Ergodic problems for viscous Hamilton-Jacobi equations with inward drift

WITH INWARD DRIFT

EMMANUEL CHASSEIGNE^∗ AND NAOYUKI ICHIHARA^†

Abstract. In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence and uniqueness of eigenfunctions associated with the generalized principal eigenvalue of the ergodic problem, (ii) relationships with the corresponding stochastic control problem of both finite and infinite time horizon, and (iii) the precise growth exponent of the generalized principal eigenvalue with respect to a perturbation of the potential function.

Key words. Ergodic problem, viscous Hamilton-Jacobi equation, stochastic ergodic control, generalized prin- cipal eigenvalue.

AMS subject classifications. 35Q93; 60J60; 93E20

1. Introduction. This paper is concerned with the ergodic problem for viscous Hamilton- Jacobi equations of the form

λ − 1

2 ∆ϕ(x) + b(x) · Dϕ(x) + H (x, Dϕ(x)) − V (x) = 0, x ∈ R

, (EP) where b : R

→ R

, H : R

× R

→ R, and V : R

→ R are given functions, Dϕ and ∆ϕ are, respectively, the gradient and the Laplacian of ϕ : R

→ R, and “ · ” stands for the inner product in R

. Throughout the paper, any pair (λ, ϕ) ∈ R × C

(R

) satisfying (EP) is called a solution of (EP), and λ, ϕ are referred to as an eigenvalue of (EP) and an eigenfunction of (EP) associated with λ, respectively.

The main results of this paper consist of three parts. In order to state them briefly, we assume in this introductory section that

b(x) = (1 + | x |

)

x, H (x, p) = 1

m | p |

, V (x) = (1 + | x |

)

, (1.1) where δ ∈ (0, 1], m > 1, and η > 0 are given constants. More precise (and more general) assumptions on (b, H, V ) will be stated in the next section. The crucial properties in (1.1) are:

(i) drift vector − b(x) is inward-pointing with the intensity of order | x |

,

(ii) Hamiltonian H(x, p) is convex in p and diverges as | p | → ∞ with the order | p |

, (iii) potential V (x) is positive and vanishes as | x | → ∞ with the order | x |

.

In the first part of this paper, we discuss the existence and uniqueness of eigenfunctions ϕ of (EP) associated with the generalized principal eigenvalue λ

defined by

λ

:= sup { λ ∈ R | (λ, ϕ) ∈ R × C

(R

) is a subsolution of (EP) } , (1.2) where (λ, ϕ) is said to be a subsolution (resp. supersolution) of (EP) if the left-hand side of (EP) is nonpositive (resp. nonnegative) for all x ∈ R

. In Theorem 2.1 we prove that there exists a unique eigenfunction ϕ

∈ C

(R

) of (EP) associated with λ

, up to an additive constant. The solvability of (EP), especially, the uniqueness of eigenfunctions associated with the generalized principal eigenvalue has been studied under a variety of conditions on (b, H, V ) (see, e.g. [3, 4, 10, 17, 18, 19, 20, 21, 22] and the references therein). Our result can be regarded as extensions of [4, 18]

and [17, 21, 22], the former of which deal with ergodic problems without drift (i.e., b(x) ≡ 0), while the latter restrict their attention to quadratic Hamiltonians (i.e., m = 2). We mention that [10]

∗Institut Denis Poisson (UMR CNRS 7013), Universit´e de Tours, Parc de Grandmont, 37200 Tours, France.

Email: emmanuel.chasseigne@lmpt.univ-tours.fr.

†Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara- shi, Kanagawa 252-5258, Japan. Email: ichihara@gem.aoyama.ac.jp.

1

studies existence of eigenfunctions of (EP) under mild assumptions including (1.1) by a different argument. However, uniqueness is not discussed there.

In the second part, we investigate some relationships between the solution (λ

, ϕ

) of (EP) obtained in the first part and the corresponding stochastic optimal control problem of both finite and infinite time horizon. More specifically, let ξ = (ξ

)

be an R

-valued control process belonging to an admissible class A specified later. Let X

= (X

)

be the controlled process governed by the stochastic differential equation

dX

= − ξ

dt − b(X

) dt + dW

, t ≥ 0, (1.3) where W = (W

)

stands for a d-dimensional standard Brownian motion defined on some prob- ability space. We set

J (T, x; ξ) := E

[∫

( 1

m

| ξ

|

+ V (X

) )

dt ]

, T > 0, x ∈ R

, (1.4) where m

:= m/(m − 1) and E

denotes the expectation conditioned on X

= x. Then we prove in Theorem 2.2 that the value function of the long-run average Λ(x) := inf

