ELECTRONIC COMMUNICATIONS in PROBABILITY
FEYNMAN-KAC PENALISATIONS OF SYMMETRIC STABLE
PRO-CESSES
MASAYOSHI TAKEDA1
Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan email: takeda@math.tohoku.ac.jp
SubmittedJune 4, 2009, accepted in final formFebruary 6, 2010 AMS 2000 Subject classification: 60J45, 60J40, 35J10
Keywords: symmetric stable process, Feynman-Kac functional, penalisation, Kato measure
Abstract
In K. Yano, Y. Yano and M. Yor (2009), limit theorems for the one-dimensional symmetricα-stable process normalized by negative (killing) Feynman-Kac functionals were studied. We consider the same problem and extend their results to positive Feynman-Kac functionals of multi-dimensional symmetricα-stable processes.
1
Introduction
In[9],[10], B. Roynette, P. Vallois and M. Yor have studied limit theorems for Wiener processes normalized by some weight processes. In [16], K. Yano, Y. Yano and M. Yor studied the limit theorems for the one-dimensional symmetric stable process normalized by non-negative functions of the local times or by negative (killing) Feynman-Kac functionals. They call the limit theorems for Markov processes normalized by Feynman-Kac functionals theFeynman-Kac penalisations. Our aim is to extend their results on Feynman-Kac penalisations to positive Feynman-Kac functionals of multi-dimensional symmetricα-stable processes.
Let Mα = (Ω,F,Ft,Px,Xt)be the symmetric α-stable process on Rd with 0< α ≤ 2, that is,
the Markov process generated by −(1/2)(−∆)α/2, and(E,D(E))the Dirichlet form of Mα(see
(2.1),(2.2)). Let µbe a positive Radon measure in the class K∞ of Green-tight Kato measures (Definition 2.1). We denote byAµt the positive continuous additive functional (PCAF in abbrevia-tion) in the Revuz correspondence toµ: for a positive Borel function f andγ-excessive function g,
〈gµ,f〉=lim
t→0 1 t
Z
Rd
Ex
Z t
0
f(Xs)dAµs
g(x)d x. (1.1)
1THE AUTHOR WAS SUPPORTED IN PART BY GRANT-IN-AID FOR SCIENTIFIC RESEARCH (NO.18340033 (B)), JAPAN
SOCIETY FOR THE PROMOTION OF SCIENCE
We define the family{Qµx,t}of normalized probability measures by
Qµx,t[B] = 1
Ztµ(x)
Z
B
exp(Aµt(ω))Px(dω), B∈ Ft,
whereZtµ(x) =Ex[exp(Aµt)]. Our interest is the limit ofQµx,t ast→ ∞, mainly in transient cases, d> α. They in[16]treated negative Feynman-Kac functionals in the case of the one-dimensional recurrent stable process,α >1. In this case, the decay rate of Ztµ(x)is important, while in our cases the growth order is.
We define
λ(θ) =inf
¨
Eθ(u,u):
Z
Rd
u2dµ=1
«
, 0≤θ <∞, (1.2)
whereEθ(u,u) =E(u,u) +θ
R
Rdu
2d x. We see from[5, Theorem 6.2.1]and[12, Lemma 3.1]that the time changed process byAµt is symmetric with respect toµandλ(0)equals the bottom of the spectrum of the time changed process. We now classify the setK∞in terms ofλ(0):
(i)λ(0)<1
In this case, there exist a positive constant θ0 >0 and a positive continuous function hin the Dirichlet spaceD(E)such that
1=λ(θ0) =Eθ0(h,h)
(Lemma 3.1, Theorem 2.3). We define the multiplicative functional (MF in abbreviation)Lh t by
Lht =e−θ0th(Xt) h(X0)
eAµt. (1.3)
(ii)λ(0) =1
In this case, there exists a positive continuous functionhin the extended Dirichlet spaceDe(E)
such that
1=λ(0) =E(h,h)
([14, Theorem 3.4]). HereDe(E)is the set of measurable functionsuonRd such that |u|<∞
a.e., and there exists anE-Cauchy sequence{un}of functions in D(E)such that limn→∞un=u
a.e. We define
Lht = h(Xt)
h(X0)e
Aµt. (1.4)
(iii)λ(0)>1
In this case, the measureµisgaugeable, that is, sup
x∈Rd
ExeAµ∞<∞
([15, Theorem 3.1]). We puth(x) =Ex[eAµ
∞]and define
Lht = h(Xt)
h(X0)e
The cases(i), (ii), and(iii) are corresponding to thesupercriticality,criticality, andsubcriticality of the operator,−(1/2)(−∆)α/2+µ, respectively ([15]). We will see thatLh
t is a martingale MF
for each case, i.e.,Ex[Lht] =1. LetM
h= (Ω,Ph
x,Xt)be the transformed process ofMαbyLht:
Phx(B) =
Z
B
Lht(ω)Px(dω), B∈ Ft.
