Here B b is the space of bounded Borel functions on. II Strong Feller Property For each t, p t B b C b, where C b is the space of bounded continuous functions on. III Tightness For any ϵ >, there exists a compact set K such that sup R K cx ϵ. x Here K c is the indicator function of the complement of the compact set K. The assumption II implies that M satisfies the absolute continuity condition, that is, its transition probability p t x, is absolutely continuous with respect to m for each t > and x. As a result, the resolvent kernel is also absolutely continuous with respect to m, R β x, dy = R β x, ymdy. By [4, Lemma 4.2.4] the density R β x, y is assumed to be a non-negative Borel function such that R β x, y is symmetric and β-excessive in x and in y. Under the absolute continuity condition, quasi everywhere statements are strengthened to everywhere ones. Moreover, we can defined notions without exceptional set, for example, smooth measures in the strict sense or positive continuous additive functional in the strict sense cf. [4, Section 5.]. Here we only treat the notions in the strict sense and omit the phrase in the strict sense. We denote S the set of positive Borel measures µ such that µ < and R µx= R x, yµdy is uniformly bounded in x. A positive Borel measure µ on is said to be smooth if there exists a sequence {E n } n= of Borel sets increasing to such that En µ S for each n and P x lim ζ =, x, n σ \E n where σ \En is the first hitting time of \ E n. The totality of smooth measures is denoted by S. If an additive functional {A t } t is positive and continuous with respect to t for each ω Λ, it is saide to be a positive continuous additive functional PCAF in abbreviation. By [4, Theorem 5..7], there exists a one-to-one correspondence between positive smooth measures and PCAF s Revuz correspondence: for each smooth measure µ, there exists a unique PCAF {A t } t such that for any

4 4 M. TAKEDA AND Y. TAWARA positive Borel function f on and γ-excessive function h γ, that is, e γt p t h h, t 2.2 lim t t E h m f s da s = fxhxµdx. Here E h m = E x hxmdx. We denote by A µ t the PCAF corresponding to the smooth measure µ. For a signed Borel measure µ = µ + µ, let µ = µ + +µ. When µ is a smooth measure, we define A µ t = A µ+ t A µ t and A µ t = A µ+ t + A µ t. Following Chen [4], we introduce classes of potentials. Definition 2.. i A signed Borel measure µ is said to be the Kato measure in notation, µ K, if µ S and lim sup E x A µ t =. t x ii A measure µ K is said to be in the class K, if for any ϵ > there exist a compact subset K and a positive constant δ > such that for all measurable set B K with µ B < δ, sup x K c B R x, y µ dy ϵ. iii A signed Borel measure µ is said to be in the class S, if for any ϵ > there exist a compact subset K and a positive constant δ > such that for all measurable set B K with µ B < δ, sup x,z \d K c B R x, yr y, z µ dy ϵ. R x, z It is known in [2] that µ belongs to K if and only if 2.3 lim sup R β x, y µ dy =, β x and in [4] that 2.4 S K K. We denote that N, H = Nx, dy, H t is the Lévy system of M, that is, N is a kernel on, B with Nx, {x} = and H is a positive continuous additive functional of M such that for any non-negative measurable function F on vanishing on the diagonal set and any x, t F s, s ; t < ζ = E x F s, yn s, dydh s. E x <s t We denote by µ H be the smooth measure corresponding to H t. Definition 2.2. Let F be a bounded measurable function on vanishing on the diagonal set. i F is said to be in the class A, if for any ϵ > there exist a compact subset K and a positive constant δ > such that for all measurable set B K with µ B < δ, sup x,z \d R x, y F y, z R z, w K\B K\B R c x, w ii F is said to be in the class A 2, if F A and µ F dx = F x, y Nx, dy µ H dx S. Ny, dzµ H dy ϵ.