ξ∈A

lim sup

T→∞

(1/T )J (T, x; ξ) coincides with λ

for all x ∈ R

, and that the value function of finite time horizon T defined by u(T, x) := inf

ξ∈A

J (T, x; ξ) satisfies

u(T, · ) − λ

T −→ ϕ

+ c as T → ∞ (1.5) for some c ∈ R on any compact subset of R

. Although convergence (1.5) is natural to expect, its validity is not trivial at all since the value of the constant c in (1.5) cannot be determined by ergodic problem (EP), only. The asymptotic behavior of the value function u(T, x) as time horizon T tends to infinity has been investigated, from both PDE and stochastic control point of view, in various settings (see, e.g. [2, 5, 6, 7, 8, 11, 12, 14, 16, 18, 21, 25, 28, 31, 32] and references therein).

The present paper can be compared with [18, 21], where either b = 0 or m = 2 is imposed. Note that equality Λ(x) = λ

has been obtained in [10], but convergence (1.5) seems to be new.

The third part of this paper concerns certain asymptotic properties of the generalized principal eigenvalue λ

with respect to a perturbation of V ; we introduce a real parameter β ≥ 0 and consider (EP) with βV in place of V . Then we easily see that the function β → λ

(β) is nondecreasing and concave. Such kind of qualitative properties of λ

(β) with respect to β have been studied by [9, 19, 20] in connection with the so-called criticality theory for ergodic problem (EP) (see also [3] for related results). In this paper, we investigate the asymptotic behavior of λ

= λ

(β) as β → ∞ . More precisely, we specify the growth exponent of λ

(β ) as β → ∞ in terms of constants δ, m, and η appearing in (1.1). It is known in [20, Proposition 6.2] that, if δ < 0 in (1.1), then λ

(β ) = 0 for all β ≥ 0. It turns out in Theorem 2.3 that this is not the case for δ ≥ 0. Indeed, one has λ

(β ) = O(β

) as β → ∞ , which particularly leads to the following asymptotic behavior:

lim

β→∞

λ

(β ) =

 

 

 



+ ∞ (δ > 0)

¯ λ (δ = 0) 0 (δ < 0),

where ¯ λ ∈ (0, ∞ ) is some constant. To the best of our knowledge, such a classification of asymptotic behaviors of λ

with the precision of its growth exponent has not been obtained in the literature.

From the stochastic control point of view, the previous result can be interpreted as follows.

From the second part, Theorem 2.2, we observe that λ

(β ) is identical with the minimum value, over all admissible controls ξ = (ξ

), of the cost functional of long-run average

J

(x; ξ) := lim sup

T→∞

1 T E

[∫

(L(ξ

) + βV (X

)) dt ]

,

where L(ξ) := (1/m

) | ξ |

with m

= m/(m − 1) for ξ ∈ R

. Roughly speaking, when β is sufficiently large, the optimal strategy for the controller is to avoid the region where V (x) is large.

In the case where δ > 0, the strength of the inward drift is such that the cost L becomes dominant compared with the potential V . Indeed, the controller needs to compensate the drift in order to avoid the regions where V is large, so that | ξ | , hence L(ξ) is large. This implies that the cost increases as β increases, which is also reflected on the fact that the optimal trajectory becomes positive recurrent (see Proposition 4.6). The growth order of λ

(β ) as β → ∞ is, therefore, determined by the balance between the intensity δ of the inward drift b, the growth exponent m

of the cost function L, and the decay rate η of the potential function V .

On the other hand, if δ = 0, then the cost incurred by L is relatively small compared to that by βV , which allows the controller to avoid the peak of the potential βV (x) without paying too much cost. As a consequence, λ

(β) remains bounded uniformly in β and converges to a finite value as β → ∞ . Let us also mention in this case that under some more assumptions we are even able to prove that λ

(β ) reaches a plateau, that is, there exists β

> 0 such that λ

(β) = λ

(β

) for any β > β

(see Proposition 6.6).

The rest of this paper is organized as follows. In the next section, we state our standing assumptions and main results precisely. Section 3 is concerned with preliminaries for later discus- sions. In Sections 4 and 5, we prove Theorems 2.1 and 2.2, respectively. The proof of Theorem 2.3 is given in Section 6. Appendices A, B, and C are devoted, respectively, to a gradient es- timate of solutions to elliptic equations, a moment estimate of controlled processes, and a PDE characterization of value functions of finite time horizon.

2. Assumptions and Main results. For x = (x

, . . . , x

) and y = (y

, . . . , y

) in R

, we set x · y := ∑

i=1

x

y

and | x | := (x · x)

. We frequently use the notation ⟨ x ⟩ := (1 + | x |

)

for x ∈ R

. Given a d × d matrix A = (a

) ∈ R

⊗ R

, we set | A | := ( ∑

i,j=1

a

)

. Let B

:= { y ∈ R

| | y | < R } and denote its closure and boundary by B

and ∂B

, respectively.

