Feynman–Kac penalizations of rotationally symmetric α -stable processes

Statistics and Probability Letters 148 (2019) 82–87

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Feynman–Kac penalizations of rotationally symmetric α -stable processes Yunke Li a , Masayoshi Takeda b , a b

∗,1

Miami Business School, 5250 University Drive, Coral Gables, Florida 33124-6520, USA Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

article

info

a b s t r a c t

Article history: Received 3 October 2018 Received in revised form 27 December 2018 Accepted 2 January 2019 Available online 15 January 2019

In Yano et al. (2009), limit theorems (so called, Feynman–Kac penalizations) for the onedimensional recurrent symmetric α -stable process weighted and normalized by negative (killing) Feynman–Kac functionals with small potential are studied. In this paper, we deal with the same problem for negative potentials diverging at infinity. © 2019 Elsevier B.V. All rights reserved.

MSC: primary 60J45 secondary 60J40 35J10 Keywords: Symmetric stable process Feynman–Kac functional Penalization Fukushima’s ergodic theorem Intrinsic ultracontractivity

1. Introduction In Roynette et al. (2006a), they gathered their results of the penalization of the 1-dimensional Brownian motion by the Feynman–Kac functionals. In Roynette et al. (2006b), they extended the results in Roynette et al. (2006a) to multid dimensional Brownian motion. More precisely, let (Ω , Ft , F∞ , PW x , Bt ) be the Brownian motion on R . Let V ≥ 0 be a borel d function on R . As a result of the Feynman–Kac penalization problem, they prove the existence of a probability QxV on (Ω , F∞ ) such that for a Fs -measurable set Γs ∈ Fs , s > 0

[

EW e− x lim

t →∞

∫t

[

0 V (Bs )ds

EW e− x

∫t

] ; Γs ] = QxV [Γs ].

(1.1)

0 V (Bs )ds

When (1.1) holds, there exists a martingale Mt on (Ω , PW x , {Ft }) such that QxV (Γs ) =

∫ Γs

Ms dPW x ∀s > 0 .

Hence, our problem is to identify the martingale Mt . ∗ Corresponding author. E-mail addresses: [email protected] (Y. Li), [email protected] (M. Takeda). 1 Supported in part by Grant-in-Aid for Scientific Research (No.18H01121(B)), Japan Society for the Promotion of Science. https://doi.org/10.1016/j.spl.2019.01.006 0167-7152/© 2019 Elsevier B.V. All rights reserved.

(1.2)

Y. Li and M. Takeda / Statistics and Probability Letters 148 (2019) 82–87

83

Yano et al. (2009) developed the penalization problem in case of the 1-dimensional recurrent symmetric α -stable process, ∫ 1 < α < 2. In particular, they studied the Feynman–Kac penalization weighted by exp(− L(t , x)V (dx)), where L(t , x) is the local time at x and V is a positive measure on R1 such that

∫ R1

(1 + |x|1−α )V (dx) < ∞.

Our objective of this paper is to extend their results to d-dimensional symmetric α -stable processes by weighted and normalized large Feynman–Kac functionals (d ≥ 2). More precisely, let X α = (Ω , F, Ft , Px , Xt ) be the symmetric α -stable process on Rd with 0 < α < 2, that is, the Markov process generated by (−∆)α/2 , and (E , D(E )) the Dirichlet form of X α ((2.1) below). Note that X α is transient. Let V be a positive Borel function in L∞ loc , the space of locally bounded functions. We denote t

∫

AVt =

V (Xs )ds, 0

and define the family {QVx,t } of probability measures on (Ω , Ft ) by

QVx,t

[B] =

∫

1 ZtV (x)

e−At (ω) Px (dω), B ∈ Ft , V

B

where ZtV (x) = Ex [exp(−AVt )] is the normalizing factor. Our problem is to identify the limit of QVx,t as t → ∞. The main theorem is Theorem 1.1. Suppose that V ∈ L∞ loc satisfies lim

V (x)

|x|→∞

(1.3)

= ∞.

log|x|

Let λ0 be the principal eigenvalue of (−∆)α/2 + V and φ0 the corresponding normalized L2 -eigenfunction, that is, (−∆)α/2 + V φ0 = λ0 φ0 with ∥φ0 ∥L2 (dx) = 1. Then

[

t →∞

φ

along (Ft ) for every x,

QVx,t −→ Px 0

]

(1.4)

that is, for any s ≥ 0 and any bounded Fs -measurable function Z ,

[

V

Ex Ze−At lim

t →∞

[

V

Ex e−At

]

] = Eφx 0 [Z ] for every x.

