Feynman-Kac Penalisations of Symmetric Stable Processes

In K. Yano, Y. Yano and M. Yor (2009), limit theorems for the one-dimensional symmetric α-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 symmetric α-stable processes.


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 the Feynman-Kac penalisations. Our aim is to extend their results on Feynman-Kac penalisations to positive Feynman-Kac functionals of multi-dimensional symmetric α-stable processes. Let M α = (Ω, , t , x , X t ) be the symmetric α-stable process on d with 0 < α ≤ 2, that is, the Markov process generated by −(1/2)(−∆) α/2 , and ( , ( )) the Dirichlet form of M α (see (2.1), (2.2)). Let µ be a positive Radon measure in the class ∞ of Green-tight Kato measures (Definition 2.1). We denote by A µ t the positive continuous additive functional (PCAF in abbreviation) in the Revuz correspondence to µ: for a positive Borel function f and γ-excessive function g, where ]. Our interest is the limit of µ x,t as t → ∞, 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 Z µ t (x) is important, while in our cases the growth order is. We define We see from [5, Theorem 6.2.1] and [12, Lemma 3.1] that the time changed process by A µ t is symmetric with respect to µ and λ(0) equals the bottom of the spectrum of the time changed process. We now classify the set ∞ in terms of λ(0): In this case, there exist a positive constant θ 0 > 0 and a positive continuous function h in the Dirichlet space ( ) such that 1 = λ(θ 0 ) = θ 0 (h, h) (Lemma 3.1, Theorem 2.3). We define the multiplicative functional (MF in abbreviation) L h t by In this case, there exists a positive continuous function h in the extended Dirichlet space e ( ) such that . Here e ( ) is the set of measurable functions u on d such that |u| < ∞ a.e., and there exists an -Cauchy sequence {u n } of functions in ( ) such that lim n→∞ u n = u a.e. We define In this case, the measure µ is gaugeable, that is, The cases (i), (ii), and (iii) are corresponding to the supercriticality, criticality, and subcriticality of the operator, −(1/2)(−∆) α/2 + µ, respectively ( [15]). We will see that L h t is a martingale MF for each case, i.e., We then see from [3, Theorem 2.6] and Proposition 3.8 below that if λ(0) ≤ 1, then M h is an h 2 d x-symmetric Harris recurrent Markov process.
To state the main result of this paper, we need to introduce a subclass This class is relevant to the notion of special PCAF's which was introduced by J. Neveu ([6]); we will show in Lemma 4.4 that if a measure µ belongs to S ∞ , then t 0 (1/h(X s ))dA µ s is a special PCAF of M h . 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.
that is, for any s ≥ 0 and any bounded s -measurable function Z, (ii) If λ(0) = 1 and µ ∈ S ∞ , then (1.7) holds. Throughout this paper, B(R) is an open ball with radius R centered at the origin. We use c, C, ..., et c as positive constants which may be different at different occurrences.

Preliminaries
is the minimal (augmented) admissible filtration and θ t , t ≥ 0, is the shift operators satisfying X s (θ t ) = X s+t identically for s, t ≥ 0. When α = 2, M α is the Brownian motion. Let p(t, x, y) be the transition density function of M α and G β (x, y), β ≥ 0, be its β-Green function, For a positive measure µ, the β-potential of µ is defined by Let P t be the semigroup of M α , Let ( , ( )) be the Dirichlet form generated by M α : for 0 < α < 2 We see from the resolvent equation that for β > 0 When d > α, that is, M α is transient, we write ∞ for ∞ (0). For µ ∈ , define a symmetric bilinear form µ by . By the Feynman-Kac formula, the semigroup P µ t is written as (2.6)
We see from Theorem 2.3 and Lemma 3.2 that if d ≤ α, then there exist θ 0 > 0 and h ∈ ( ) such that We can assume that h is a strictly positive continuous function (e.g. Section 4 in [14]).
Then we see from the Doléans-Dade formula that L h t is expressed by Here M c t is the continuous part of M t and ∆M s = M s − M s− . By Itô's formula applied to the semi-martingale h(X t ) with the function log x, we see that L h t has the following expression: Let d > α and suppose that θ 0 = 0, that is, We then see from [14,Theorem 3.4] that there exists a function h ∈ e ( ) such that (h, h) = 1. We can also assume that h is a strictly positive continuous function and satisfies (see (4.19) in [14]). We define the MF L h t by We denote by

Proposition 3.3. The transformed process
x -a.s., (3.8) where m is the Lebesgue measure.
by [5,Theorem 4.6]. Moreover, since the Markov process M h has the transition density function

Penalization problems
In this section, we prove Theorem 1.1.
In the remainder of this section, we consider the case when λ(0) = 1. It is known that a measure µ ∈ ∞ is Green-bounded, To consider the penalisation problem for µ with λ(0) = 1, we need to impose a condition on µ.
We denote by S ∞ the set of special measures. (II) A PCAF A t is said to be special with respect to M h , if for any positive Borel function g with A Kato measure with compact support belongs to S ∞ . The set S ∞ is contained in ∞ , Indeed, since for any R > 0

Lemma 4.3. Let B t be a PCAF. Then
Proof. We have Then since Y t is a right continuous process, its optional projection is equal to x [Y t | t ] (e.g. [7,Theorem 7.10]). Hence the right hand side equals Since X t e A µ s−t h(X s−t ) = h(X t ), the right hand side equals Hence the proof is completed by letting s → ∞.
The next theorem was proved in [15].
Then the following conditions are equivalent: (ii) There exists the Green function G µ (x, y) < ∞ (x = y) of the operator − 1 2 (−∆) α/2 + µ such that We see from (4.19) in [14] that if one of the statements in Theorem 4.1 holds, then G µ (x, y) satisfies Proof. We may assume that g is a bounded positive Borel function with compact support. Note that by Lemma 4.3 If the measure µ satisfies λ(0) = 1, then µ − g · d x ∈ ∞ − ∞ satisfies Theorem 4.1 (iii), and G µ−g·d x (x, y) is equivalent with G(x, y) by (4.5). Therefore the equation (3.6) implies that (4.3) is equivalent to that We note that by Lemma 4.3 Thus for a finite positive measure ν, where ν h = h · ν/〈ν, h〉. For a positive smooth function k with compact support, put Then lim t→∞ ψ(t) = ∞ by the Harris recurrence of M h . Moreover, Indeed, Remark 4.5. We suppose that d > α and λ(0) = 1. If d > 2α, then h ∈ L 2 ( d ) on account of (3.6). Hence M h is an ergodic process with the invariant probability measure h 2 d x, and thus for a smooth function k with compact support, Hence we see that for µ ∈ S ∞ lim t→∞ 1 t x e A µ t = h(x)〈µ, h〉. (4.10)