Given an integer k ≥ 0, we denote by C

(R

) the set of all functions u : R

→ R which, together with all their partial derivatives D

f of orders | α | ≤ k, are continuous on R

, where α = (α

, . . . , α

) denotes a multi-index and | α | := α

+ · · · + α

. We set C(R

) := C

(R

) and C

(R

) := ∩

k=0

C

(R

). We also use the notation C

(R

) to denote the set of continuous functions u : R

→ R that are polynomially growing as | x | → ∞ , that is, | u(x) | ≤ C ⟨ x ⟩

in R

for some C, q > 0.

We say that a sequence of functions { u

} converges to u in C

(R

) as n → ∞ if, for any compact subset K ⊂ R

and any multi-index α with | α | ≤ k, max

x∈K

| D

u

(x) − D

u(x) | → 0 as n → ∞ . In particular, { u

} converges to u in C(R

) as n → ∞ if and only if { u

} converges to u as n → ∞ uniformly on any compact subset of R

. A subset G ⊂ C

(R

) is said to be precompact in C

(R

) if any sequence { u

} ⊂ G contains a subsequence which converges in C

(R

) to a function u ∈ C

(R

).

Throughout the paper, we assume that b = (b

, . . . , b

) and H satisfy the following (A1) and (A2), respectively:

(A1) b

∈ C

(R

) for all 1 ≤ i ≤ d, and there exists a 0 ≤ δ ≤ 1 such that sup

x∈R^d

| b(x) |

⟨ x ⟩

< ∞ , sup

x∈R^d

| Db(x) | < ∞ , lim inf

|x|→∞

b(x) · x

⟨ x ⟩

> 0, where Db = (∂b

/∂x

) : R

→ R

⊗ R

.

(A2) There exist m > 1 and Σ = (Σ

(x)) : R

→ R

⊗ R

with Σ

∈ C

(R

) for all i, j ∈ { 1, . . . , d } such that

H(x, p) = 1

m | Σ

(x)p |

, inf

x,ζ∈R^d,ζ̸=0

| Σ

(x)ζ |

| ζ | > 0, sup

x,ζ∈R^d,ζ̸=0

| Σ

(x)ζ |

| ζ | < ∞ ,

where Σ

(x) denotes the transposed matrix of Σ(x).

Note that, under condition (A2), one can easily see that

| D

H(x, p) | ≤ K | p |

, x, p ∈ R

, (2.1) for some K > 0, where D

H (x, p) denotes the gradient of H (x, p) with respect to p.

As to the assumption on V , we consider three types of conditions:

(A3a) V ∈ C

(R

), sup

R^d

| V | < ∞ , and sup

R^d

| DV | < ∞ . (A3b) V ∈ C

(R

), sup

R^d

| DV | < ∞ , V ≥ 0 in R

with V ̸≡ 0, and lim

|x|→∞

V (x) = 0.

(A3c) V ∈ C

(R

), sup

R^d

| DV | < ∞ , and there exists an η > 0 such that inf

x∈R^d

(V (x) ⟨ x ⟩

) > 0, sup

x∈R^d

(V (x) ⟨ x ⟩

) < ∞ .

Clearly, (A3c) implies (A3b), and (A3b) yields (A3a). Hereafter, δ, m, and η always stand for the constants appearing in (A1), (A2), and (A3c), respectively. As a typical example of (b, H, V ) satisfying (A1), (A2), and (A3c), we have in mind functions given in (1.1).

We are in position to state our first main result. We set

Φ

:= { ϕ ∈ C

(R

) | lim sup

|x|→∞

ϕ(x)

⟨ x ⟩

< ∞ , lim inf

|x|→∞

ϕ(x)

⟨ x ⟩

≥ 0, ∀ γ > 1 − δ } , (2.2) Φ

:= { ϕ ∈ C

(R

) | lim sup

|x|→∞

ϕ(x)

⟨ x ⟩

< 0, lim inf

|x|→∞

ϕ(x)

⟨ x ⟩

≥ 0, ∀ γ > 1 − δ } . (2.3) Note that Φ

⊂ Φ

. Roughly speaking, functions in Φ

look like − c ⟨ x ⟩

(1 + o(1)) for some c > 0 as | x | → ∞ . In particular, any function in Φ

is bounded above in R

. The class Φ

contains functions ϕ such that − C ⟨ x ⟩

≤ ϕ(x) ≤ C ⟨ x ⟩

in R

for some C > 0.