φ

φ

Here Ex 0 is the expectation with respect to the probability measure Px 0 defined by φ

φ

φ

Px 0 (dω) = Lt 0 (ω) · Px (dω), Lt 0 = eλ0 t

φ0 (Xt ) −AV e t. φ0 (X0 ) φ

Our main idea for the proof of Theorem 1.1 is as follows: Let X φ0 = (Ω , Xt , Px 0 , ζ ) be the transformed process of X . We then see from Lemma 6.3.2 in Fukushima et al. (1994) that X φ is an irreducible, conservative φ02 dx-symmetric Markov process on Rd . For the proof of Theorem 1.1, Fukushima’s ergodic theorem is crucial: For any bounded Borel function g φ

lim Ex 0 (g(Xt )) =

t →∞

∫ Rd

g(y)φ0 (y)2 dy, ∀x ∈ Rd . φ

The conservativeness of X φ0 implies that Lt 0 is a martingale on (Ω , Ft , Px ) and thus Theorem 1.1 says that the martingale in φ (1.2) is identified with Lt 0 . Finally, we remark that in Takeda (2010) we deal with Feynman–Kac penalizations for multi-dimensional transient symmetric α -stable processes weighted and normalized by positive (creation) Feynman–Kac functionals with small potential. 2. Preliminaries Let X α = (Ω , F, Ft , θt , Px , Xt ) be the symmetric α -stable process on Rd with d ≥ 2 and 0 < α < 2, that is, the Markov process generated by (−∆)α/2 . Here {Ft }t ≥0 is the minimal (augmented) admissible filtration and θt , t ≥ 0, is the shift operators satisfying Xs (θt ) = Xs+t for s, t ≥ 0. Let p(t , x, y) be the transition density function of X α and Gβ (x, y), β ≥ 0, be its β -Green function, Gβ (x, y) =

∞

∫

e−β t p(t , x, y)dt . 0

84

Y. Li and M. Takeda / Statistics and Probability Letters 148 (2019) 82–87

As d > α , X α is transient, and G0 (x, y) =

A(d, α )

, |x − y|d−α

A(d, α ) =

α 2d−2 Γ ( α+2 d ) ( ). π d/2 Γ 1 − α2

Let (E , D(E )) be the Dirichlet form associated to X α :

∫∫ ⎧ (u(x) − u(y))(v (x) − v (y)) ⎪ dxdy E (u , v ) = A (d , α ) ⎪ ⎨ |x − y|d+α Rd ×Rd \∆ { } ∫∫ ⎪ (u(x) − u(y))2 ⎪ ⎩ D(E ) = u ∈ L2 (Rd ) : dxdy < ∞ , |x − y|d+α Rd ×Rd \∆

(2.1)

where ∆ = {(x, x) : x ∈ Rd } (Fukushima et al., 1994, Example 1.4.1). Let De (E ) denote the extended Dirichlet space (Fukushima et al., 1994, p. 35). As d > α , De (E ) is a Hilbert space with respect to inner product E (Fukushima et al., 1994, Theorem 1.5.3). For a non-negative, locally bounded Borel function V on Rd (abbreviated by V ∈ L∞ loc ), define a symmetric bilinear form E V by E (u, u) = E (u, u) + V

∫ Rd

u2 Vdx, u ∈ D(E V ),

(2.2)

where D(E V ) = D(E ) ∩ L2 (Rd ; Vdx). We see from Albeverio et al. (1991, Theorem 4.1) that (E V , D(E V )) becomes a Dirichlet form. Denote by X V = (Xt , PVx , ζ ) and HV the Markov process and the self-adjoint operator associated to (E V , D(E V )) respectively: E V (u, v ) = (HV u, v ). Let pVt be the Markov semigroup of X V , pVt f (x) = EVx [f (Xt )], f ∈ Bb (Rd ). Then pVt coincides on the intersection of their domains with the L2 -semigroup generated by HV : pVt f (x) = exp(−t HV )f (x), a.e. for f ∈ L2 (Rd ) ∩ Bb (Rd ). Moreover, by the Feynman–Kac formula, the semigroup pVt is expressed by

[

pVt f (x) = Ex e−

∫t

0 V (Xs )ds

]

f (Xt ) .