5 LARGE DEVIATION PRINCIPLE 5 For properties and examples of A and A 2, see [4], [5]. In the remainder of this paper, we assume that F is symmetric, F x, y = F y, x. We write µ + F K + A 2 if µ K and F A 2. For µ + F K + A 2 define the AF A t by A t = A µ t + <s t F s, s, and the generalized Feynman-Kac semigroup {p t } t by p t fx = E x e A t f t ; t < ζ, f B b. For F A 2, we define the symmetric Dirichlet form E F, DE as follows: for u, v DE 2.5 E F u, v = E c u, v + E k u, v + ux uyvx vye F x,y Nx, dyµ H dx, 2 where E c and E k are the local part and the killing part of the Dirichlet form E, DE in Beurling-Deny formula [4, Theorem 3.2.]. Fundamental properties of non-local Feynman-Kac transforms were earlier studied by J. Ying [3], [32]. It is known in [8] that {p t } t is the semigroup generated by the Schrödinger form E, DE: 2.6 E u, v = E F u, v uxvxdµ F x uxuydµx, where F = expf. The form E is also written as E u, v = Eu, v uxvyf x, ynx, dydµ H y uxvxdµx, u, v DE. Let P be the set of probability measures on equipped with the weak topology. We define the function I on P by { I E f, f if ν = f m, f DE ν = otherwise. Let µ + F K + A 2 and define κµ + F by κµ + F = lim log p t,. t t We see from [] that κµ + F is finite. If > κµ + F and f B b, we define the resolvent R by fx = E x e t+a t f t dt. We set R D + H = { R f : > κµ + F, f L 2 ; m C b, f and f }, Each function ϕ = R f D + H is strictly positive because P x σ O < ζ > for any x by the assumption I. Here O is a non-empty open set {x : fx > } and σ O = inf{t > : t O}. We define the generator H by H u = u f, u = R f D + H.

6 6 M. TAKEDA AND Y. TAWARA Let h be the function defined by hx = E x exp A ζ. We may assume that is gaugeable, that is, sup x hx <. In fact, it is enough to prove Theorem 2. and Theorem 4. below for the β-subprocess, P β x = e βt P x. Moreover, we see that every K +A 2 becomes gaugeable with respect to the β-subprocess of M for a large enough β. In fact, we see from [5, Theorem 3.4] that K +A 2 is gaugeable with respect to the β-subprocess if and only if { inf E F u, u + ux 2 µ + µ F dx + β ux 2 mdx 2.7 } : ux 2 µ + + µ F + dx = >, where F + and F is the positive and negative part of F. Since by 3. E F u, u + ux 2 µ + µ F dx + β ux 2 mdx e F Eu, u + β ux 2 e F mdx R β µ +, + µ F + and the right hand side tends to as β because of µ + + µ F + holds for a large β. We define the function I h on P by 2.8 I h ν = inf ϕ D + H ϵ> H ϕ ϕ + ϵh dν. K, 2.7 The gauge function hx satisfies < c hx C <. Indeed, it follows from Proposition 2.2 in [4] and 2.4 that for µ K and F A 2, sup x E x A µ + F ζ <. Hence, by Jensen s inequality, inf E x exp A x ζ exp sup E x A µ + F ζ >. x Let us define the function I on P by R I ν = inf log Lemma 2.. It holds that u B + b ϵ> I ν I hν, ν P. Proof. For u = R f D + H and ϵ >, set R ϕ = log dν. Then, noting that d d dϕ d = Since R R R 2 u R u u = R R u R 2 u R dν u = R dν = 2 R 2 R 2 u, we have H R 2 u R dν. 2 u R R u

7 equals R LARGE DEVIATION PRINCIPLE 7 2 u R u 2, we have R 2 u R u R R 2 2 u R u 2, R and thus H R 2 u R dν H R 2 u 2 dν 2 R = H R 2 u 2 2 dν R u + ϵh 2 Therefore which implies Since βr β 2.9 ϕ ϕ = inf u D + H ϵ> 2 I hν. log log R R dν I hν, dν I hν. f C f, β >, and βr β fx fx as β, R βr β f + ϵh log βr dν β R f + ϵh log β f + ϵh f + ϵh Define the measure ν by ν A = R x, Adνx, A B. Given v B + b, take a sequence {g n} n= C + b L2 ; m such that v g n dν + ν as n. We then have R dν. v R g n dν R v g n dν = v g n dν as n, and so R 2. log g n + ϵh g n + ϵh Hence, combining 2.9 and 2. R inf log u D + H which implies the lemma. dν n R log v + ϵh dν = inf log u B + b v + ϵh R dν. Lemma 2.2. If I h ν <, then ν is absolutely continuous with respect to m. dν,