Let λ

be the generalized principal eigenvalue of (EP) defined by (1.2). Note that λ

>

−∞ since (λ, ϕ) := ( − sup

| V | , 0) is a subsolution of (EP) under (A1), (A2), and (A3a). Then one has the following characterization of solutions to (EP).

Theorem 2.1. Assume (A1) with δ > 0, (A2), and (A3a). Then the following (a)-(c) hold.

(a) A solution (λ, ϕ) ∈ R × C

(R

) of (EP) satisfies ϕ ∈ Φ

if and only if λ = λ

.

(b) There exists a unique solution (λ

, ϕ

) ∈ R × C

(R

) of (EP), up to an additive constant with respect to ϕ

, such that ϕ

∈ Φ

. In particular, λ

= λ

, and ϕ

is the unique eigenfunction of (EP) associated with λ

, up to an additive constant.

(c) If we assume (A3b) instead of (A3a), then ϕ

∈ Φ

.

In order to state our second result, we formulate the stochastic control problem associated with (EP). We say that the 6-tuple

π = (Ω, F , ( F

)

, P ; (W

)

; (ξ

)

)

is an admissible control if (Ω, F , P ) is a complete probability space, ( F

)

a family of sub-σ- algebras satisfying N ⊂ F

⊂ F

and F

= ∩

u>t

F

for all 0 ≤ t ≤ s, where N is the collection of all P -null sets, (W

)

a d-dimensional ( F

)-Brownian motion, and ξ = (ξ

)

an R

-valued ( F

)-progressively measurable process such that

E [∫

0

| ξ

|

dt ]

< ∞ for all T > 0,

where m

:= m/(m − 1). Let A denote the collection of all admissible controls π. As usual, we

use the conventional notation ξ ∈ A , instead of π ∈ A .

Given an admissible control ξ ∈ A , we denote by X

= (X

)

the associated controlled process governed by (1.3). It is known (e.g. [30, Section 1]) that, for any initial point x ∈ R

, there exists a unique solution of (1.3) which does not explode in finite time.

We next introduce the cost functional to be optimized which is slightly general than (1.4). Let g : R

→ R be a given terminal cost belonging to the class Ψ

with α = 1 +

, where

Ψ

:= { g ∈ C

(R

) | lim inf

|x|→∞

g(x)

⟨ x ⟩

≥ 0 } , α ≥ 0. (2.4)

For T > 0, x ∈ R

, and ξ ∈ A , we set J (T, x; ξ) := E

[∫

(L(X

, ξ

) + V (X

)) dt + g(X

) ]

, (2.5)

where L : R

× R

→ R is the convex conjugate of H defined by L(x, ξ) := sup

p∈R^d

(ξ · p − H(x, p)), x, ξ ∈ R

. (2.6) Note that, if we denote by Σ

(x) the inverse matrix of Σ(x) appearing in (A2), then L(x, ξ) can be explicitly written as

L(x, ξ) := 1

m

| Σ

(x)ξ |

, x, ξ ∈ R

. (2.7) One can also verify from (2.6) that H (x, p) + L(x, ξ) ≥ ξ · p for all x, p, ξ ∈ R

, and that H(x, p) + L(x, ξ) = ξ · p if and only if ξ = D

H (x, p).

We finally define the value functions of finite time horizon T and of long-run average by u(T, x) := inf

ξ∈A

J(T, x; ξ), Λ(x) := inf

ξ∈A

lim sup

T→∞

J (T, x; ξ)

T , (2.8)

respectively. Then, our second result can be stated as follows.

Theorem 2.2. Assume (A1) with δ > 0, (A2), (A3b), and one of the following (i)-(ii):

(i) g ∈ Ψ

;

(ii) g ∈ Ψ

and δ = 1 in (A1).

Then, Λ(x) = λ

for all x ∈ R

. Moreover, there exists a constant c ∈ R such that

u(T, x) − λ

T −→ ϕ

(x) + c in C(R

) as T → ∞ , (2.9) where ϕ

is the eigenfunction of (EP) associated with λ

, given in Theorem 2.1.

Now, we state our third result. Let us introduce the real parameter β ≥ 0 and consider ergodic problem (EP) with βV instead of V , that is,

λ − 1

2 ∆ϕ + b(x) · Dϕ + H (x, Dϕ) − βV = 0 in R

. (EP

) Let λ

= λ

(β) denote the generalized principal eigenvalue of (EP

) defined similarly as (1.2).

Then λ

(β ) has the following properties.

Theorem 2.3. Assume (A1), (A2), and (A3c). Then λ

(0) = 0, and λ

(β) is strictly positive, nondecreasing, and concave in β ∈ (0, ∞ ). Moreover, there exists some ν > 0 such that

νβ

− ν

≤ λ

(β) ≤ ν

(1 + β

), β ≥ 0.