(2.3)

Noting that X α satisfies (Albeverio et al., 1991, Assumption 2.11) and V in L∞ loc belongs to the local Kato class, we see from Albeverio et al. (1991, Theorem 7.5 (v)) that pVt admits a symmetric integral kernel pV (t , x, y) which is jointly continuous function on (0, ∞) × Rd × Rd (For the definition of the local Kato class, see Kuwae and Takahashi (2007, Definition 3.1, Example 5.1)). 3. Feynman–Kac penalization In this section, we shall prove Theorem 1.1. We first remind you of Fukushima’s ergodic theorem. Theorem 3.1 (Fukushima, 1982, p. 201, Corollary). Let X = (Px , Xt ) be an irreducible, conservative m-symmetric Markov process on E. Assume that m(E) < ∞. Then for f ∈ Lp (E ; m), 1 < p < ∞ lim pt f (x) =

t →∞

1 m(E)

∫

fdm, m-a.e. x and in Lp (E ; m).

(3.1)

E

Moreover, if X has a transition density with respect to m, then for f ∈ L∞ (E ; m) the limit (3.1) holds for every x ∈ E. It is well-known that if lim V (x) = ∞,

(3.2)

|x|→∞

then pVt is a compact operator on L2 (Rd ). Let φ0 be the L2 -normalized principal eigenfunction corresponding to the eigenvalue λ0 , pVt φ0 = exp(−λ0 t)φ0 . φ0 has a strictly positive continuous version. We then see from Kaleta and Kulczycki (2010, Theorem 1) that

φ0 (x) ≤ C1 ·

ϕ V (x) for some constant C1 , (1 + |x|)d+α

Here

ϕ V (x) = Ex

τB(x,1)

[∫

V

e−At dt

]

0

and τB(x,1) is the first exit time from the ball with center x and radius 1. In particular, φ0 (x) ≤ C2 /(1 + |x|)d+α for some

Y. Li and M. Takeda / Statistics and Probability Letters 148 (2019) 82–87

85

constant C2 because supx∈Rd ϕ V (x) ≤ E0 [τB(1) ] < ∞. As a result, we see that 1

φ0

∈ Lp (Rd ; φ02 dx) for 1 ≤ p <

d + 2α d+α

,

(3.3)

in particular, φ0 ∈ L1 (Rd ). φ We define the multiplicative functional Lt 0 by φ

Lt 0 = eλ0 t

φ0 (Xt ) −AV e t. φ0 (X0 )

(3.4)

φ

φ

We denote by X φ0 = (Px 0 , Xt ) the transformed process of X α by Lt 0 , φ

φ

Px 0 (dω) = Lt 0 (ω) · Px (dω). We know from Fukushima et al. (1994, Theorem 4.7.1) that X φ0 is a φ0 2 dx-symmetric recurrent Markov process and for a Borel set A with m(A) > 0

Px [σA ◦ θn < ∞, ∀n ≥ 0] = 1, q.e. x ∈ Rd ,

(3.5)

where m is the Lebesgue measure. Moreover, we have Proposition 3.2. The transformed process X φ0 is Harris recurrent, that is, for a non-negative function f with m({x : f (x) > 0}) > 0, ∞

∫

φ

f (Xt )dt = ∞, Px 0 -a.s.

(3.6)

0

Proof. Set A = {x : f (x) > 0}. Since the Markov process X φ0 has the transition density function pV (t , x, y)

e λ0 t ·

φ0 (x)φ0 (y)

with respect to φ0 2 dx, (3.5) holds for all x ∈ Rd by Fukushima et al. (1994, Problem 4.6.3). Using the strong Feller property and the proof of Revuz and Yor (1998, Chapter X, Proposition (3.11)), we see from (3.5) that X φ0 is Harris recurrent. □ Lemma 3.3. Suppose that V (x)

lim

|x|→∞

log|x|

(3.7)

= ∞.

Then, for each x ∈ Rd

[

eλ0 t Ex e−At

V

]

∫

−→ φ0 (x)

Rd

φ0 (y)dy, t −→ ∞.

(3.8)

Proof. Note that

[

eλ0 t Ex e−At

V

]

φ

= φ0 (x)Ex 0

[

1

]

φ0 (Xt )

by (3.4) and that φ

[

Ex 0

]

1

φ0 (Xt )

φ

1

φ

= pt −0 1 (p1 0 (

φ0

))(x), t > 1

(3.9)

by the semigroup property. Under the condition (3.7), the intrinsic ultracontractivity of pVt follows from Kaleta and Kulczycki (2010, Theorem 3), and thus by (3.3) 1

φ

∥p1 0 (

φ0

)∥∞ ≤ c ∥

1

φ0

∫ ∥L1 (φ0 2 dx) = c

Rd

φ0 dx < ∞ for some constant c .

