For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup
TtAf(x):=Ex[eAtf(Xt)],x∈Rn,t>0,0,\]]]>
where A is a continuous additive functional of X associated with some signed measure. Under the assumption that X admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to TtA possesses the density ptA(x,y) with respect to the Lebesgue measure and construct upper and lower bounds for ptA(x,y). Some examples are provided.
Transition probability densitycontinuous additive functionalKato classFeynman–Kac semigroup60J3560J4560J5560J5760J75Introduction
Let (Xt)t≥0 be a Markov process with the state space Rn. For a Borel measurable function V:Rn→R, we can define the functional At of X by
At:=∫0tV(Xs)ds,t>0.0.\]]]>
Suppose that limt→0supxEx|At|=0. Then, by the Khasminski lemma there exist constants C,b>00$]]>, such that
supxExe|At|≤Cebt;
see, for example, [11, Lemma 3] or [12, Lemma 3.3.7]. Estimate (1.2) allows us to define the operator
TtAf(x):=Ex[eAtf(Xt)],x∈Rn,t>0,0,\]]]>
where the function f is bounded and Borel measurable. The family of operators (TtA)t≥0 forms a semigroup, called the Feynman–Kac semigroup.
Feynman–Kac semigroup is well studied in the case of a Brownian motion (see [23, 24, 12, 3]); in particular, in [3] more general functionals are treated. The case of a general Markov process is much more complicated; see, however, [12, Chap. 3.3.2] and [24]. The essential condition on the process, stated in the papers cited, is that the Markov process X is symmetric and possesses a transition probability density pt(x,y).
In this paper, we construct and investigate the Feynman–Kac semigroups for a wider class of Markov processes. First, we construct the Feynman–Kac semigroup for a (non-symmetric) Markov process, admitting a transition density. We also treat a more general class of functionals At, that is, in our setting the functional At is not necessarily of the form (1.1), but is constructed by means of some measure ϖ, which is in the Kato class with respect to the transition probability density of X (cf. (2.3)). The approach used in [8] allows us to show the existence of the kernel ptA(x,y) of the semigroup (TtA)t≥0 and to give its representation. The method from [8] relies on the construction of the Markov bridge density, which in turn employs the regularity properties of the transition probability density of the initial process X rather than its symmetry.
In such a way, this prepares the base for the main result of the paper, which is devoted to the investigation of the Feynman–Kac semigroup for the particular class of processes constructed in [18]. In [20, 19], we develop the approach that allows us to relate to a pseudo-differential operator of certain type a Markov process possessing a transition probability density pt(x,y) and construct for this density two-sided estimates. In particular, such estimates provide an easily checkable condition when a measure ϖ belongs to the Kato class with respect to pt(x,y). This allows us to describe the respective continuous additive functional At and to show (1.2). Starting with the class of processes investigated in [18], we construct (see Theorem 3) the upper and lower estimates for the Feynman–Kac density ptA(x,y). In particular, we show that the structure of such estimates is “inherited” from the structure of the estimates on pt(x,y). In some cases when the upper bound on pt(x,y) can be written in a rather compact way, we can describe explicitly the Kato class of measures. For example, this is the case if pt(x,y) is comparable for small t with the density of a symmetric stable process; see also [4, Cor. 12] for refined results. In Proposition 4 we show that if the initial transition probability density possesses an upper bound of a rather simple (polynomial) form, this form is inherited by the Feynman–Kac density ptA(x,y).
Up to the author’s knowledge, in general, the results on two-sided estimates of ptA(x,y) are yet unavailable. For X being an α-stable-like process, the estimates of the kernel ptA(x,y) are obtained in [22]; see also [10] and the references therein for more recent results in this direction, including two-sided estimates on ptA(x,y) in the case when the functional A is not necessarily continuous. The approach used in [22, 10] to construct the Feynman–Kac semigroup is based on the Dirichlet form technique. See also [5] for yet another approach to investigate Feynman–Kac semigroups.
The paper is organized as follows. In Section 2, we give the basic notions and introduce the main results. Proofs are given in Sections 3 and 4. In Section 5, we illustrate our results with examples.
Notation
For functions f, g, by f≍g we mean that there exist some constants c1,c2>00$]]> such that c1f(x)≤g(x)≤c2f(x) for all x∈Rn. By x·y and ‖x‖ we denote, respectively, the scalar product and the norm in Rn, and Sn denotes the unit sphere in Rn. By Bb(Rn) we denote the family of bounded Borel functions on Rn. By C∞k(Rn) we denote the space of k-times differentiable functions, with derivatives vanishing at infinity. By ci, c and C we denote arbitrary positive constants. The symbols ∗, □, and ♢ denote, respectively, the convolutions
(f∗g)(x,y):=∫Rnf(x−z)g(z−y)dz,(f□g)(x,y):=∫Rnf(x−z)g(z−y)ϖ(dz),
and
(f♢g)t(x,y):=∫0t∫Rnft−s(x,z)gs(z,y)ϖ(dz)ds,
where ϖ is a (signed) measure.
Settings and the main results
Let X be a Markov process with the state space Rn. We call X a Feller process if the corresponding operator
Ttf(x):=Exf(Xt)
maps the space C∞(Rn) of continuous functions vanishing at infinity into itself. Assume that X possesses a transition probability density pt(x,y) which satisfies the following assumption.
For fixed x∈Rn, the mapping y↦ps(x,y) is continuous for all s∈(0,t], and the mapping s↦ps(x,y) is continuous for all x,y∈Rn.
Recall some notions on the Kato class of measures and related continuous additive functionals.
We say that a functional φt of a Markov process Xt is a W-functional (see [13, §6.11]) if φt is a positive continuous additive functional, almost surely homogeneous, and such that supxExφt<∞. By additivity we mean that φt satisfies the following equality:
φt+s=φt+φs∘θt,
where θt is the shift operator, that is, Xs∘θt=Xt+s. The function vt(x):=Exφt is called the characteristic of φt and determines φt in the unique way; see [13, Thm. 6.3].