In particular, as β → ∞ , λ

(β) diverges to + ∞ for δ > 0, while λ

(β) converges to a finite

value ¯ λ ∈ (0, ∞ ) for δ = 0.

In view of Theorem 2.3, in the case where δ > 0, since β 7→ λ

is concave and diverges to ∞ as β → ∞ , we deduce that λ

(β ) is strictly increasing with respect to β. We will see in Section 6 that this is not the case, in general, for δ = 0. Indeed, under some additional assumptions on (b, H, V ), it happens that there exists a β

> 0 such that λ

(β) = λ

(β

) for all β ≥ β

. This kind of “flattening” phenomena have been observed and discussed by [19, 20] in connection with the criticality theory for ergodic problems. See [19, 20] for details.

We close this section with some remarks on the extension of Theorems 2.1 and 2.2 to the case where δ = 0. In that case, it is also possible to obtain similar results under certain additional assumptions. For instance, under the hypothesis of Theorem 2.3 with δ = 0, Theorems 2.1 and 2.2 still hold provided λ

(β ) < ¯ λ := sup { λ

(β

) | β

≥ 0 } . This is a direct consequence of [20, Theorem 2.2] for Theorem 2.1 and of [20, Theorem 2.1] for the first half of Theorem 2.2, i.e., for the validity of Λ(x) = λ

. For the second half of Theorem 2.2, one can apply the same argument developed in Section 5 of this paper to establish the convergence (2.9). In any case, it is crucial to assume the condition λ

(β ) < λ, which is the key to the ergodicity of the optimal trajectory as ¯ well as the uniqueness of eigenfunctions associated with the generalized principal eigenvalue (see Section 4). As is mentioned, this condition is trivial if δ > 0 since ¯ λ = ∞ , whereas, if δ = 0, it may occur that λ

(β) = ¯ λ for some β, and that the optimal trajectory becomes transient (cf.

[19, 20]). We refer to [3] for related results in this direction.

3. Preliminaries. In this section, we always assume (A1) and (A2). For a given ϕ ∈ C

(R

), we define the second order partial differential operator A

by

A

:= 1

2 ∆ − b(x) · D − D

H (x, Dϕ(x)) · D, where D = (∂/∂x

, · · · , ∂/∂x

) and ∆ := ∑

j=1

∂

/∂x

. We consider the stochastic differential equation

{ dX

= − D

H(X

, Dϕ(X

)) dt − b(X

) dt + dW

, t ≥ 0,

X

= x. (3.1)

It is well known (see, for instance, [30, Theorem 11.1]) that, for any ϕ ∈ C

(R

), there exists a filtered probability space (Ω, F , ( F

)

, P ) and an ( F

)-Brownian motion W = (W

)

such that (3.1) has a unique solution up to its explosion time τ

:= lim

n→∞

τ

, where τ

:= inf { t > 0 | | X

| > n } for n ≥ 1. In this paper, we shall call the solution of (3.1) the A

-diffusion. The rest of this section is devoted to recalling some fundamental results on A

-diffusions that are well known in the literature not only for A

-diffusions but also for more general diffusion processes in R

.

We begin with the following fact that will be used in later discussions.

Proposition 3.1. Let X = (X

)

be the A

-diffusion for some ϕ ∈ C

(R

).

(a) Suppose that X is positive recurrent. Let µ = µ(dy) be its invariant probability measure on R

. Then, for any measurable f : R

→ R such that ∫

R^d

| f(y) | µ(dy) < ∞ , E

[f (X

)] −→

∫

R^d

f (y)µ(dy) as T → ∞ ,

where the convergence is uniform, with respect to x, on any compact subset of R

.

(b) Suppose that X is not positive recurrent. Then, for any f : R

→ R with compact support, and for any x ∈ R

,

E

[f (X

)] −→ 0 as T → ∞ .

Proof. Claim (a) has been proved in [21, Proposition 2.6]. For the proof of (b), see, for instance,

[24, Theorem 1.3.10].

The recurrence and transience of A

-diffusions can be verified by using the following proposi- tion, which we call the Lyapunov method in this paper.

Proposition 3.2. Let X = (X

)

be the A

-diffusion for some ϕ ∈ C

(R

).

(a) Suppose that there exist R > 0, u ∈ C

(R

\ B

), and K > 0 such that

|x

lim

u(x) = ∞ , A

u ≤ Ku in R

\ B

. Then X does not explode in finite time, i.e., P

(τ

= ∞ ) = 1 for all x ∈ R

. (b) Suppose that there exist R > 0, u ∈ C

(R

\ B

), and x

̸∈ B

such that

inf

R^d\B_R

u > −∞ , u(x

) < min

∂B_R

u, A

u ≤ 0 in R

\ B

. Then X is transient.