We then see from [3, Theorem 2.6] and Proposition 3.8 below that ifλ(0)≤1, then Mh is an h2d x-symmetric Harris recurrent Markov process.
To state the main result of this paper, we need to introduce a subclass KS
∞ of K∞; a measure µ∈ K∞is said to be inKS
∞if sup
x∈Rd
|x|d−α
Z
Rd
dµ(y)
|x−y|d−α
<∞. (1.6)
This class is relevant to the notion ofspecialPCAF’s which was introduced by J. Neveu ([6]); we will show in Lemma 4.4 that if a measureµbelongs toKS
∞, then
Rt
0(1/h(Xs))dA
µ
s is aspecialPCAF
of Mh. This fact is crucial for the proof of the main theorem below. In fact, a key to the proof
lies in the application of the Chacon-Ornstein type ergodic theorem for special PCAF’s of Harris recurrent Markov processes ([2, Theorem 3.18]).
We then have the next main theorem. Theorem 1.1. (i)Ifλ(0)6=1, then
Qµx,t t−→→∞ Phx along(Ft), (1.7) that is, for any s≥0and any boundedFs-measurable function Z,
lim
t→∞
ExZexp(Aµt)
Exexp(Aµt)
=Ehx[Z].
(ii)Ifλ(0) =1andµ∈ KS
∞, then (1.7) holds.
Throughout this paper,B(R)is an open ball with radiusRcentered at the origin. We usec,C, ...,et c as positive constants which may be different at different occurrences.
2
Preliminaries
Let Mα= (Ω,F,F
t,θt,Px,Xt)be the symmetricα-stable process onRd with 0< α≤2. Here
{Ft}t≥0 is the minimal (augmented) admissible filtration andθt, t ≥ 0, is the shift operators satisfying Xs(θt) = Xs+t identically fors,t ≥0. When α= 2,Mαis the Brownian motion. Let
p(t,x,y)be the transition density function ofMαandGβ(x,y),β≥0, be itsβ-Green function,
Gβ(x,y) =
Z∞
0
e−βtp(t,x,y)d t. For a positive measureµ, theβ-potential ofµis defined by
Gβµ(x) =
Z
Rd
LetPt be the semigroup ofMα,
Ptf(x) =
Z
Rd
p(t,x,y)f(y)d y=Ex[f(Xt)].
Let(E,D(E))be the Dirichlet form generated byMα: for 0< α <2
E(u,v) =1
2A(d,α)
ZZ
Rd×Rd\∆
(u(x)−u(y))(v(x)−v(y))
|x−y|d+α d x d y
D(E) =
(
u∈L2(Rd):
ZZ
Rd×Rd\∆
(u(x)−u(y))2
|x−y|d+α d x d y<∞
)
,
(2.1)
where∆ ={(x,x):x∈Rd}and
A(d,α) = α2
d−1Γ(α+d
2 ) πd/2Γ(1−α
2)
([5, Example 1.4.1]); forα=2
E(u,v) = 1
2D(u,v), D(E) =H
1(Rd), (2.2)
whereDdenotes the classical Dirichlet integral andH1(Rd)is the Sobolev space of order 1 ([5,
Example 4.4.1]). LetDe(E)denote the extended Dirichlet space ([5, p.35]). Ifα <d, that is, the processMαis transient, thenDe(E)is a Hilbert space with inner productE ([5, Theorem 1.5.3]). We now define classes of measures which play an important role in this paper.