A positive Borel measure ϖ is said to belong to the Kato class SK with respect to pt(x,y) if
limt→0supx∈Rn∫0t∫Rnps(x,y)ϖ(dy)ds=0.
By [13, Thm. 6.6], the condition ϖ∈SK implies that the function
χt(x):=∫0t∫Rnps(x,y)ϖ(dy)ds
for which the mapping x↦χt(x) is measurable for all t≥0, is the characteristic of some W-functional φt.
Let ϖ=ϖ+−ϖ− be a signed measure such that ϖ±∈SK with respect to pt(x,y). Then
χt±:=∫0t∫Rnps(x,y)ϖ±(dy)ds
are the characteristics of some W-functionals At±, respectively, that is, there exist At± such that χt±(x)=ExAt±. Since for such functionals we have
limt→0supxExAt±=0,
then estimate (1.2) holds true, and thus the Feynman–Kac semigroup (TtA)t≥0 for At:=At+−At− is correctly defined.
To show that the semigroup (TtA)t≥0 can be written as
TtAf(x)=∫Rnf(y)ptA(x,y)dy,f∈Bb(Rn),
and to find the representation of the density ptA(x,y) in terms of the probability density of the initial process, recall some notions on Markov bridge measures.
Denote by (Ft)t≥0 the admissible filtration related to X. A Markov bridgeXtx,y of Xt is a Markov processes conditioned by X0=x and Xt=y. In the proof of [8, Thm. 1], it is shown that under P1 there exists the corresponding Markov bridge measure Px,yt on Ft− for (t,x,y) such that pt(x,y)>00$]]>. We denote by Ex,yt the expectation with respect to Px,yt.
The next proposition is essentially contained in [8, Thm. 1], but we reformulate the result in the way convenient for our purposes.
Let X be a Feller process, admitting the transition probability densitypt(x,y), for which assumption P1 holds. Letϖ=ϖ+−ϖ−be a signed Borel measure,ϖ±∈SK, andAt=At+−At−, whereA±are continuous additive functionals with characteristics (2.5), respectively. ThenTtAf(x)=∫{y:pt(x,y)>0}f(y)ptA(x,y)dyfor any f∈Bb(Rn),0\}}f(y){p_{t}^{A}}(x,y)dy\hspace{1em}\textit{for any }f\in B_{b}\big({\mathbb{R}}^{n}\big),\]]]>whereptA(x,y)=pt(x,y)Ex,yteAt,x,y∈Rn, t>0.0\textit{.}\]]]>
When X is a Brownian motion, the statement of Proposition 1 is known, see [23] and also [3]. The construction from [3, 23] can be extended to the case of a symmetric Markov process, see [24]. On the contrary, the construction presented in [8] relies on P1 and does not require the symmetry of the initial process.
Proposition 1 implicitly gives the representation of the function ptA(x,y). However, when one wants to get quantitative information about ptA(x,y), like the upper bound on ptA(x,y), estimation of the expectation Ex,yteAt in (2.6) appears to be non-trivial. Instead, for some class of Feller processes, we can use another approach, which enables us to get explicitly an upper estimate of ptA(x,y). Namely, in [18] we formulated the assumptions under which one can construct a Feller process possessing the transition probability density pt(x,y) satisfying assumption P1 and admitting upper and lower bounds of certain form. In order to make the presentation self-contained, we quote this result below.
Let
Lf(x):=a(x)·∇f(x)+∫Rn(f(x+u)−f(x)−u·∇f(x)1{‖u‖≤1})m(x,u)μ(du),
where f∈C∞2(Rn), and μ is a Lévy measure, that is, a Borel measure such that
∫Rn(‖u‖2∧1)μ(du)<∞.
Assume that μ satisfies the following assumption.
There exists β>11$]]> such that
supℓ∈SnqU(rℓ)≤βinfℓ∈SnqL(rℓ)for allr>0large enough,0\hspace{2.5pt}\text{large enough,}\hspace{2.5pt}\]]]>
where
qU(ξ):=∫Rn[(ξ·u)2∧1]μ(du),qL(ξ):=∫|u·ξ|≤1(ξ·u)2μ(du).
Assume that the functions a(x) and m(x,u) in (2.7) satisfy the assumptions A2–A4 given below.
The functions m(x,u) and a(x) are measurable, and satisfy with some constants b1,b2,b3>00$]]>, the inequalities
b1≤m(x,u)≤b2,|a(x)|≤b3,x,u∈Rn.
There exist constants γ∈(0,1] and b4>00$]]> such that
|m(x,u)−m(y,u)|+‖a(x)−a(y)‖≤b4(‖x−y‖γ∧1),u,x,y∈Rn.
In the case β>22$]]>, we assume that a(x)=0 and the kernel m(x,u)μ(du) is symmetric with respect to u for all x∈Rn.
Denote by flow and fup the functions of the form
flow(x):=a1(1−a2‖x‖)+,fup(x):=a3e−a4‖x‖,x∈Rn,
where ai>00$]]>, 1≤i≤4, are some constants.
Finally, define q∗(r):=supℓ∈SnqU(rℓ), r>00$]]>. It was shown in [17] (see also [20]) that condition A1 implies that
q∗(r)≥r2/β,r≥1.
Note also that the continuity of qU in ξ implies the continuity of q∗ in r. Therefore, we can define its generalized inverse
ρt:=inf{r:q∗(r)=1/t},t∈(0,1].
([18]).
Under assumptions A1–A4, the operator(L,C∞2(Rn)extends to the generator of a Feller process, admitting a transition probability densitypt(x,y). This density is continuous in(t,x,y)∈(0,∞)×Rn×Rn, and there exist constantsai>00$]]>,1≤i≤4, and a family of sub-probability measures{Qt,t≥0}such thatρtnflow((x−y)ρt)≤pt(x,y)≤ρtn(fup(ρt·)∗Qt)(x−y),t∈(0,1],x,y∈Rn,whereflowandfupare functions of the form (2.10) with constantsai, andρtis defined in (2.11).