(c) Suppose that there exist R > 0 and u ∈ C

(R

\ B

) such that

|x

lim

u(x) = ∞ , A

u ≤ 0 in R

\ B

. Then X is recurrent.

(d) Suppose that X does not explode in finite time, and that there exist R > 0, u ∈ C

(R

\ B

), and ε > 0 such that

inf

R^d\B_R

u > −∞ , A

u ≤ − ε in R

\ B

.

Then X is positive recurrent with an invariant probability measure µ = µ(dy) which has a positive and continuous density in R

. More generally, suppose that there exist some u ∈ C

(R

), f ∈ C(R

), K > 0, and R > 0 such that

inf

R^d

u > −∞ , inf

R^d\B_R

f > 0, A

u ≤ − f 1

+ K1

in R

. Then the invariant probability measure µ of X satisfies ∫

R^d

f (y)µ(dy) < ∞ .

Proof. We refer to [30, Theorems 6.7.1, 6.1.1, 6.1.2] for the proofs of (a)-(c). The positive recurrence of X stated in (d), as well as the existence of a positive continuous density of µ are proved in [30, Theorem 6.1.3] and [30, Section 4.8], respectively (see, especially, Theorem 4.8.4 for the latter). Here, we only check the integrability of f with respect to µ, which is also classical but is not explicitly mentioned in [30].

We may assume without loss of generality that u ≥ 0 in R

. Applying Ito’s formula to u(X

), we have

E

[u(X

)] − u(x) = E

[∫

A

u(X

) dt ]

≤ E

[∫

( − f (X

)1

(X

) + K1

(X

)) dt

] (3.2)

for all x ∈ R

, T > 0, and n ≥ 1, where τ

:= inf { t > 0 | | X

| > n } . Since u and f are bounded below in R

and X does not explode in finite time, we see by letting n → ∞ that

E

[∫

f (X

)1

(X

) dt ]

≤ u(x) + KT.

Dividing both sides by T and then sending T → ∞ , we observe in view of the ergodic theorem for the A

-diffusion that

∫

R^d\BR

f (y)µ(dy) ≤ lim inf

T→∞

1 T E

[∫

f (X

)1

(X

) dt ]

≤ K,

which implies the desired estimate. Hence, we have completed the proof of (d).

Remark 3.3. In view of (3.2) and Fatou’s lemma, we observe that sup

n≥1

E

[u(X

)] ≤ u(x) + KT, E

[u(X

)] ≤ u(x) + KT for all x ∈ R

and T > 0. These estimates will be used in the proof of Proposition 4.6.

Hereafter, for the simplicity of notation, we set F[ϕ](x) := − 1

2 ∆ϕ(x) + b(x) · Dϕ(x) + H(x, Dϕ(x)), x ∈ R

, ϕ ∈ C

(R

).

The next proposition is the key to the construction of Lyapunov functions u appearing in Proposition 3.2.

Proposition 3.4. Let D ⊂ R

be a domain, and let ϕ, ψ ∈ C

(D).

(a) u := ϕ − ψ satisfies A

u ≤ F[ψ] − F [ϕ] in D.

(b) Suppose that m ≥ 2 in (A2). Then, for any ε > 0, there exists a κ > 0 such that u := e

satisfies

A

u ≤ κu(F [ψ] − F[ϕ] + ε) in D.

Moreover, if m = 2 in (A2), then the above inequality holds with ε = 0.

(c) Suppose that 1 < m < 2 in (A2). Then, for any K > 0, there exists a κ > 0 such that, for any ϕ, ψ with | Dϕ | + | Dψ | ≤ K in D, the function u := e

satisfies

A

u ≤ κu(F[ψ] − F [ϕ]) in D.

Proof. We reproduce a sketch of the proof for the convenience of the reader (see [20, Proposition 3.10] for a complete proof). Claim (a) is a direct consequence of the convexity of p 7→ H(x, p).

Indeed, noting that H(x, q) − H(x, p) ≥ D

H(x, p) · (q − p), we see that u := ϕ − ψ satisfies A

u = Aϕ − Aψ − D

H (x, Dϕ) · (Dϕ − Dψ)

≤ Aϕ − H (x, Dϕ) − (Aψ − H (x, Dψ)) = F [ψ] − F [ϕ] ,

where A := (1/2)∆ − b(x) · D. Hence, (a) is valid. We next prove (b). Fix any m ≥ 2 and ε > 0.