Hence, by Theorem 3.1, the right hand side of (3.9) converges for every x to φ0

symmetry of pt with respect to φ02 dx

∫

φ

Rd

p1 0 (

1

φ0

)φ02 dx =

∫

1 Rd

φ0

φ0 2 dx =

the proof of this lemma is completed. □

∫ Rd

φ0 dx,

∫

φ

Rd

p1 0 (1/φ0 )(y)φ0 (y)2 dy, t → ∞. Since by the

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Y. Li and M. Takeda / Statistics and Probability Letters 148 (2019) 82–87

Lemma 3.3 leads us to Theorem 1.1. Indeed,

[

V

Ex e−At |Fs

[

V

Ex e−At

]

[

eλ0 t Ex e−At |Fs

=

]

V

[

V

eλ0 t Ex e−At

[

]

] V

eλ0 s e−As eλ0 (t −s) EXs e−At −s V

=

[

V

eλ0 t Ex e−At eλ0 s e−As φ0 (Xs ) V

−→

φ0 (x)

]

]

∫

Rd

φ0 (x)dx

φ0 (x)dx

∫

Rd

φ

= Ls 0 , Px -a.s. φ

φ

as t → ∞. The transformed process X φ0 is recurrent and consequently conservative, pt 0 1 = 1. As a result, E(Lt 0 ) = 1, and φ consequently, Lt 0 is martingale. Scheffé’s lemma leads us to Theorem 1.1 (e.g. Roynette et al. (2006a)). Remark 3.4. Suppose that V satisfies only (3.2). We then see from Theorem 3.1 and (3.3) that

[

V

⏐ ⏐

Ex e−At ⏐ Fs

[

V

Ex e−At

]

] φ

−→ Ls 0 , Px -a.s., a.e. x.

Hence, the statement (1.4) holds for a.e. x. Remark 3.5. We compare the (killing) Feynman–Kac penalization with the Yaglom limit. It follows from Theorems 1.1 and 3.1 that for a bounded Borel function g

[

⎛

V

Ex g(Xs )e−At

lim ⎝ lim

s→∞

t →∞

[ Ex

V e−At

]⎞ ⎠ = lim Eφx 0 [g(Xs )]

]

(3.10)

s→∞

∫ = Rd

g(x)φ02 (x)dx.

On the other hand, by replacing g(Xs ) in (3.10) with g(Xt ), we have

[

V

Ex g(Xt )e−At lim

t →∞

[

V

Ex e−At

]

φ

]

e−λ0 t φ0 (x)Ex 0

[

e−λ0 t φ0 (x)Ex

[

= lim

t →∞

∫ =

Rd

∫

g(x)φ0 (x)dx Rd

φ0 (x)dx

g(Xt ) φ0 (Xt ) 1

]

]

(3.11)

φ0 (Xt )

.

Eq. (3.11) tells us that for V ∈ L∞ loc with (3.7) V

lim PVx (Xt ∈ A | t < ζ ) = lim Ex (1A (Xt )e−At | t < ζ )

t →∞

t →∞

= ν φ0 (A), ∀A ∈ B(Rd ), (3.12) ∫ where ν φ0 (A) = A φ0 (x)dx/ Rd φ0 (x)dx. Eq. (3.12) implies that ν φ0 is the Yaglom limit. We prove in Takeda (0000) that ν φ0

∫

is a unique quasi-stationary distribution of X V (For definition of Yaglom limit and quasi-stationary distribution , see Mélárd and Villemonais (2012, Definition 2, Definition 3)). In paragraphs after Pinsky (1985, Proposition 1.10), Pinsky explained this difference. Acknowledgment The authors wish to thank the anonymous reviewer for several helpful suggestions of this paper. References Albeverio, S., Blanchard, P., Ma, Z.M., 1991. Feynman–Kac semigroups in terms of signed smooth measures. In: Hornung, U., et al. (Eds.), Random Partial Differential Equations. Birkhäuser. Fukushima, M., 1982. A note on irreducibility and ergodicity of symmetric Markov processes. Springer Lecture Notes in Phys. 173, 200–207. Fukushima, M., Oshima, Y., Takeda, M., 1994. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin. Kaleta, K., Kulczycki, T., 2010. Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians. Potential Anal. 33, 313–339. Kuwae, K., Takahashi, M., 2007. Kato class measures of symmetric Markov processes under heat kernel estimates. J. Funct. Anal. 250, 86–113.

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