The constructed process is a Lévy type process. In the “constant coefficient case,” that is, where a(x)≡const and m(x,u)=const, (2.7) is just the representation of the generator of a Lévy process; in other words, a Lévy type process is the process with “locally independent increments.” It is known (cf. the Courrège–Waldenfels theorem, see [16, Thm. 4.5.21]) that if the class Cc∞(Rn) of infinitely differentiable compactly supported functions belongs to the domain D(A) of the generator A of a Feller process, then on this set Cc∞(Rn) the operator A coincides with L+“Gaussian component.” Thus, the class of processes satisfying the conditions of Theorem 2 is rather wide.
Let us show that, under the conditions of Theorem 2, we have
pt(x,y)>0for allt>0,x,y∈Rn.0\hspace{1em}\text{for all}\hspace{2.5pt}t>0,\hspace{2.5pt}x,y\in {\mathbb{R}}^{n}.\]]]>
We find the minimal N such that the distance from x to y can be covered by N balls of the radius smaller than (2a2ρt/N)−1 (where a2>00$]]> is the constant appearing in flow in (2.12)), that is, the minimal N for which
‖x−y‖N≤1a2ρt/N.
Observe that q∗(r)≤c1r2, r≥1, implying c2t−1/2≤ρt for all t small enough. Hence, (2.13) holds with N≥(a2c2‖x−y‖)2t. Therefore, putting y0=x and yN=y, we get
pt(x,y)=∫Rn⋯∫Rn(∏i=1Npt/N(yi−1,yi))dy1…dyN≥∫B(y0,(2a2ρt/N)−1)⋯∫B(yN−1,(2a2ρt/N)−1)∏i=1Npt/N(yi−1,yi)dyi≥c0ρt/NNn,
where in the last line we used that
pt/N(yi−1,yi)≥2−1a1ρt/Nnfor allyi∈B(yi−1,(2a2ρt/N)−1).
Thus, the transition probability density pt(x,y) is strictly positive.
Finally, for a signed Borel measure ϖ, define
h(r):=supx|ϖ|{y:‖x−y‖≤r},
where |ϖ|:=ϖ++ϖ− is the total variation of ϖ. Denote by hˆ the Laplace transform of h.
The following theorem is the main result of the paper. Let t0∈(0,1] be small enough.
Let X be the Feller process constructed in Theorem2. Take a signed Borel measure ϖ such that its volume function (2.14) satisfies∫0tρsn+1hˆ(ρs)ds≤Ctζ,t∈[0,1],with some constantsC,ζ>00$]]>, whereρtis given by (2.11). Then
There exists a continuous functionalAtsuch thatExAt=∫0t∫Rnps(x,y)ϖ(dy)ds;
The semigroup(TtA)t≥0is well defined, and its kernel possesses a densityptA(x,y)with respect to the Lebesgue measure onRn;
There exist constantsai>00$]]>,1≤i≤4, and a family of sub-probability measures{Rt,t≥0}such that fort∈(0,t0]andx,y∈Rn,ρtnflow((x−y)ρt)≤ptA(x,y)≤ρtn(fup(ρt·)∗Rt)(y−x);
hereflowandfupare the function of the form (2.10) with some constantsai,1≤i≤4.
In general, ai in estimate (2.16) are some constants, that may not coincide with those in estimate (2.12). In order to simplify the notation, we assume that in Theorem 2, a1=a3=1, a2=a, and a4=b.
Assumption (2.15) can be relaxed, provided that more information about the initial transition probability density is available. Put
gt(x):=1tnα(1+‖x‖/t1/α)d+α,t>0,x∈Rn.0,\hspace{2.5pt}x\in {\mathbb{R}}^{n}.\]]]>
Note that for d=n, this function is equivalent to the transition probability density of a symmetric α-stable process in Rn (that is, the process whose characteristic function is e−t‖ξ‖α). Denote by Kn,α the class of Borel signed measures such that
limt→0supx∫0t|ϖ|{y:‖x−y‖≤s}sn+1−αds=0.
The following lemma shows that for d>n−αn-\alpha $]]> the Kato class of measures with respect to gt(x−y) coincides with Kn,α. The proof uses the idea from [4], and will be given in Appendix A.
A finite Borel signed measure ϖ belongs toSKwith respect togt(x−y), given by (2.17) withd>n−αn-\alpha $]]>, if and only if|ϖ|∈Kn,α.
In particular, it follows from Lemma1thatϖ∈SKwith respect to the transition probabiility density of a symmetric α-stable process if an only ifϖ∈Kn,α.
In the proposition below, we state the “compact” upper bound for ptA(x,y).
Let X be a Feller process satisfying the conditions of Proposition1, and in addition assume that the transition densitypt(x,y)of X is such that for allt∈(0,1],x,y∈Rn, the inequalitypt(x,y)≤cgt(x−y),t∈(0,1],x,y∈Rn,where the functiongt(x)is defined in (2.17) withd>n−αn-\alpha $]]>. Suppose thatϖ∈Kn,α. ThenptA(x,y)≤Cgt(x−y),t∈(0,1],x,y∈Rn.
a) For X being a symmetric α-stable-like process, such a result is known, see [22]. In particular, the upper bound (2.20) holds with n=d. In our case, X is from a wider class; in particular, we do not assume the symmetry of the initial process, and the method of constructing the Feynman–Kac semigroup is completely different.
b) In view of Lemma 1, under the assumptions of this proposition, we can take ϖ∈Kn,α rather than ϖ∈SK with respect to gt, which is more convenient for usage.
In Section 5, we provide examples that illustrate Theorem 3 and Proposition 4.
Discussion and overview
On continuous additive functionals. Loosely speaking, there are two approaches for constructing continuous additive functionals. One approach, which we described previously, relies on the Dynkin theory of W-functionals. Another approach, based on the Dirichlet form technique, establishes the one-to-one correspondence between the class of positive continuous additive functionals and the class of smooth measures, see [14, Lemmas 5.1.7, 5.1.8] or [15, Thm. 5.1.4] in the case when the process under consideration is symmetric; see also [21, Thm. 2.4] for the non-symmetric case. In this paper, we use Dynkin’s approach as more appropriate in our situation, in particular, we do not assume that the initial Markov process X is symmetric. Our standard reference in this paper is [13].