Then we observe from (A2) that

H(x, p + q) − H(x, p) − D

H (x, p) · q ≥ κ

2 | q |

− ε, x, p, q ∈ R

, (3.3) for some κ > 0 (see, e.g., [20, Proposition 3.7]). Using this inequality, we see by direct computations similar to (a) that u := e

with w := ϕ − ψ satisfies

A

u = κu(A

w + κ

2 | Dw |

) ≤ κu(F[ψ] − F [ϕ] − κ

2 | Dw |

+ ε + κ 2 | Dw |

)

= κu(F [ψ] − F [ϕ] + ε).

If m = 2 in (A2), then (3.3) holds with ε = 0. This leads to the inequality A

u ≤ κu(F [ψ] − F [ϕ]) in D. Hence, (b) is valid. We finally verify (c). Fix any 1 < m < 2 and K > 0. Then, we observe that

H(x, p + q) − H (x, p) − D

H (x, p) · q ≥ κ

2 | q |

, x ∈ R

, | p | , | q | ≤ K, (3.4) for some κ > 0 (see, e.g., [20, Proposition 3.7]). By the same argument as (b), we conclude that u := e

with w := ϕ − ψ satisfies

A

u = κu(A

w + κ

2 | Dw |

) ≤ κu(F[ψ] − F [ϕ]).

Hence, we have completed the proof.

4. Proof of Theorem 2.1. This section is devoted to the proof of Theorem 2.1. Unless otherwise specified, we assume (A1) with δ > 0, (A2), and (A3a). Note that, for any solution (λ, ϕ) ∈ R × C

(R

) of (EP), ϕ is indeed of C

-class. This is a direct consequence of classical Schauder type estimates for linear elliptic equations (e.g. [15, Theorem 6.17]). Hence, as far as solutions (λ, ϕ) ∈ R × C

(R

) of (EP) are concerned, we may assume without loss of generality that ϕ ∈ C

(R

).

We begin with an a priori gradient estimate for eigenfunctions ϕ of (EP) which plays a crucial role throughout the paper.

Proposition 4.1. There exists a constant K > 0 such that, for any solution (λ, ϕ) ∈ R × C

(R

) of (EP),

| Dϕ(x) | ≤ K(1 + | x |

+ (λ

)

), x ∈ R

, where r

:= max {± r, 0 } for r ∈ R. In particular,

sup

x∈R^d

| ϕ(x) |

⟨ x ⟩

< ∞ .

Proof. Let (λ, ϕ) ∈ R × C

(R

) be a solution of (EP). Then, in view of Theorem A.2 in Appendix A, there exists a K > 0 depending only on m and d such that, for any r > 0,

sup

B_r

| Dϕ | ≤ K { 1 + sup

B_r+1

( | b |

+ | Db |

+ (V − λ)

1 m

+

+ | DV |

) } . From this estimate, together with (A1) and (A3a), we obtain the claim.

We next state an existence result for a solution of (EP).

Proposition 4.2. Let λ

be the generalized principal eigenvalue of (EP) defined by (1.2).

Then, for any λ ≤ λ

, there exists an eigenfunction ϕ ∈ C

(R

) of (EP) associated with λ.

Proof. We only give a sketch of the proof since the proof of this proposition is standard (cf.

[9, 19, 20]). We first claim that, if (λ

, ϕ

) ∈ R × C

(R

) is a subsolution of (EP) and ε > 0 is arbitrary, then there exists an eigenfunction ϕ ∈ C

(R

) of (EP) associated with λ

− ε. This implies that, for any λ < λ

, there exists an eigenfunction of (EP) associated with λ. In order to show this claim, fix any R > 0 and consider the Dirichlet problem

λ

− ε + F [u] − V = 0 in B

, u = ϕ

on ∂B

. (4.1) Then, in view of Theorem A.1 in Appendix A, there exists a solution u

∈ C

(B

) ∩ C(B

) of (4.1). Furthermore, by Proposition 4.1, togerther with the standard regularity estimates (e.g. [27, Theorem 4.6.1] and [15, Theorem 4.6]), one can see that, along a suitable diverging sequence { R

} , the family { u

− u

(0) }

converges to a function ϕ in C

(R

) as j → ∞ , and that ϕ is of C

-class and enjoys (EP) with λ = λ

− ε. In order to construct an eigenfunction ϕ ∈ C

(R

) of (EP) associated with λ

, we choose any increasing sequence { λ

} such that λ

→ λ

as n → ∞ , and let { ϕ

} denote a sequence of associated eigenfunctions of (EP). Then, similarly as above, one can see that, along a suitable subsequence, { ϕ

− ϕ

(0) } converges to a function ϕ in C

(R

) as n → ∞ . In particular, ϕ satisfies (EP) with λ = λ

, and therefore ϕ ∈ C

(R

).

