Revuz Measures,Energy Functionals and Capacities Under Girsanov Transform Induced by α-Excessive Function?

Ruili SONG

1 Introduction and Preliminaries

In the theory of Markov processes,it is always interesting and important to explore various kinds of relationships between X and the transformed process of X under some kind of transform.Getoor and Ste ff ens[1]studied the relationships among the energy functional,the balayage operators and the capacities for a Markov process and the same objects for its qsubprocesses and u-transform;Ying[2]studied the homogeneous multiplicative functionals of Lévy processes and gave their characterizations;Ying[3]studied the additive and multiplicative functionals of a right Markov process systematically in the setting of weak duality and proved that any subprocess of a nearly symmetric Markov process was also nearly symmetric and gave a generalized Feynman-Kac formula;Ying[4]proved that the killing transform of Markov processes was equivalent to strong subordination of the respective Dirichlet forms and gave a characterization of so-called bivariate smooth measures;Ying[5]studied the htransform of symmetric Markov processes and corresponding Dirichlet spaces and discussed the drift transformation of Fukushima and Takeda’s type;Ying[6]studied the relationships of Revuz measures,several formulas on the energy functional and capacity between any process and its subprocesses.Chen et al.[7]investigated the most general Girsanov transformation leading to another symmetric Markov process.Fukushima et al.[8]extended the classical Douglas integral by using the approach of time change of Markov processes.Song and Ying[9]established a representation formula for transition density function of a certain type of rightprocesses under Girsanov transform and gave the in finitesimal generator of the transformed process.He and Ying[10]studied how energy functionals and Revuz measures change under time change of Markov processes and provided an intuitive and direct approach of the Lévy system and jumping measure of time changed process;Song[11]studied how Revuz measures,energy functional,capacity and Lévy system change under some Girsanov transform of Hunt processes.

Ying[5]considered a drift transformation which was in the setting of symmetric right process.In this paper,we consider a Girsanov transformation of a right Markov process which is formally like the drift transformation in[5].However,the right Markov process in our paper is not symmetric and the transformation is not the same.We will extend the excessive function in[11]to α-excessive function and generalize the results of[11].In the following,we will give the specific explanations.

We start from a right Markov process X=(Ω,F,Ft,θt,Px)t∈(0,+∞)on the state space(E,E)with transition semigroup(Pt){t>0}and resolvent(Uq){q≥0}.E is,at least,a separable Radon space and E is the Borel σ- field of E.A cemetery point Δ is adjoined to E as an isolated point of E and EΔ:=E∪{Δ},EΔ:= σ(E ∪{Δ})(the symbol‘:=’is always read as‘is defined to be’).Let ζ:=inf{t:Xt= Δ}be the lifetime of X.The filtration(F,Ft)is the augmented natural filtration of X.Denote by Excq(X)and Sq(X)(q≥0)the cones of q-excessive measures and q-excessive functions of X,respectively.As usual we write S:=S0,Exc:=Exc0.

For α >0,let h ∈ Sα(X)and 0

whereis a local martingale,is a continuous increasing additive functional.Refer to[12]for the theory of additive functionals.

Let

which is a martingale additive functional of X.Note that for t<ζ,we have

Letbe the decomposition as continuous and purely discontinuous parts.Denote by Ltthe Doléan-Dade exponential martingale of.Then it admits a representation as the following:

which is a martingale of X.Consequently,the formula

uniquely determines a family of probability measure on(Ω,F∞).It is known that X is a right Markov process on E under these new measures(cf.[13,Section 62]).We will use(Y,F,Ft,Qx,x ∈ E)to denote the transformed process.Here Yt(ω)=Xt(ω)but we use Ytfor emphasis when working with Qx.By It?o’s formula,we have

Let Qtand Vtbe the semigroup and resolvent of Y,respectively,that is,for any f∈E,we have

The process Y is called the Girsanov transform of X by(Lt){t>0}or induced by α-excessive function h.

In this article,we will get some potential objects under the transform,such as Revuz measures,energy functionals,capacities and Lévy systems.

2 Revuz Measures,Energy Functionals,Capacities

In this section,we will get the relationships of some potential objects between the process X and the transformed process Y which is defined in Section 1.In the following,we denote by Sq(Y)and Excq(Y)(q≥0)the cones of q-excessive measures and q-excessive functions of Y,respectively.Particularly,denote by S(Y)and Exc(Y)the cones of excessive measures and excessive functions of Y,respectively.

Lemma 2.1For any q≥0,if u∈Sq(Y),then hu∈Sq+α(X).Particularly,if u∈S(Y),then hu∈Sα(X).

ProofBy the definition of q-excessive function,we have

By(1.1),we have

Thus it follows that

and

Then also by the definition of q-excessive function,we have hu∈Sq(X).

Lemma 2.2For any q ≥ 0,if μ ∈ Excq(Y),then∈ Excq+α(X).Particularly,if μ∈Exc(Y),then∈Excα(X).

ProofWe assume that f∈pE(positive measurable function with respect to E)in the following.By the definition of q-excessive measure,we have μ∈Excq(Y)if and only if μe?qtQt≤ μ.