On the generator of(TtA)t≥0. Suppose that the Markov process X and the positive functional At are as in Proposition 1. In this case, the semigroup (TtA)t≥0 is contractive, and thus there exists a sub-Markov process with transition sub-probability density ptA(x,y). Formally, we can describe the generator of (TtA)t≥0 as
LA=L−ϖ,
where L is the generator of the semigroup associated with X, and ϖ is the measure appearing in the characteristic of At (cf. (2.4)), see [13, Thms. 9.5, 9.6] for the (equivalent) formulation. Nevertheless, in this framework the problem of defining the domain D(LA) of LA still remains open. In the general case, that is, when A can attain negative values, in order to define the generator of (non-contractive) semigroup (TtA)t≥0, we can use the quadratic form approach, see [1, 2], and also [9].
Proof of Theorem 3Proof of statements a) and b)
a) By the upper bound in (2.12) on pt(x,y) (see also Remark 2), (2.15) implies that ϖ∈SK:
supx∈Rn∫0t∫Rnps(x,y)|ϖ|(dy)ds≤supx∈Rn∫0t∫Rn∫Rnρsnfup((y−x−z)ρs)Qs(dz)|ϖ|(dy)ds≤bsupx∈Rn∫0t∫Rn∫0∞ρsn|ϖ|{y:‖y−x−z‖≤v/ρs}e−bvdvQs(dz)ds≤b∫0tρsn+1hˆ(bρs)ds→0,t→0.
Hence, applying [13, Thm. 6.6], we derive the existence of a continuous functional At with claimed characteristic.
Statement b) is already contained in Proposition 1.
Outline of the proof of c)
For the proof of Theorem 3(c), we use the Duhamel principle. First, we show that the function ptA(x,y) satisfies the integral equation
ptA(x,y)=pt(x,y)+∫0t∫Rnpt−s(x,z)psA(z,y)ϖ(dz)ds,
provided that the integral on the right-hand side converges. We show that if the series
πt(x,y):=∑k=1∞pt♢k(x,y)
converges, then it satisfies Eq. (3.1). We derive an upper estimate for the convolutions pt♢k(x,y), which guarantees the absolute convergence of the series and allows to find the upper estimate for πt(x,y).
Second, we show that on (0,t0]×Rn×Rn the solution (3.2) to (3.1) is unique in the class of non-negative functions {f(t,x,y)≥0,t∈(0,t0],x,y∈Rn} such that
∫Rnf(t,x,y)dy≤Cfor allt∈(0,t0],x∈Rn.
We use the standard method, based on the Gronwall–Bellman inequality.
Finally, observe that the kernel ptA(x,y) of TtA belongs to the class of functions satisfying (3.3). Indeed, since for At we have (1.2), it follows that
|TtAf(x)|≤c1Exe|At|≤c2,f∈Bb(Rn),x∈Rn,t∈(0,t0].
Thus, ptA(x,y)≡πt(x,y) on (0,t0]×Rn×Rn.
Before we prove that (3.2) is the solution to Eq. (3.1) on (0,t0]×Rn×Rn, let us discuss a simple case when ϖ is the Lebesgue measure on Rn. In this case h(r)=cnrn, and thus assumption (2.15) is satisfied:
∫0tρsn+1hˆ(ρs)ds=cnt.
Therefore, the procedure of estimation of convolutions reduces to those treated in [18, Lemmas 3.1, 3.2].
Rewrite the upper bound in (2.12) as
pt(x,y)≤C1t−1/2(gt(1)∗Qt)(y−x),
where C1>00$]]> is some constant,
gt(1)(x):=t1/2gt(x),
and (cf. Remark 2)
gt(x):=ρtnfup(ρtx)=ρtne−bρt|x|.
This modification is technical, but proves to be useful for estimating the convolutions pt♢k(x,y). Let us estimate pt♢k(x,y). Take now a sequence (θk)k≥1 such that 0<θk+1<θk, θ1=1, and put
gt(k)(x):=tk/2gt(θkx),k≥1.
Since ρt is monotone decreasing, for 0<s<t2, we have ρt−s≤ρt/2. Note that ρt≍ρt/2; this follows from condition A1 and the definition of ρt; see [20] for the detailed proof. Then, for 0<s<t/2,
(gt−s(k−1)∗gs(1))(x)≤tk/2∫Rngt−s(θk−1x−θk−1y)gs(θk−1y)dy=tk/2θk−1−n∫Rngt−s(θk−1x−y)gs(y)dy≤tk/2θk−1−n∫Rnρt−snρsne−bρtθkθk−1(|θk−1x−z|+|z|)−bρs(1−θkθk−1)|z|dz≤c1tk/2θk−1−nρtne−bρtθk|x|∫Rnρte−bρs(1−θkθk−1)|z|dz=Dkgt(k)(x),
where Dk=c(θk−1−θk)−n, c=c1∫Rne−b|z|dz, and in the second line from below, we used the triangle inequality and monotonicity of ρt. In the case t/2≤s≤t, calculation is similar.
By induction we can get
|pt♢k(x,y)|≤Cktk2−1(gt(k)∗Qt(k))(y−x),k≥2,
where
Ck:=ck−1C1kΓk(1/2)Γ(k/2)∏j=2k1(θj−1−θj)n,
and for k≥2Qt(k)(dw):=1B(k−12,12)∫01∫R(1−r)(k−1)/2−1/2r−1/2Qt(1−r)(k−1)(dw−u)Qtr(1)(du)dr.
Since {Qt(k),t>0,k≥1}0,\hspace{0.1667em}k\ge 1\}$]]> is the sequence of sub-probability measures and gt(k)(x)≤ρtntk/2, we obtain
|pt♢k(x,y)|≤Cktk−1ρtn.