8 8 M. TAKEDA AND Y. TAWARA Proof. By a similar argument in the proof of [2, Lemma 4.], we obtain this lemma. Indeed, for a > and A B, set ux = a A x + B + b. Then R ar x, A + R x, + ϵh log dν = log dν. a A x + + ϵh Define the measure ν as in the proof of Lemma 2.. Put c = R x, dνx= ν. We see from Lemma 2. and Jensen s inequality that log aν A + c + ϵh νa loga + + ϵh + νa c log + ϵh I h ν/, and by letting ϵ log aν A + c νa loga + I h ν/. Since log x x for x >, we have and so aν A + c νa loga + I h ν/, ν A νa I hν/ + νaloga + a + c. a Noting that loga + a <, we have ν A νa I hν/ + loga + a + c a for all A B and νa ν A = c + ν A c νa c I hν/ + loga + a + c a + a for all A B. Therefore we can conclude that sup A B νa ν A a loga + + I hν/ + c a +. a Note that c as. Then since lim sup sup A B νa ν A a loga + a and the right-hand side converges to as a, the lemma follows. Proposition 2.. It holds that for ν P I h ν = I ν. Proof. We follow the argument of the proof of [2, Theorem 5]. Suppose that I h ν = l <. By Lemma 2.2, ν is absolutely continuous with respect to m. Let us denote by f its density and let f n = f n. Since log x x for < x < and < f n R f n f n <, + ϵh we have log R f n + ϵh f n + ϵh fdm = log f n R f n + ϵh f n R f n + ϵh f n f n fdm, fdm

9 LARGE DEVIATION PRINCIPLE 9 and then f n R f n + ϵh By letting n and ϵ, we have f f R f n fdm I f m. fdm I f m I hf m, which implies that f DE and E f, f I h f m. Let ϕ D + H and define the semigroup P ϕ t by P ϕ ϕ + ϵh t fx = E x e A t t t ϕ + ϵh exp H ϕ ϕ + ϵh sds f t. Then, P ϕ t is ϕ + ϵh 2 m-symmetric and satisfies P ϕ t. Given ν = f m P with f DE, set t S ϕ H t fx = Ex e ϕ ft A t exp ϕ + ϵh sds Then 2 S ϕ t f 2 dm = ϕ + ϵh P 2 ϕ f t dm ϕ + ϵh f 2 ϕ + ϵh 2 P ϕ t dm ϕ + ϵh 2 f ϕ + ϵh 2 dm ϕ + ϵh = fdm. Hence lim t t f S ϕ t f, fm = E f, H ϕ f + ϕ + ϵh fdm, and thus E f, f I h f m. We now obtain a generalization of Theorem. in exactly the same way as the proof of it cf. [], [28]: Theorem 2. [24]. Assume I, II and III. Suppose that µ + F K + A 2. i For each open set G P lim inf t t log E x e A t ; L t G, t < ζ inf ν G I ν. ii For each closed set K P lim sup log sup E x t t x e A t ; L t K, t < ζ inf ν K I ν. 3. The existence of ground states We first recall an inequality [9]: for µ K, 3. ũ 2 dµ R µ Eu, u + u, u m u DE. Let λ 2 µ + F be the bottom of the spectrum of H : 3.2 λ 2 µ + F = inf { E u, u : u DE, u 2 = }.