Definition 2.1. (I) A positive Radon measureµonRd is said to be in theKato class(µ∈ K in notation), if
lim
β→∞xsup∈Rd
Gβµ(x) =0. (2.3)
(II) A measureµis said to beβ-Green-tight(µ∈ K∞(β)in notation), ifµis inK and satisfies
lim
R→∞sup
x∈Rd
Z
|y|>R
Gβ(x,y)µ(d y) =0. (2.4)
We see from the resolvent equation that forβ >0
K∞(β) =K∞(1).
Whend> α, that is,Mα is transient, we writeK∞forK∞(0). Forµ∈ K, define a symmetric bilinear formEµby
Eµ(u,u) =E(u,u)−
Z
Rd
e
u2dµ, u∈ D(E), (2.5)
whereeuis a quasi-continuous version of u([5, Theorem 2.1.3]). In the sequel, we always as-sume that every functionu∈ De(E)is represented by its quasi continuous version. Sinceµ∈ K
[1, Theorem 4.1]that(Eµ,D(E))becomes a lower semi-bounded closed symmetric form. Denote
byHµ the self-adjoint operator generated by(Eµ,D(E)):Eµ(u,v) = (Hµu,v). LetPµ
t be theL2
-semigroup generated byHµ:Ptµ=exp(−tHµ). We see from[1, Theorem 6.3(iv)]thatPtµadmits a symmetric integral kernelpµ(t,x,y)which is jointly continuous function on(0,∞)×Rd×Rd. Forµ∈ K, letAµt be a PCAF which is in the Revuz correspondence toµ(Cf. [5, p.188]). By the Feynman-Kac formula, the semigroupPtµis written as
Ptµf(x) =Ex[exp(Aµt)f(Xt)]. (2.6) Theorem 2.2([11]). Letµ∈ K. Then
Z
Rd
u2(x)µ(d x)≤ kGβµk∞Eβ(u,u), u∈ D(E), (2.7)
whereEβ(u,u) =E(u,u) +βR
Rdu
2d x.
Theorem 2.3. ([14, Theorem 10],[13, Theorem 2.7])Ifµ∈ K∞(1), then the embedding ofD(E)
into L2(µ)is compact. If d> αandµ∈ K∞, then the embedding ofDe(E)into L2(µ)is compact.
3
Construction of ground states
Ford≤α(resp.d> α), letµbe a non-trivial measure inK∞(1)(resp.K∞). Define λ(θ) =inf
¨
Eθ(u,u):
Z
Rd
u2dµ=1
«
, θ≥0. (3.1)
Lemma 3.1. The functionλ(θ)is increasing and concave. Moreover, it satisfieslimθ→∞λ(θ) =∞. Proof. It follows from the definition ofλ(θ)that it is increasing. Forθ1,θ2≥0, 0≤t≤1
λ(tθ1+ (1−t)θ2) =inf
¨
Etθ
1+(1−t)θ2(u,u):
Z
Rd
u2dµ=1
«
≥tinf
¨
Eθ1(u,u):
Z
Rd
u2dµ=1
«
+ (1−t)inf
¨
Eθ2(u,u):
Z
Rd
u2dµ=1
«
=tλ(θ1) + (1−t)λ(θ2).
We see from Theorem 2.2 that foru∈ D(E)withRRdu
2dµ=1,E
θ(u,u)≥1/kGθµk∞. Hence we have
λ(θ)≥ 1 kGθµk∞
. (3.2)
By the definition of the Kato class, the right hand side of (3.2) tends to infinity asθ→ ∞. Lemma 3.2. If d≤α, thenλ(0) =0.
Proof. Note that foru∈ D(E)
λ(0)
Z
Rd
u2dµ≤ E(u,u).
Since (E,D(E)) is recurrent, there exists a sequence {un} ⊂ D(E) such that un ↑ 1 q.e. and
We see from Theorem 2.3 and Lemma 3.2 that ifd≤α, then there existθ0>0 andh∈ D(E)such that
λ(θ0) =inf
¨
Eθ0(h,h):
Z
Rd
h2dµ=1
«
=1.