Thus it follows that

Thus we get the desired result.

The above two lemmas give the relationships of excessive functions and excessive measures of X and Y.

Revuz measures were first introduced for ordinary additive functionals in[14].Let m∈Exc(X)and A∈RAF(raw additive functional).Then the Revuz measure of A relative to m is defined by

where ↑ (↓)means increasing(decreasing).Let Xαdenote the α-subprocess of X,that is,the semigroup of Xαis defined byWe have

Let ρX, ρXαand ρYdenote the Revuz measures of X,Xαand Y,respectively.

Theorem 2.1Let μ∈Exc(Y)and A be an increasing continuous additive functional which isfinite on[0,ζ[.Then

Particularly,if∈Exc(X),then

ProofIf At=(Xs)ds,where a is a bounded positive Borel function,then for any μ ∈ Exc(X),we have=a ·μ.Since μ ∈ Exc(Y),by Lemma 2.2,we have∈ Exc(Xα).

Thus

Then

For a general A which is an increasing continuous additive functional,define a strictly increasing continuous additive functional H by Ht:=At+t∧ ζ,then

By time-change,let

Since AtHt,thendt.By Motoo’s theorem(cf.[13,(66.2)]),there is a bounded positive Borel functionsuch that=(Xs)ds for all t≥ 0 a.s..By the discussion in the first part of the proof,we have

And by time-change(cf.[10]),we have

Thus it follows that

If∈Exc(X),by[15,(8.10)],we have

then

This theorem gives the relationship of Revuz measures between X and Y.In the proof,we use the method in[16].We see that time-change plays an important role in the above proof.

The energy functional L(of X)is defined on Exc(X)×S(X)by

Refer to[15]for the basic properties of energy functional L.

Let LX,Lαand LYdenote the energy functional of X,Xαand Y,respectively.

Before we get our results,we will explain our ideas as follows.Since h ∈ Excα(X)and Excα(X)=Exc(Xα),then h∈ Exc(Xα).Since 00,

From(1.1),we see that the process Xα,hcan be seen as the killing process of Y by the decreasing multiplication functional M,where

We will use the process Xα,has a bridge to get the relationships of energy functional and capacity between the process X and the process Y.We will get the relationship between Lα(LX)and LYin the following Theorem 2.2.

Theorem 2.2Let μ ∈ Exc(Y),u ∈ S(Y),then

Particularly,if∈Exc(X),hu∈S(X),then

ProofBy Lemmas 2.1–2.2,we have

Due to[1,(4.8)],we have

The process Xα,hcan be seen as the M-subprocess of Y,where

Due to Theorem 3.3 in[6],we have

By(2.1)–(2.2),we have

By Theorem 2.1,we have

The Stieltjes logarithm of M is

Due to[6,Corollary 2.8],we have

Using(2.3)–(2.5),it follows that

If∈Exc(X),hu∈S(X),since

and

then we have

The Hunt’s balayage operationis defined on Excq(q>0).That is,for any terminal time T,

For μ∈Exc(X),

LetIf B ∈Eeand μ ∈Excq(q≥0),we shall writein place of

For B∈Eeand q≥0,the q-capacity of B with m∈Exc(X)is defined by

That is

An exact terminal time T is called strict if T ? θT=0 a.s.In the case that T=TB,the hitting time of B∈Ee,T is strict if and only if XT∈Br,where Bris the set of regular points of B.The set B is called strict if TBis.Let ΓXand ΓYdenote the capacity of X and Y,respectively.

Theorem 2.3Let B∈E and be strict,μ∈Exc(Y).Then

Particularly,if,∈Exc(X)and∈S(X),then

ProofSince μ∈Exc(Y),h∈S(Xα),then RBμ∈Exc(Y),∈S(Xα).By Lemma 2.2,we have,∈ Exc(Xα).

By the definition of capacity,we have

In the above reasoning,we use(2.1)to get the second equality and use[1,(5.4)(i)]to get the third equality.

Since the process Xα,his the M-subprocess of Y,and by using[6,(4.11)],we have

By(2.4)–(2.5),we get

and

By(2.8)–(2.11),we have the result(2.6).

Particularly,if,∈Exc(X)and∈S(X),we have

By(2.6)and(2.12)–(2.13),we have the result(2.7).

3 Lvy System

A Lévy system for X is a pair(N,H),where N is a kernel on(E,Eu)(where Euis the σ-algebra of universally measurable subsets of E)with N(x,{x})=0 for any x∈ E and H is a continuous additive functional of X having bounded 1-potential,such that for any F ∈pEu×Euvanishing on the diagonal and any predictable process Z,we have

where

Refer to[13, §73]for the existence of Lévy systems.Now we state the relationship of Lévy system of X and Y.

Theorem 3.1If(N,H)is a Lévy system of X,then(N?,H)is a Lévy system of Y,where

ProofBy the definition of Lévy system,for any F ∈ pEu× Eu,we have

The fifth equation is due to the continuity of H.Then also by the definition of Lévy system,we have that(N?,H)is a Lévy system of the process Y.

AcknowledgementThe author is indebted to two anonymous referees for their constructive comments.

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