Thus, to show the absolute convergence of the series ∑k=1∞pt♢k(x,y), we may check that ∑k=1∞Ck<∞. However, the behaviour of Ck as k→∞ is rather complicated. To see this, take, for example, θk=12+12k. Then
Ck=ck−1C1kΓk(1/2)Γ(k/2)(2kk!(k−1)!)n,
and thus Ck explodes as k→∞. Therefore, this procedure of estimation of convolutions is too rough, and needs to be modified. For this, we change the estimation procedure after some finite number of steps; this allows us to control the decay of coefficients and, in such a way, to prove that ∑k=1∞pt♢k(x,y)<∞.
In the next subsection, we handle the general case, in particular,
We give the generic calculation, which allows us to estimate the convolution (gt−s□gs)(x);
We estimate the convolutions pt♢k(x,y), k≥2;
We change the estimation procedure after k0 steps, where k0 is properly chosen, and estimate pt♢(k0+ℓ)(x,y), ℓ≥1.
The change of the estimation procedure could be unnecessary if we would know that pt(x,y) possesses a more regular upper bound than (2.12). In this case, we obtain a sufficient control on the coefficients Ck, k≥1. This is exactly the case under the conditions of Proposition 4.
Representation lemma, generic calculation, and estimation of convolutions
The functionptA(x,y)given by (2.6) satisfies Eq. (3.1).
In the case when X is a symmetric stable-like process and ϖ∈SK with respect to the transition probability density of X, the sketch of the proof is given in [22]. In the general case, the proof is the same; in order to make the presentation self-contained, we present it below. Using the equality
eAt=∫0teAt−AsdAs+1,
the strong Markov property of X, and the additivity of At (cf. (2.2)), we write
TtAf(x)=Ex[f(Xt)eAt]=Exf(Xt)+Ex[∫0t[f(Xt)eAt−As]dAs]=Exf(Xt)+Ex[∫0tEXs[f(Xt−s)eAt−s]dAs]=Exf(Xt)+Ex∫0tTt−sAf(Xs)dAs.
Observe that for f∈Bb(Rn), we have
Ex∫0tf(Xs)dAs=∫0t∫Rnf(y)ps(x,y)ϖ(dy)ds.
Indeed, since χt=χt+−χt− with χt± given by (2.5) is the characteristic of At, Eq. (3.11) holds for a finite linear combination of indicators. Approximating f∈Bb(Rn) by such linear combinations and passing to the limit, we get (3.11). □
For θ∈[0,1], put
gt,θ(x):=gt(θx),
where gt(x) is defined in (3.7), and
ϕν(s):=ρsn+1hˆ(νρs),ν>0,0,\]]]>
where h is the volume function (cf. (2.14)) appearing in condition (2.15). Lemma below gives the generic calculation, needed for the proof of Theorem 3.
Forθ∈(0,1), we have(gt−s□gs)(x)≤C[ϕ(1−θ)b(t−s)+ϕ(1−θ)b(s)]gt,θ(x),x∈Rn,0<s<t≤1,whereC>00$]]>is some constant, independent of θ, andb>00$]]>comes from the definition ofgt, see (3.7).
Take θ∈(0,1). Since by definition the function ρt is decreasing, we have
‖x−z‖ρt−s+‖z‖ρs≥‖x‖ρt,
which implies
(gt−s□gs)(x)≤e−θb‖x−y‖ρtρt−snρsn∫Rn[fup((z−x)ρt−s)fup((y−z)ρs)](1−θ)|ϖ|(dz).
By integration by parts we derive, using that ρt is monotone decreasing, that
∫Rnρt−snρsn[fup((x−z)ρt−s)fup((z−y)ρs)]1−θ|ϖ|(dz)≤ρt/2n∫Rnρsnfup1−θ((z−y)ρs)|ϖ|(dz)≤c1ρtnρsn∫0∞|ϖ|{z:e−b(1−θ)‖z−y‖ρs≥e−v}e−vdv=(1−θ)bc1ρtnρsn∫0∞|ϖ|{z:‖z−y‖≤v/ρs}e−b(1−θ)vdv≤(1−θ)bc1ρtnρsn∫0∞h(v/ρs)e−b(1−θ)vdv=c1ρtnρsn+1hˆ(b(1−θ)ρs)=c1ρtnϕb(1−θ)(s).
Similar estimate holds true for s>t2\frac{t}{2}$]]>, which finishes the proof of (3.14). □
Take a sequence (θk)k≥1 such that
θ1=1,θk>0,θk−1>θk,k≥2.0,\hspace{2em}\theta _{k-1}>\theta _{k},\hspace{1em}k\ge 2.\]]]>
Let
k0:=[nαζ],
where ζ is the parameter appearing in (2.15). Define
κ:=min{b(θj−1−θj),1≤j≤k0},F(t):=∫0tϕκ(r)dr,
and
g˜t(k)(x):=gt,θk(x)Fk−1(t),1≤k≤k0,e−bθk0ρt‖x‖Fk−k0(t),k>k0,k_{0},\end{array}\right.\]]]>
where gt,θ(x) is defined in (3.12).
Finally, define inductively the sequence of measures
Rt(1)(dw):=Qt(dw)ifk=1,Rt(k)(dw):=(2F(t))−1∫0t∫Rn[ϕκ(t−s)+ϕκ(s)]Qt−s(dw−u)Rs(k−1)(du)ds
if k≥2. Since (Qt)t≥0 is the family of sub-probability measures (see Theorem 2), we have
Rt(2)(Rn)≤(2F(t))−1∫0t[ϕκ(t−s)+ϕκ(s)]Qt−s(Rn)Qs(Rn)ds≤1,
and we can see by induction that Rt(k)(Rn)≤1, t∈[0,1], for all k≥2.