We can assume thathis a strictly positive continuous function (e.g. Section 4 in[14]). LetMt[h]be the martingale part of the Fukushima decomposition ([5, Theorem 5.2.2]):
h(Xt)−h(X0) =Mt[h]+Nt[h]. (3.3) Define a martingale by
Mt =
Z t
0 1 h(Xs−)
d Msh
and denote byLht the unique solution of the Doléans-Dade equation:
Zt=1+
Zt
0
Zs−d Ms. (3.4)
Then we see from the Doléans-Dade formula thatLh
t is expressed by
Lht = exp
Mt−
1 2〈M
c〉 t
Y
0<s≤t
(1+ ∆Ms)exp(−∆Ms)
= exp
Mt−
1 2〈M
c〉 t
Y
0<s≤t
h(Xs)
h(Xs−)exp
1− h(Xs)
h(Xs−)
.
Here Mc
t is the continuous part of Mt and∆Ms = Ms−Ms−. By Itô’s formula applied to the semi-martingaleh(Xt)with the function logx, we see thatLht has the following expression:
Lht =e−θ0th(Xt) h(X0)exp(A
µ
t). (3.5)
Letd> αand suppose thatθ0=0, that is, λ(0) =inf
¨
E(u,u):
Z
Rd
u2dµ=1
«
=1.
We then see from[14, Theorem 3.4]that there exists a functionh∈ De(E)such thatE(h,h) =1. We can also assume thathis a strictly positive continuous function and satisfies
c
|x|d−α ≤h(x)≤
C
|x|d−α, |x|>1 (3.6)
(see (4.19) in[14]). We define the MFLh t by
Lht = h(Xt)
h(X0)
exp(Aµt). (3.7)
We denote byMh= (Ω,Ph
x,Xt)the transformed process ofMαbyLht,
Proposition 3.3. The transformed process Mh = (Ph
x,Xt)is Harris recurrent, that is, for a
non-negative function f with m({x:f(x)>0})>0,
Z∞
0
f(Xt)d t=∞ Phx-a.s., (3.8)
where m is the Lebesgue measure.
Proof. SetA={x:f(x)>0}. SinceMhis anh2d x-symmetric recurrent Markov process,
Px[σA◦θn<∞, ∀n≥0] =1 for q.e. x∈Rd (3.9) by[5, Theorem 4.6]. Moreover, since the Markov processMhhas the transition density function
e−θ0t· p
µ(t,x,y)
h(x)h(y)
with respect to h2d x, (3.9) holds for all x ∈Rd by[5, Problem 4.6.3]. Using the strong Feller property and the proof of[8, Chapter X, Proposition (3.11)], we see from (3.9) thatMhis Harris
recurrent.
We see from [14, Theorem 4.15] : If θ0> 0, thenh∈ L2(Rd)andMhis positive recurrent. If θ0 = 0 and α < d ≤ 2α, then h 6∈ L2(Rd) Mhis null recurrent. If θ0 = 0 and d ≥ 2α, then h∈L2(Rd)Mhis positive recurrent.
4
Penalization problems
In this section, we prove Theorem 1.1. (1◦)Recurrent case(d≤α)
Theorem 4.1. Assume that d≤α. Then there existθ0>0and h∈ D(E)such thatλ(θ0) =1and
Eθ0(h,h) =1. Moreover, for each x∈R
d
e−θ0tE
x
h
eAµt
i
−→h(x)
Z
Rd
h(x)d x as t−→ ∞.. (4.1)
Proof. The first assertion follows from Theorem 2.3 and Lemma 3.2. Note that e−θ0tE
x
h
eAµt
i
=h(x)Ehx
1 h(Xt)
Then by[13, Corollary 4.7]the right hand side converges toh(x)RRdh(x)d x.