10 M. TAKEDA AND Y. TAWARA Proposition 3.. Assume I, II and III. There exists a unique ground state ϕ DE: λ 2 µ + F = E ϕ, ϕ. Proof. Let {u n } be a minimizing sequence of the right-hand side of 3.2, i.e., u n 2 = and λ 2 µ + F = lim n E u n, u n. Put µ = µ + µ F. Since Eu n, u n c E F u n, u n c = exp F and u2 ndµ R µ Eu n, u n +, E u n, u n = E F u, u u 2 ndµ c Eu n, u n R µ Eu n, u n + = c R µ Eu n, u n R µ. Taking large enough so that c R µ < on account of 2.3, we have sup Eu n, u n csup n E u n, u n + R µ n c R µ <. We see from the assumption III that for any ϵ > there exists a compact set K such that sup u 2 n dm R K c sup Eu n, u n + < ϵ. n K c n As a result, the subset {u 2 n m} of P is tight. Hence there exists a subsequence u 2 n k m which converges to a probability measure ν weakly. Since the function I is lower semi-continuous by Proposition 2., I ν lim inf k I u 2 n k m = lim inf E u nk, u nk <. k Therefore ν can be written as ν = ϕ 2 m, ϕ DE by Proposition 2. and λ 2 µ + F = E ϕ, ϕ, that is, ϕ is the ground state. The uniqueness of the ground state follows from the irreducibility I e.g. [9, Proposition.4.3]. We also know from the proof above that the level set {ν P : I ν+f ν l} is compact. 4. Large deviations from ground states Given ω Ω with < t < ζω, we define the occupation distribution L t ω P by L t ωa = A s ωds t for a Borel set A of, where A is the indicator function of the set A. Define the probability measure Q x,t on P by 4. Q x,t B = E xe A t ; L t B, t < ζ, B BP. E x e A t ; t < ζ We define the function J on P by t 4.2 Jν = I ν λ 2 µ + F. We then have the next lemma by Proposition 2. and Proposition 3.. Lemma 4.. The function J satisfies: i Jν. ii J is lower semicontinuous. iii For each l <, the set {ν P : Jν l} is compact. iv Jϕ 2 m = and Jν > for ν ϕ 2 m.

11 LARGE DEVIATION PRINCIPLE Remark 4.. Let E ϕ, DE ϕ the bilinear form on L 2 ; ϕ 2 m defined by { E ϕ u, v = E uϕ, uϕ λ 2 µ + F uϕ, uϕ m DE ϕ = {u L 2 ; ϕ 2 m : uϕ DE}. We then see that E ϕ, DE ϕ is a Dirichlet form and E ϕ is expressed by E ϕ u, v = ϕ 2 dµ c u,v + ux uyvx vyϕ xϕ yjdx, dy. Here µ c u,v \ is the local part of energy measure [6]. We then see that Jν = I E ϕ ν, where I E ϕ is defined by { E ϕ f, f if ν = f ϕ I E ϕ ν = m, f DE ϕ otherwise. We then have the main theorem: Theorem 4.. Assume I, II and III. Let µ + F K + A 2. Let {Q x,t } t> be a family of probability measures defined in 4.. Then {Q x,t } t> obeys a large deviation principle with rate function J: i For each open set G P lim inf t ii For each closed set K P lim sup t t log Q x,t G inf ν G Jν. t log Q x,t K inf ν K Jν. Corollary 4.. The measure Q x,t converges to δ ϕ 2 m weakly. Proof. If a closed set K does not contain ϕ 2 m, then inf x K Jx > by Lemma 4. iv. Hence Theorem 4. ii says that lim t Q x,t K = and lim t Q x,t K c =. For a positive constant δ and a bounded continuous function f on the set of P, define the closed set K P by K = {ν P : fν fϕ 2 m δ}. Then we have fνq x,t dν fϕ 2 m fν fϕ 2 m Q x,t dν P P = fν fϕ 2 m Q x,t dν + fν fϕ 2 m Q x,t dν K K c δq x,t K c + 2 f Q x,t K δ as t. Since δ is arbitrary, the weak convergence follows. On account of Corollary 4., we can regard Theorem 4. as a genuine large deviation principle from the ground state. 5. Quasistationary distribution In this section, we consider the existence of quasi-stationary distributions as an application of the existence of ground states. We continue with the setting of the preceding section. Define the semigroup {p ϕ t } t on L 2 ; ϕ 2 m generated by E ϕ, DE ϕ, that is 5. p ϕ t fx = e λ 2 t ϕ x E x e A t ϕ t f t.