Fork≥2we have|pt♢k(x,y)|≤C˜k(g˜t(k)∗Rt(k))(y−x),x,y∈Rn,t∈(0,1],where the sequence(g˜t(k))k≥1is given by (3.20),Rt(k)is defined in (3.21),k≥2, and fork>k0k_{0}$]]>, the constantsC˜kcan be expressed asC˜k=Ck−k0M,whereM,C>00$]]>are some constants.
We use induction. Rewrite the upper estimate on pt(x,y) in the form (3.5). For k=2 we get, using (3.5) and (3.15), the following estimates:
|pt♢2(x,y)|≤C12∫0t∫R2n[∫Rng˜s(1)(z−x−w1)g˜t−s(1)(y−z−w2)|ϖ|(dz)]·Qt−s(dw1)Qs(dw2)ds≤C2∫Rngt,θ2(x−w){∫0t[ϕb(θ1−θ2)(t−s)+ϕb(θ1−θ2)(s)]·∫RnQt−s(dw−u)Qs(du)ds}≤C2∫Rngt,θ2(x−w){∫0t[ϕκ(t−s)+ϕκ(s)]·∫RnQt−s(dw−u)Qs(du)ds}≤2C2F(t)(gt,θ2∗R(2))(y−x)=2C2(g˜t(2)∗R(2))(y−x),
where C1>00$]]> comes from (3.5), and in the third line from below we used that by the definition of κ and monotonicity of ϕν in ν,
ϕb(θj−1−θj)(t)≤φκ(t),t∈(0,1].
Suppose that (3.22) holds for some 2≤k≤k0. Then
|pt♢(k+1)(x,y)|≤2k−1CkC1∫0t∫Rn(g˜t−s(1)∗Qt−s)(z−x)·(g˜s(k)∗Rs(k))(y−z)dzds=2k−1CkC1∫0t∫Rn∫Rn(g˜t−s(1)□g˜s(k))(y−x−w1−w2)·Qt−s(dw1)Rs(k)(dw2)ds.
By the same argument as those used in the proof of Lemma 3, we have
(g˜t−s(1)□g˜s(k))(x)≤(gt−s,θk□gs,θk)(x)Fk−1(t)≤ck+1gt,θk+1(x)Fk−1(t)[ϕb(θk−1−θk)(t−s)+ϕb(θk−1−θk)(s)]=ck+1(F(t))−1[ϕκ(t−s)+ϕκ(s)]g˜t(k+1)(x).
Substituting this estimate into (3.27), performing the change of variables and normalizing, we get (3.22) for 2≤k≤k0.
Take c0>00$]]>. Note that for some c1>00$]]>, we have c0ρt≤ρc1t, t∈(0,1]. Then, by (2.15),
∫0tρtn+1hˆ(c0ρt)dt≤c2∫0tρc1tn+1hˆ(ρc1t)dt≤c3∫0c1tρtn+1hˆ(ρt)dt≤c4tζ.
Therefore, taking k0 as in (3.17), we get
ρtnFk0(t)≤c5t−n/α+k0ζ≤c6,t∈[0,1].
In such a way, on the (k0+1)-th step, we obtain
(g˜t−s(k0)□g˜s(1))(x)≤ce−bθk0ρt‖x‖∫Rne−bρs(1−θk0)‖z−x‖|ϖ|(dz)=ce−bθk0ρt‖x‖∫0∞|ϖ|{z:ρsb(1−θk0)‖z−x‖≤r}e−rdr≤ce−bθk0ρt‖x‖ϕb(1−θk0)(s)≤cg˜t(k0+1)(x)ϕκ(s)F−1(t)
(cf. (3.15)), where in the last line we used the inequality κ<b(1−θk0) and the monotonicity of ϕν in ν. Using this estimate, we derive
pt♢(k0+1)(x,y)≤Ck0C1∫0t∫Rn∫Rn(g˜t−s(k0)□g˜s(1))(y−x−w1−w2)·Qs(dw1)Rt−s(k0)(dw2)ds≤2cC1Ck0·(g˜t(k0+1)∗Rt(k0+1))(y−x).
Then (3.22) follows by induction. Indeed, assume that (3.22) holds for k=k0+ℓ−1. For ℓ≥2 we get
(g˜t−s(k0+ℓ−1)□g˜s(1))(x)≤cFℓ−1(t)e−bθk0ρt‖x‖ϕκ(s)=cF−1(t)g˜t(k0+ℓ)(x)ϕκ(s).
Therefore,
|pt♢(k0+ℓ)(x,y)|≤(2C1c)ℓ−1C1Ck0∫0t∫Rn(g˜t−s(k0+ℓ−1)∗Rt−s(k0+ℓ−1))(z−x)·(g˜s(1)∗Qs)(y−z)dzds=Ck0(2C1c)ℓ(g˜t(k0+ℓ)∗Rt(k0+ℓ))(y−x).
□
As we observed in the proof, the estimation procedure depends on condition H1, which guarantees the existence of the number k0 such that (3.25) holds. In general, without H1 we cannot guarantee the existence of such a number, which is crucial in our approach. For example, suppose that ρs≍s−1 for small s, and take the measure ϖ such that
h(r)≍1ln2r,r∈(0,1].
By the Tauberian theorem, we have hˆ(λ)≍[λln2λ]−1 for large λ. Therefore, ϕν(t)∼|lnt|−1 as t→0, and thus the integral F(t) diverges. Nevertheless, assumption H1 can be dropped, if the function pt(x,y) possesses a more precise upper bound. We discuss this question later in Section 4.
Proof of statement c)
From (3.27) we get for all x,y∈Rn,
|pt♢(k0+ℓ)(x,y)|≤M(CF(t))ℓ,ℓ≥1,
where M=Ck0 and C=2C1c. Without loss of generality, assume that C≥1. Since F(t)→0 as t→0, there exists t0>00$]]>, such that
CF(t)<1/2,t∈(0,t0].
Thus, for t∈(0,t0], the series (3.2) converges absolutely and is the solution to (3.1).
Let us show that the integral equation (3.1) possesses a unique solution in the class of functions {f(t,x,y)≥0,t∈(0,t0],x,y∈Rn}, such that
∫Rnf(t,x,y)dy≤c,t∈(0,t0],x∈Rn.