Theorem 4.1 implies (1.7). Indeed, Ex
exp(Aµt)|Fs
Exexp(Aµt) =
e−θ0tE
x
exp(Aµt)|Fs
e−θ0tE
x
exp(Aµt) = e
−θ0sexp(Aµ
s)e
−θ0(t−s)E
Xs
exp(Aµt−s)
e−θ0tE
x
exp(Aµt)
−→ e
−θ0sexp(Aµ
s)h(Xs)
R
Rdh(x)d x
h(x)RRdh(x)d x
We showed in[3, Theorem 2.6 (b)]that the transformed processMhis recurrent. We see from 4.1 below), that is,
sup
Hence for anys≥0 and anyFs-measurable bounded functionZ ExZ eAµt
In the remainder of this section, we consider the case whenλ(0) =1. It is known that a measure µ∈ K∞is Green-bounded,
To consider the penalisation problem forµwithλ(0) =1, we need to impose a condition onµ. Definition 4.2. (I) A measureµ∈ K is said to bespecialif
We denote byK∞S the set of special measures.
(II) A PCAFAt is said to bespecialwith respect to Mh, if for any positive Borel function g with
R
A Kato measure with compact support belongs toKS
∞. The setK∞S is contained inK∞,
K∞S ⊂ K∞. (4.4)
Indeed, since for anyR>0
we have Hence the proof is completed by lettings→ ∞.
The next theorem was proved in[15].
Theorem 4.1. ([15])Suppose d> α. Forµ=µ+−µ−∈ K∞− K∞, let A
Lemma 4.4. Ifµ∈ KS
Proof. We may assume thatg is a bounded positive Borel function with compact support. Note that by Lemma 4.3
Ehx
We note that by Lemma 4.3
Ex Thus for a finite positive measureν,
andRt
h, we see from Chacon-Ornstein
type ergodic theorem in[2, Theorem 3.18]that 1
For a boundedFs-measurable functionZ, define a positive finite measureνby ν(B) =Ex Then by the Markov property,
Ex
By (4.7) and (4.8), the right hand side equals
(〈ν,h〉〈µ,h〉)/R (3.6). HenceMhis an ergodic process with the invariant probability measureh2d x, and thus for a smooth functionkwith compact support,
ψ(t)
[1] Albeverio, S., Blanchard, P., Ma, Z.M.: Feynman-Kac semigroups in terms of signed smooth measures, in "Random Partial Differential Equations" ed. U. Hornung et al., Birkhäuser, (1991). MR1185735
[2] Brancovan, M.: Fonctionnelles additives speciales des processus recurrents au sens de Harris, Z. Wahrsch. Verw. Gebiete.47, 163-194 (1979). MR0523168
[4] Fitzsimmons, P.J., Absolute continuity of symmetric diffusions, Ann. Probab. 25, 230-258 (1997). MR1428508
[5] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin (1994). MR1303354
[6] Neveu, J.: Potentiel Markovien recurrent des chaines de Harris. Ann. Inst. Fourier22, 85-130 (1972). MR0380992
[7] Rogers, L., Williams, D.: Diffusions, Markov Processes, and Martingales, Vol. 2, John Wiley (1987). MR0921238
[8] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Third edition, Springer-Verlag, Berlin (1999). MR1725357
[9] Roynette, B., Vallois, P., Yor, M.: Some penalisations of the Wiener measure. Jpn. J. Math.1, 263-290 (2006). MR2261065
[10] Roynette, B., Vallois, P., Yor, M.: Limiting laws associated with Brownian motion per-turbed by normalized exponential weights. I. Studia Sci. Math. Hungar.43, 171-246 (2006). MR2229621
[11] Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures, Potential Analysis5, 109-138 (1996). MR1378151
[12] Takeda, M.: Exponential decay of lifetimes and a theorem of Kac on total occupation times, Potential Analysis11, 235-247, (1999). MR1717103
[13] Takeda, M.: Large deviations for additive functionals of symmetric stable processes. J. The-oret. Probab.21, 336-355 (2008). MR2391248
[14] Takeda, M., Tsuchida, K.: Differentiability of spectral functions for symmetricα-stable pro-cesses, Trans. Amer. Math. Soc.359, 4031-4054 (2007). MR2302522
[15] Takeda, M., Uemura, T.: Subcriticality and gaugeability for symmetric α-stable processes, Forum Math.16, 505-517 (2004). MR2044025