12 2 M. TAKEDA AND Y. TAWARA Let M ϕ = Ω, t, P ϕ x be the ϕ 2 m-symmetric Markov process generated by the Markov semigroup p ϕ t in 5.. Set { } P = ν P : p x, xdνx <, ϕ xdνx <. We then have Theorem 5.. Assume that m <. Then for ν P and B B lim t eλ 2 t E ν e A t ; t B = ϕ dν ϕ dm. Proof. Note that e λ2 t E ν e A t B ; t B = ϕ xe ϕ x t dνx. ϕ Let {E λ, λ < } be the spectral family of E ϕ, F ϕ. Then lim t p ϕ t f = E f in L 2 ; ϕ 2 m. Since E ϕ E f, E f =, E f equals fϕ2 dm, m-a.e. by the irreducibility of E ϕ, F ϕ cf. [7, Theorem 5.2.3]. Note that p ϕ t x, L 2 ; ϕ 2 m because pϕ t x, y 2 ϕ 2 ydmy = p ϕ 2t x, x <. Put c = B ϕ dm. We then have ϕ xe ϕ B x t ϕ 5.2 = ϕ x p ϕ x, y 2 dνx E ϕ y The right-hand side is dominated by ϕ x Since ydmydνx p ϕ x, y 2 ϕ 2 2 p ϕ t ϕ dν B ϕ dm B ϕ t c 2 x, y = e λ 2 t p t x, y ϕ xϕ y, the first factor is equal to ϕ x p ϕ x, xdνx = e 2 λ 2 B ϕ y 2 dmy dνx. 2 E ϕ B y ϕ t c ϕ 2 2 ydmy. p x, xdνx and is finite by the assumption that ν P. Hence the right-hand side of 5.2 converges to zero as t because B /ϕ L 2 ; ϕ 2 m. Let η and R ν,t be probability measures on defined by 5.3 ηb = B ϕ xdmx E ν e A t ; t B ϕ xdmx, R ν,tb = for B B. E ν e A t ; t < ζ Corollary 5.. For ν P and B B 5.4 lim t R ν,t B = ηb. Note that the Dirac measure δ x belongs to P and so the distribution R δx,t converges to η for all x. Hence Corollary 5. says that the semigroup {p t } t is conditionally ergodic and η is a quasi-stationary distribution of the semigroup {p t } t : for any t > 5.5 R η,t = η.

13 LARGE DEVIATION PRINCIPLE 3 e.g [6]. If the semigroup {p t } t is ultracontractive, p t x, y c t, then p t x, x and ϕ x are bounded and P equals P. Consequently, for any ν P, the distribution R ν,t converges to η. When the measure m is not finite, we assume the intrinsic ultracontractivity of } t, that is, {p t 5.6 p t x, y C t ϕ xϕ y. In [6], they proved that for a not necessary symmetric Markov process, the intrinsic ultracontractivity is a sufficient condition for the measure η being a unique quasi-stationary distribution, and the equation 5.4 holds for any initial distribution. We would like to give another proof of this fact by using the next theorem due to Fukushima [3]. Theorem 5.2. Assume that m < and M is conservative, p t =, t >. Then for f L ; m, lim p tfx = fxdmx, m-a.e. and in L ; m. t m Note that M ϕ satisfies the assumptions in Theorem 5.2. Theorem 5.3. Assume that {p t } t is intrinsically ultracontractive. Then for any ν P and any B B lim t eλ 2 t E ν e A t Consequently, the equation 5.4 follows. ; t B = ϕ dν ϕ dm. B Proof. First note that the upper bound 5.6 implies the lower bound [9, Theorem 4.2.5]: 5.7 c t ϕ xϕ y p t x, y. As a result, sup ϕ x ϕ ydmy p t <. x c t Hence ϕ belongs to L ; m L ; m and B /ϕ L ; ϕ 2 m. Applying Theorem 5.2 to M ϕ, we have B E ϕ y t ϕ dm, m-a.e. y and L ; ϕ 2 ϕ m B as t. Since p ϕ x, is bounded by the ultracontractivity, it follows from the 2 equation 5.2 that lim t ϕ xe ϕ x B t dνx = ϕ dν ϕ dm. ϕ B We finally consider the exponential integrability of hitting times of compact sets. Let K be a compact set and D the complement of K, D = \ K. We define the part or absorbing process D on D by { t D t t < τ D = τ D = inf{t : t D}. t τ D, Define the regular Dirichlet form E D, DE D on D by { E D = E, DE D = {u DE : u = q.e. on K}.