Then the series (3.2) is a unique representation of the Feynman–Kac kernel ptA(x,y) for t∈(0,t0], x,y∈Rn.
Suppose that there are two solutions pt(1),A(x,y) and pt(2),A(x,y) to (3.1). Put p˜tA(x,y):=|pt(1),A(x,y)−pt(2),A(x,y)| and vt(x):=∫Rnp˜tA(x,y)dy. Then, by (3.1) we have
vt(x)≤∫0t∫Rnpt−s(x,z)vs(z)ϖ(dz)ds.
By induction we get
vt(x)≤∫0t∫Rnpt−s♢k(x,z)vs(z)ϖ(dz)ds.
Note that there exists c>00$]]> such that pt♢(k0+1)(x,y)≤c for all t∈(0,t0], x,y∈Rn (cf. (3.26)). In such a way, by the finiteness of measure ϖ, we get
vt(x)≤c1∫0t∫Rnvs(z)ϖ(dz)ds≤c2∫0tv˜sds,
where v˜s:=supz∈Rnvs(z). Taking supx∈Rn in the left-hand side of (3.33), we derive
v˜t≤c2∫0tv˜sds,t∈(0,t0].
Applying the Gronwall–Bellman lemma, we derive v˜t≡0 for all t∈(0,t0]. Thus, the solution to (3.1) is unique in the class of functions
{f(t,x,y)≥0,t∈(0,t0],x,y∈Rn}
satisfying (3.30).
Estimating series (3.2) from above, we get an upper bound in (2.16) with fup of the form (2.10) and
Rt(dw)=c0∑k≥1ckRt(k)(dw),
with some c∈(0,1) and the normalizing constant c0>00$]]> chosen so that Rt(Rn)≤1 for all t∈(0,t0].
For the lower bound, observe that by (3.20) we have
|pt♢k(x,y)|≤C(k0)ρtnF(t),2≤k≤k0.
By (3.28) and (3.29) we get
∑ℓ≥1pt♢(k0+ℓ)(x,y)≤2MCF(t),t∈(0,t0],
which, together with (3.35) and the observation that ρt is decreasing, yields the estimate
|∑k=2∞pt♢k(x,y)|≤C0F(t)ρtn,t∈(0,t0],
where C0>00$]]> is some constant. Therefore, choosing t0 small enough, we have by the lower bound in (2.12) the inequalities
ptA(x,y)≥ρtnflow((y−x)ρt)−C0F(t)ρtn≥cρtnflow((y−x)ρt),t∈(0,t0].
□
Proof of Proposition 4
Since the proof of the proposition follows with minor changes from the proof of the upper estimate in [22, Thm. 3.3], we only sketch the argument. For (t,x,y)∈(0,t0]×Rn×Rn, put
I0(t,x,y):=gt(x−y),Ik(t,x,y)=∫0t∫Rngt−s(x−z)Ik−1(s,z,y)ϖ(dz)ds.
By the same argument as in [22], we can get
|Ik(t,x,y)|≤ckgt(y−x),k≥1,t∈(0,t0],
where c∈(0,1) is some constant. Thus, for k≥1, we have
|pt♢k(x,y)|≤ckgt(x−y),x,y∈Rn,t∈(0,t0].
This proves the convergence of the series (3.2) and the upper estimate (2.20). □
Let us briefly discuss the crucial difference between the proofs of Theorem 3 and Proposition 4. We changed the procedure of estimation of pt♢k(x,y) after a certain step, which was possible due to (2.15). In the case when we have a single-kernel estimate for pt(x,y), for example, (2.19), we can drop condition (2.15). In fact, it is enough to require that ϖ∈SK with respect to gt(y−x). This happens because in the case of the single-kernel estimate of type (2.19), it is possible to show that the convolutions pt♢k(x,y) satisfy the upper bound (4.1) with c∈(0,1), which implies the convergence of the series (3.2).
Examples
As one might observe, the scope of applicability of Theorem 3 heavily relies on the properties of the initial process X. To assure the existence of such a process, we applied Theorem 2. Below we give the examplesin which condition A1 is satisfied. Since conditions A2–A4 are easy to check, we may assume that the functions a(x) and m(x,u) are appropriate. We confine ourselves to the case when the measure μ in the generator of X is “discretized α-stable; up to the author’s knowledge, in this case the corresponding Feynman–Kac semigroup was not investigated. Examples below illustrate that our approach is applicable also in the situation when the “Lévy-type measure” m(x,u)μ(du) related to the initial process X is not absolutely continuous with respect to the Lebesgue measure.
a) Consider a “discretized version” of an α-stable Lévy measure in Rn. Let mk,υ(dy) be the uniform distribution on a sphere Sk,υ centered at 0 with radius 2−kυ, υ>00$]]>, k∈Z. Consider the Lévy measure
μ(dy)=∑k=−∞∞2kγmk,v(dy),
where 0<γ<2υ. In [17], it is shown that for such a Lévy measure condition A1 is satisfied, and
ρt≍t−1/α,t∈(0,1],
where α=γ/υ.
Take some functions a(·):Rn→R and a non-negative bounded function m(·,·) defined on Rn×Rn satisfying assumptions A2–A4. By Theorem 2 the operator of the form (2.7) with μ, a(x), and m(x,u) as before can be extended to the generator of a Feller process X that admits the transition density pt(x,y) satisfying (2.12).
Let ϖ be a finite Borel measure, and let h be its volume function, see (2.14). Let us show that if the inequality
∫0th(v)vn+1−αdv≤c1tζ,t∈(0,1],
for some ζ>00$]]>, then we have (2.15). Using (5.2), changing variables, and applying the Fubini theorem, we derive
∫0tρsn+1hˆ(ρs)ds≤∫0ts−n+1αhˆ(c2s−1/α)ds=α∫0∞[∫0t1/αvh(τ)τn+1−αdτ]vn−αe−c2vdv.
