Long time asymptotics of unbounded additive functionals of Markov processes*

Under hypercontractivity and Lp-integrability of transition density for some p > 1, we use the perturbation theory of linear operators to obtain existence of long time asymptotics of exponentials of unbounded additive functionals of Markov processes and establish the moderate deviation principle for the functionals. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang’s Harnack inequality. As an application of our general results, we obtain the existence of these asymptotics and the moderate deviation principle of additive functionals with quadratic growth for the stochastic differential equations with multiplicative noise under some explicit conditions on the coefficients and prove that these asymptotics solve the related ergodic Hamilton-Jacobi-Bellman equation with nonsmooth and quadratic growth cost in viscosity sense.


Introduction
The purpose of this paper is to study the existence of the following limits (1.1) and (1.2) for general Markov processes {X t , t ≥ 0} and unbounded functions c: (1. 2) The existence of the limits (1.1) and (1.2) is closely related to spectral theory, large deviations and functional inequalities. For bounded functions c, Wu [27] proved the limits (1.1) and (1.2) exist under a logarithmic Sobolev inequality, and Kontoyiannis and Meyn ( [13]) studied the existence of the two limits by the spectral theory under the geometric ergodicity assumption. In [14], the results were extended to a large class of functions whose growth at infinity are strictly less than quadratic when their results are applied to diffusion processes. Cattiaux, Dai Pra and Roelly ( [3]) proposed an approach based on cluster expansion (a method in statistical mechanics, see [15]) to establish the existence of (1.2) for a class of unbounded and nonsmooth functions. Their result can be applied to a class of ergodic HJB equations with constant diffusion coefficient and quadratic growth cost. But the condition p(t, ·, ·) Lp(µ×µ) < ∞ for some p > 2 in [3] restricts its applications, where p(t, x, ·) is a transition density. One motivation of this paper is to improve the condition to p(t, ·, ·) Lp(µ×µ) < ∞ for some p > 1.
Using the perturbation theory of linear operators, we establish existence of the pair (λ, V ) of limits (1.1) and (1.2) of unbounded additive functionals for general Markov processes under hypercontractivity and L p -integrability of transition density p(t, x, ·) for some p > 1. This improves the condition p(t, ·, ·) Lp(µ×µ) for some p > 2 in [3]. The improvement plays an important role in application to stochastic differential equations with multiplicative noise. This method can give a precise representation of V by the projection operator. The second motivation is to study moderate deviations of additive functional t 0 c(X s )ds with unbounded function c. Wu ([26]) studied the problem by the perturbation theory, a key step ((2.11) in [26]) is to check when t → ∞, converges to a holomorphic function in a neighborhood of 0 ∈ C. But this is far from trivial because it is very difficult to verify that {Λ t (z), t ≥ t 0 } is a family of holomorphic functions in a common complex neighborhood of 0 ∈ C for some t 0 > 0. To overcome these difficulties, we restrict the logarithmic function on R to avoid the logarithmic function of complex Feynman-Kac operator in [26] and use the Taylor expansion of the largest modulus eigenvalue of the Feynman-Kac semigroup to replace C 2 -regularity in [26]. For stochastic differential equations with multiplicative noise, we show the hypercontractivity and the integrability based on Wang's Harnack inequality ( [25]). As an application of our general results, we establish the existence of the pair (λ, V ) for stochastic differential equations with multiplicative noise under some explicit conditions on the coefficients and the moderate deviation principle for additive functionals with quadratic growth of the solutions and prove that the pair (λ, V ) solves an ergodic HJB equation with nonsmooth and quadratic growth cost in viscosity sense.
Let us first introduce some notations. Let (E, E) be a Polish space and let {P (t, x, A), t ∈ [0, ∞), x ∈ E, A ∈ E} be a transition probability function. P x denotes the probability measure on F such that under P x the coordinate process {X t , t ∈ [0, ∞)} is a right-continuous Markov process with the transition probability functions P (t, x, A) and starting from x. We assume that P (t, x, A) has a stationary distribution µ. Set Long time asymptotics of additive functionals We introduce the following conditions: has the strong Feller property, i.e., P t f ∈ C b (E) for any f ∈ bE.
(C3). There exists transition density function p(t, x, y), i.e., and there existt 1 > 0 and p > 1 such that for any t ≥t 1 , any compact set K ⊂ E, (1.4) (C4). Let c : E → R be a mesurable function which satisfies that for any a > 0, there exist δ a > 0 and a locally bounded function M a (x) such that Our two abstract results are the following two theorems.
Then for any z ∈ R, and for any compact K ⊂ E, (1.9) In particular, P µ ( 1 a(t) t 0 (c(X s ) − µ(c))ds ∈ ·) satisfies a large deviation principle in R with speed t/a 2 (t) and rate function J(y) = y 2 2σ 2 (c) , and P x ( 1 a(t) t 0 (c(X s ) − µ(c))ds ∈ ·) satisfies a local uniform large deviation principle in R. Namely, for any compact K ⊂ E, for any closed set F in R, (1.10) and for any open set G in R, (1.11) Remark 1.1. (1). The local uniformity in Theorem 1.2 is with respect to initial points rather than initial measures in [26], Theorem 2.4.
(2). Using the method in this paper, that is the Taylor expansion of the largest modulus eigenvalue of the Feynman-Kac semigroup which can avoid the logarithmic function of complex Feynman-Kac operator in [26], we can prove the main results in [26].
Next, we apply Theorem 1.1 and Theorem 1.2 to the following SDEs with multiplicative noise: (1.12) where the coefficients of the SDEs (1.12) satisfy the following conditions (see [25]): where A HS = tr(AA τ ), tr(AA τ ) denotes the trace of square matrix AA τ and A τ denotes transpose of matrix A, and y, z = d i=1 y i z i for any y, z ∈ R d . (A2). There exist constants 0 < κ 1 ≤ κ 2 < ∞ such that where a(x) := σ(x)σ τ (x). (A3). There exists a constant ϑ > 0 such that almost surely Then under (A1) and (A2), there exists a constant L such that and the fact that a is continuous and uniform positive definite yields the existence and uniqueness for the weak solution of (1.12) (see [21],Theorem 7.2.1 and localization), and the weak solution has the strong Feller property. On the other hand, the assumption (A1) ensures the uniqueness of the solution of (1.12). Thus, by Yamada-Watanabe theorem (see [10] ), we obtain the existence and uniqueness for the strong solution of (1.12) and the strong solution has the strong Feller property. Assume c b > K + σ,b κ 2 2 /κ 2 1 , where K + σ,b = max{K σ,b , 0}. Then (1). The solution {X t , t ≥ 0} of the SDEs (1.12) has the following properties: (a) There exist a unique stationary distribution µ and a constant β > K + σ,b /κ 2 (1.13) In particular, {P t , t > 0} is µ-hypercontractive.
if V is a viscosity supersolution of (1.19), that is, for any ψ ∈ C 2 (R d ) and any local maximum point EJP 22 (2017), paper 94. and V is a viscosity subsolution of (1.19), that is, for any ψ ∈ C 2 (R d ) and any local minimum point  [3] to non-constant diffusion coefficients. They can not be obtained by the method in [3] because some rigor conditions for the diffusion coefficient are needed when the method in [3] is applied to general diffusion processes (see Remark 3.1).

Remark 1.2. Theorem 1.3 and Theorem 1.4 extend Theorem 2 and Proposition 2 in
The existence of the solution of (1.19) can date back to [7] and [19], where (1.19) takes the form Under sufficient ergodicity assumption on {X t , t ≥ 0}, if c is bounded and sufficient smooth, then (1.20) has a solution. The case of c unbounded was considered in [16].
Recently, under some smooth conditions, Ichihara and Sheu [9] considered the solution of a type of ergodic HJB equation in the classical sense. When a ∈ C 2+γ , and b ∈ C 1+γ , and c ∈ C 1+γ for some γ ∈ (0, 1], Robertson and Xing ([22]) studied the solution of ergodic HJB equation (1.19) in the classical sense. The advantage of our method is that it can be applied to general nonsmooth and quadratic growth cost functions.
The proofs of Theorem 1.1 and Theorem 1.2, and The proof of Theorem 1.3 will be given in Section 2 and Section 3, respectively. The main tool of the proof of Theorem 1.1 is the perturbation theory of linear operators. The proof of Theorem 1.3 is based on Theorem 1.1 and the Harnack inequality of stochastic differential equations with multiplicative noise in [25]. A sketch proof of Theorem 1.4 is in Section 3. We introduce briefly Kato's perturbation theory of linear operators ( [12]) in Appendix.

Long time asymptotics and moderate deviations
In this section, we prove Theorem 1.1 and Theorem 1.2 using the perturbation theory of linear operators. we restrict the logarithmic function on R to avoid the logarithmic function of complex Feynman-Kac operator in [26] and use the Taylor expansion of the largest modulus eigenvalue of the Feynman-Kac semigroup to replace C 2 -regularity in [26].

Step 2. Strong Feller property and its density estimates for the Feynman-Kac operators.
Firstly, let us show that for any t ∈ [0, t 0 ], x → P zc t g(x) is continuous, where g is a bounded function on E and z ∈ (−2 , 2 ).
By the Markov property, we can write that and so By (C4), x → P ε ((P zc t−ε g) 2 )(x) is locally bounded. Therefore, for each x ∈ E, for any sequence x n → x, EJP 22 (2017), paper 94.
Next, let us give some estimates for the density of the Feynman-Kac operators. Let z ∈ (− , ) Since for any A ∈ E, it is shown that x → P 2zc there exists (x, y) → p z (t 0 , x, y) non-negative measurable such that P zc t0 I A (x) = A p z (t 0 , x, y)µ(dy).
Let r be the conjugated number of 2p , i.e., r = 2p p+1 . Then Thus, by (C3) and (C4), for any compact set K ⊂ E, is a tight family of probability measures on E.

The proof of Theorem 1.2
Set T (z) = P z(c−µ(c)) t0 and B = L C q (E, µ). Let G(z) be the complex number with the largest modulus in the spectrum of P z(c−µ(c)) t0 regarded as an operator on B and Then, it is similar to (2.10) that for any z ∈ R, and for any compact set K, In particular, log G a(t)z t = 0; (2.11) and lim t→∞ sup x∈K t a 2 (t) log log G a(t)z t = 0. (2.12) We can write By the definition of S (see (A.6)), for any g ∈ B with µ(g) = 0, . (2.13) For l ≥ 1, let λ be an eigenvalue the operator T (l) S (0) and let g be a eigenvector of λ.

Applications to SDEs
In this section, we show Theorem 1.  [3]. We only give a sketch proof in the section.

Proofs of Theorem 1.3
By the assumption (DC), if we take g(x) = |x| 2 , then Thus, the existence and uniqueness of the invariant measure µ follow from Theorem 3.7 in [11].
where, for α > 1 + ϑκ 1 Integrating the above inequality for P t with respect to µ(dy) on U (0, t), and by Hölder's inequality we get Let p be the conjugated number of α. Then, Since for any t > 0, when α → ∞, 2α(t) α → 0, thus when α large enough, The proof of (1.15) is completed.

Remark 3.1.
For the diffusion process (1.12), we only obtain that (3.1) holds for some p > 1 which is due to the restriction of the Harnack-inequality. In order to obtain that (3.1) holds for some p > 2, using the method on P.2609 in [3], an additional rigor restriction condition ϑ < ( 5/2 − 1)κ 1 is needed. The improved condition (1.4) plays an important role in application to stochastic differential equations with σ = I.

The proof of Theorem 1.4
We only give a sketch proof, because its proof is similar to Proposition 2.2 in [3].
It is sufficient to show that v(x) = e V (x) is a viscosity solution of the following linear Firstly, as the same in [3] or Step 2 in the proof of Theorem 1.1, we can show continuity of the function ϕ(t, x) defined by Next, we show that v T (t, x) := ϕ(T − t, x) is a viscosity solution of the parabolic equation Let (t, x) ∈ [0, T ] × R d be given. Let ψ : [0, t) × R d → R be a smooth function such that ψ(t, x) = v T (t, x) and v T − ψ has a local extreme at (t, x). We can assume that v T − ψ has a strict local extreme in (t, x). We write Now we use (3.5) to prove that v T has the subsolution property. Without loss of generality, we assume that ψ is a smooth function with compact support such that v T − ψ has a local minimum at (t, x), and ψ(t, x) = v T (t, x) at (t, x). In the same way as [3], we can get On the other hand, by Itô's formula, we have that Thus, from (3.5),(3.6) and (3.7), we get and so, the subsolution property holds. The supersolution property is proved in the same way.
Moreover,ṽ T (x) → v(x) as T → +∞ uniformly on compact sets. In particular, v is continuous.
Finally, we show that v is a viscosity solution of (3.2). Let x ∈ R d , and ψ : R d → R be a smooth solution such that v(x) = ψ(x) and υ − ψ has a local minimum at x. Fix t > 0, and defineψ(s, y) := ψ(y) − |y − x| 4 − (s − t) 2 . Then v −ψ has a strict minimum at (t, x), (3.10) Letting n → +∞ and using (3.9) and lower-semicontinuity of c * , we obtain The subsolution property is proved. The supersolution property can be shown in the same way.

A Perturbation theory of linear operators
In this section, we introduce briefly Kato's perturbation theory of linear operators (see Chapter 7 in [12]).
Let B be a Banach space. Let T be an operator on B and let D(T ) and R(T ) denote its domain of definition and range, respectively. The set of closed operators on B will be denoted by C (B, B).  C(B, B); χ ∈ D} be boundedholomorphic and let P (T (χ)) be the resolvent set of T (χ). Assume that the assumption (GP) hold. Then (1). The resolvent R(ζ, χ) = (T (χ)−ζ) −1 is bounded-holomorphic in the two variables on the set of all ζ, χ such that ζ is in the resolvent set of T (0) and |χ| is sufficiently small (depending on ζ). (2). The spectrum Σ(T (0)) of T (0) can be separated in two parts Σ (0) = {λ 0 }, Σ (0) by the curve Γ in the manner previously described.
Thus, r 0 is a lower bound for the convergence radius of (A.12), and the following result holds.