14 4 M. TAKEDA AND Y. TAWARA By [4, Theorem 4.4.3] the part process D is regarded as a Hunt process generated by E D, DE D. We see from [4, Theorem 4.2] that m is in K. We write K R for K to show the dependence. Let R D be the -resolvent of D. The restriction m D of m on D is in K R D. Indeed, let a compact set K and a positive constant δ in the definition of K Definition 2.. We can suppose K K. Let G be a relatively compact open set such that K G Ḡ K and mg \ K < δ. Then K G c is a compact subset of D and R D K G c c = R D Kc G\K R Kc + R G\K 2ϵ. Moreover, R D B R B for any Borel set B K G c. Hence we have m D K R D. If D satisfies the irredicibility I, it follows from [4, Theorem 4.] that sup E x e λτ D < λ < λ D, x D where λ D is the bottom of the spectrum of E D, DE D. Noting that by 3. we see from III that R D λ D +, 5.8 λ D as K. We can conclude that if for any compact set K, the part process D D = \ K is irreducible, then for any λ > there exists a compact set K such that 5.9 sup E x e λτ D <. x If M is conservative, τ D equals the first hitting time σ K of K, σ K = inf{t > : t K}. Then the property 5.9 is called the uniform hyper-exponential recurrence in [3]. Example 5. One-dimensional diffusion processes. Let us consider a one-dimensional diffusion process M = t, P x, ζ on an open interval I = r, r 2 such that P x ζ = r or r 2, ζ < = P x ζ <, x I, and P a σ b < > for any a, b I. The diffusion M is symmetric with respect to its canonical measure m and it satisfies I and II. The boundary point r i of I is classified into four classes: regular boundary, exit boundary, entrance boundary and natural boundary [5, Chapter 5]: a If r 2 is a regular or exit boundary, then lim x r2 R x =. b If r 2 is an entrance boundary, then lim r r2 sup x r,r 2 R r,r2 x =. c r 2 is a natural boundary, then lim x r2 R r,r2x = and thus sup x r,r 2 R r,r2x =. Therefore, III is satisfied if and only if no natural boundaries are present. As a corollary of the equation 5.8, If r 2 is entrance, for any λ > there exists r < r < r 2 such that sup E x expλσ r <, x>r where σ r is the first hitting time of {r}. The statement above implies a uniqueness of quasi-stationary distributions [3].