Denote by I(t) the right-hand side in this expression. Applying (5.3), we get
I(t)≤c1∫0∞(t1/αv)ζvn−αe−c2vdv≤c3tζ/α.
In particular, if h(v)≤cvd, d>n−αn-\alpha $]]>, then (5.3) holds.
Thus, by Theorem 3, the Feynman–Kac semigroup (TtA)t≥0 is well defined, and the kernel ptA(x,y) satisfies (2.16) with some constants ai, 1≤i≤4, and some family of sub-probability measures (R(k))t≥0.
b) Consider now the one-dimensional situation. In this case, the Lévy measure μ from (5.1) is just
μ(dy)=∑n=−∞∞2nγ(δ2−nυ(dy)+δ−2−nυ(dy)).
Let X be a Lévy process with characteristic exponent
ψ(ξ):=∫Rn(1−cos(ξu))μ(du).
In [20] we show that if 1<α<2, then the transition probability density pt(x,y) of X, X0=x, is continuous in (t,x,y)∈(0,∞)×R×R and admits the following upper bound:
pt(x,y)≤ct−1/α(1+|y−x|/t1/α)−α,t∈(0,1],x,y∈R.
Note that the right-hand side of (5.5) is of the form (2.17) with d=0. Thus, the conditions of Proposition 4 are satisfied, and we can construct the Feynman–Kac semigroup for the related functional At and the transition density pt(x,y), and get the upper bound for the function ptA(x,y) with ρt≍t−1/α, t∈(0,1].
To end this example, we remark that it is still possible to construct the upper bound for such pt(x,y) for α∈(0,1) of the form t−n/αf(xt−1/α), but the function f in this upper bound might not be integrable; see [20] for details. Note that the upper bound (5.5) is non-integrable in Rn for n≥2.
Consider the Lévy measure
ν0(A)=∫Rn∫0∞1A(rv)r−1−αdrμ0(dv),α∈(0,2),
where α∈(0,2), μ0 is a finite symmetric non-degenerate (that is, not concentrated on a linear subspace of Rn) measure on the unit sphere Sn in Rn. Suppose that there exists d>00$]]> such that for small r we have
ν0(B(x,r))≤Crd,‖x‖=1.
For d+α>nn$]]>, it is shown in [6] that the corresponding Lévy process X, X0=x, admits the transition probability density pt(x,y), which satisfies
pt(x,y)≤ct−n/α(1+‖y−x‖t−1/α)−d−α,t>0,x,y∈Rn.0,\hspace{2.5pt}x,y\in {\mathbb{R}}^{n}.\]]]>
In the forthcoming paper [7], we construct a class of Lévy-type processes that admit the transition densities bounded from above by the left-hand side of (5.7). Thus, taking ϖ∈Kn,α, we may apply Proposition 4.
Acknowledgments
The author thanks Alexei Kulik and Niels Jacob for valuable discussions and comments, and the anonymous referee for helpful remarks and suggestions. The DFG Grant Schi 419/8-1 and the Scholarship of the President of Ukraine for young scientists (2012–2014) are gratefully acknowledged.
Appendix
We follow the idea of the proof of [4, Lemma 11]. Without loss of generality, assume that ϖ is non-negative. Suppose first that ϖ∈Kn,α. Using integration by parts and the Fubini theorem, we get
∫0t∫Rngs(x−y)ϖ(dy)ds≍∫0t∫Rns−n/α(1∧(s1/α/‖x−y‖))α+dϖ(dy)ds=∫0ts−n/αϖ{y:‖x−y‖≤s1/α}ds+∫0ts−n/α∫‖x−y‖>s1/α(s1/α‖x−y‖)d+αϖ(dy)ds=α(1+d+αd+2α−n)∫0t1/αϖ{y:‖x−y‖≤v}vn+1−αdv+α(d+α)d+2α−ntd+2α−nα∫t1/α∞ϖ{y:‖x−y‖<v}vd+1+αdv.{s}^{1/\alpha }}{\bigg(\frac{{s}^{1/\alpha }}{\| x-y\| }\bigg)}^{d+\alpha }\varpi (dy)ds\\{} & \displaystyle \hspace{1em}=\alpha \bigg(1+\frac{d+\alpha }{d+2\alpha -n}\bigg){\int _{0}^{{t}^{1/\alpha }}}\frac{\varpi \{y:\hspace{0.1667em}\| x-y\| \le v\}}{{v}^{n+1-\alpha }}dv\\{} & \displaystyle \hspace{2em}+\frac{\alpha (d+\alpha )}{d+2\alpha -n}{t}^{\frac{d+2\alpha -n}{\alpha }}{\int _{{t}^{1/\alpha }}^{\infty }}\frac{\varpi \{y:\hspace{0.1667em}\| x-y\|
Since ϖ∈Kn,α, the first term tends to 0 as t→0. Further, since d>n−αn-\alpha $]]> and the measure ϖ is finite, we have
td+2α−nαsupx∫1∞ϖ{y:‖x−y‖<v}vd+1+αdv→0,t→0.
Let us show that
supxtd+2α−nα∫t1/α1ϖ{y:‖x−y‖<v}vd+1+αdv.
Let K0≡K0(t):=[t−1/α]+1; note that K0(t)t1/α→1 as t→0. We have
td−ϵ+2α−nα∫t1/α1ϖ{y:‖x−y‖<v}vd+1+αdv≤∑k=1K0(1k)(d−n+2α)/α∫kt1/α(k+1)t1/αϖ{y:‖x−y‖<v}vn+1−αdv.
Since d>n−αn-\alpha $]]>, we have ∑k=1∞k−(d−n+2α)/α<∞. Since ϖ∈Kn,α, we have
max1≤k≤K0(t)supx∫kt1/α(k+1)t1/αϖ{y:‖x−y‖<v}vn+1−αdv⟶0,t→0.
Thus, we arrive at (A.2). This proves that (2.18) implies that ϖ∈SK with respect to gt(y−x).
The converse is straightforward. □
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