Hence, we have completed the proof.

The above existence result does not give any useful information on the asymptotic behavior of eigenfunctions ϕ as | x | → ∞ . In what follows, we construct a special class of solution (λ

, ϕ

) to (EP) such that ϕ

∈ Φ

. To this end, we begin with the following abstract result.

Proposition 4.3. Suppose that there exists a triplet (ψ

, ψ

, ψ

) of C

-functions which satisfies the following (i)-(iii):

(i) ψ

≤ ψ

≤ ψ

in R

, lim

|x|→∞

(ψ

− ψ

)(x) = ∞ ,

(ii) lim sup

|x|→∞

(λ

+ F [ψ

] − V )(x) < 0, lim sup

|x|→∞

(λ

+ F [ψ

] − V )(x) < 0, (iii) inf

R^d

F [ψ

] > −∞ .

Then, there exists a solution (λ

, ϕ

) of (EP) such that inf

R^d

(ϕ

− ψ

) > −∞ , and the associated A

-diffusion is positive recurrent. Moreover, if ψ

satisfies

(iii)

lim inf

|x|→∞

(λ

+ F [ψ

] − V )(x) > 0, then inf

R^d

(ψ

− ϕ

) > −∞ .

Proof. Fix any R > 0 and consider the following Dirichlet problem with discount factor α > 0:

αv + F [v] − V = αψ

in B

, v = ψ

on ∂B

. (4.2) Since F[ψ

] ≤ K and F[ψ

] ≥ − K in R

for some K > 0, we observe, by setting C := K + sup

| V | , that ψ

− α

C and ψ

+ α

C are, respectively, a subsolution and a supersolution of (4.2). In particular, by virtue of Theorem A.1 in Appendix A, there exists a solution v

∈ C

(B

) ∩ C(B

) of (4.2) such that ψ

− α

C ≤ v

≤ ψ

+ α

C in B

. Noting that { v

}

is precompact in C

(R

) in view of classical regularity estimates for elliptic equations, one can find a function v

∈ C

(R

) which satisfies ψ

− α

C ≤ v

≤ ψ

+ α

C in R

and

αv + F[v] − V = αψ

in R

. (4.3)

Let { α

} be any positive decreasing sequence such that α

→ 0 as n → ∞ , and set λ

:=

α

v

(0) and w

:= v

− v

(0). Then, (λ

, w

) satisfies

λ

+ α

w

+ F [w

] − V = α

ψ

in R

. (4.4) Since α

ψ

(0) − C ≤ λ

≤ α

ψ

(0) + C in R

, we see that { λ

} is bounded in R. Furthermore, by Theorem A.2 in Appendix A, we observe that sup

sup

| Dw

| < ∞ for all R > 0. This and the standard Schauder estimates imply that { w

} is precompact in C

(R

). Thus, there exists a subsequence of { α

} , denoted again by { α

} , and a pair (λ

, ϕ

) ∈ R × C

(R

) such that λ

→ λ

as n → ∞ and w

→ ϕ

in C

(R

) as n → ∞ . Letting n → ∞ in (4.4), we conclude that (λ

, ϕ

) is a solution of (EP) and ϕ

∈ C

(R

).

Next, we verify that the above (λ

, ϕ

) is the desired solution. In view of assumption (ii), we observe that there exist some ρ > 0 and R > 0 such that, for any x ∈ R

\ B

,

λ

+ F[ψ

](x) − V (x) ≤ − ρ, λ

+ F[ψ

](x) − V (x) ≤ − ρ.

Since λ

→ λ

as n → ∞ , we may assume without loss of generality that | λ

− λ

| < ρ for all n ≥ 1. Then, noting that λ

≤ λ

, we obtain

λ

+ F [ψ

](x) − V (x) < 0, x ∈ R

\ B

, n ≥ 1, i = − 1, 0.

We now set ψ

:= (1 − θ)ψ

+ θψ

for θ ∈ (0, 1), and M := sup

B_R

(sup

n≥1

| w

| + | ψ

| + | ψ

| + | ψ

| ) < ∞ .

We show that ψ

− M ≤ w

in R

for any θ ∈ (0, 1) and n ≥ 1. From the definition of M , it is clear that ψ

− M ≤ w

in B

. One can also see that, as | x | → ∞ ,

w

− ψ

= w

− ψ

+ θ(ψ

− ψ

) = (v

− ψ

) − v

(0) + θ(ψ

− ψ

)

≥ − C

α

− v

(0) + θ(ψ

− ψ

) ≥ − ψ

(0) − 2C

α

+ θ(ψ

− ψ

) → ∞ .

In particular, for each n ≥ 1, there exists an R

> R such that ψ

− M ≤ w

in R

\ B

.