15 LARGE DEVIATION PRINCIPLE 5 References [] S. Albeverio, P. Blanchard and Z.-M. Ma: Feynman-Kac semigroups in terms of signed smooth measures, in Random Partial Differential Equations ed. U. Hornung et al., Birkhäuser, 99. [2] S. Albeverio, Z.-M. Ma: Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms, Osaka J. Math , [3] P. Cattiaux, P. Collet, A. Lamberrt, S. Martinez, S. Meleard, J. San Martin: Quasistationary distributions and diffusion models in population dynamics, Ann. Probab. 3729, [4] Z.-Q. Chen: Gaugeability and Conditional Gaugeability, Trans. Amer. Math. Soc., 35422, [5] Z.-Q. Chen: Analytic characterization of conditional gaugeability for non-local Feynman- Kac transforms, J. Funct. Anal., 2223, [6] Z.-Q. Chen, P.J. Fitzsimmons, M. Takeda, J. Ying and T.S. Zhang: Absolute continuity of symmetric Markov processes, Ann. Probab., 3224, [7] Z.-Q. Chen, M. Fukushima: Symmetric Markov Processes, Time Change and Boundary Theory, book manuscript, 29. [8] Z.-Q. Chen, R. Song: Conditional gauge theorem for non-local Feynman-Kac transforms, Probab. Theory Related Fields 2523, [9] E.B. Davies: Heat Kernels and Spectral Theory, Cambridge Univ. Press, Cambridge, U.K., 989. [] G. De Leva, D. Kim and K. Kuwae: L p -independence of spectral bounds of Feynman-Kac semigroups by continuous additive functionals, J. Funct. Anal., 2592, [] J-D. Deuschel and D. W. Stroock: Large deviations, Academic Press, Boston, MA, 989. [2] M.D. Donsker and S.R.S. Varadhan: Asymptotic evaluation of certain Markov process expectations for large time, I, Comm. Pure Appl. Math., 28975, -47. [3] M. Fukushima: A note on irreducibility and ergodicity of symmetric Markov processes, Lecture Notes in Phys. 73, Springer 982, [4] M. Fukushima, Y. Oshima and M. Takeda: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 2nd rev. and ext. ed. 2. [5] K. Itô: Essentials of stochastic processes, American Mathematical Society, Providence, 26. [6] R. Knobloch and L. Partzsch: Uniform conditional ergodicity and intrinsic ultracontractivity, Potential Anal., 332, [7] E.H. Lieb and M. Loss, : Analysis, Second edition, Graduate Studies in Math. 4, American Mathematical Society, 2. [8] Z.M. Ma and M. Röckner: Introduction to the Theory of Non-Symmetric Dirichlet Forms, Springer-Verlag, Berlin, 992. [9] P. Stollmann and J. Voigt: Perturbation of Dirichlet forms by measures, Potential Anal., 5996, [2] M. Takeda: Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal., 9998, [2] M. Takeda: Conditional gaugeability and subcriticality of generalized Schrödinger operators, J. Funct. Anal., 922, [22] M. Takeda: Large deviation principle for additive functionals of Brownian motion corresponding to Kato measures, Potential Anal., 923, [23] M. Takeda: Large deviations for additive functionals of symmetric stable processes, J. Theoret. Probab., 228, [24] M. Takeda: A large deviation principle for symmetric Markov processes with Feynman-Kac functional, to appear in J. Theoret. Probab. [25] M. Takeda and Y. Tawara: L p -independence of spectral bounds of non-local Feynman-Kac semigroups. Forum Math., 229, [26] M. Takeda and K. Tsuchida: Differentiability of spectral functions for symmetric -stable processes, Trans. Amer. Math. Soc., 35927, [27] M. Takeda, T. Uemura: Subcriticality and Gaugeability for Symmetric -Stable Processes, Forum Math., 624, [28] Y. Tawara: L p -independence of spectral bounds of Schrödinger operators with non-local potentials, J. Math. Soc. Japan, 622, [29] S.R.S. Varadhan: Asymptotic probabilities and differential equations, Comm. Pure Appl. Math., 9966, [3] L. Wu: Some notes on large deviations of Markov processes, Acta Math. Sin. Engl. Ser. 62,

16 6 M. TAKEDA AND Y. TAWARA [3] J. Ying: Bivariate Revuz measures and the Feynman-Kac formula, Ann. Inst. H. Poincare Probab. Statist , [32] J. Ying: Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math., 34997, M. Takeda Mathematical Institute, Tohoku University, Aoba, Sendai, , Japan Y. Tawara Division of General Education, Nagaoka National College of Technology, 888 Nishikatakai, Nagaoka, Niigata , Japan address, M. Takeda: address, Y. Tawara: