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 (1.1) At:=∫0tV(Xs)ds,t>0. 0.\]]]> Suppose that limt→0supxEx|At|=0 . Then, by the Khasminski lemma there exist constants C,b>0 0$]]>, such that (1.2) supxExe|At|≤Cebt; see, for example, [11, Lemma 3] or [12, Lemma 3.3.7]. Estimate (1.2) allows us to define the operator (1.3) 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.

Let X be a Markov process with the state space Rn . We call X a Feller process if the corresponding operator (2.1) 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.

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: (2.2) 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 (2.3) limt→0supx∈Rn∫0t∫Rnps(x,y)ϖ(dy)ds=0. By [13, Thm. 6.6], the condition ϖ∈SK implies that the function (2.4) 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 (2.5) 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 bridge Xtx,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)>0 0$]]>. 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.

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 (2.7) 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.

A1.

There exists β>1 1$]]> such that supℓ∈SnqU(rℓ)≤βinfℓ∈SnqL(rℓ)for allr>0large enough, 0\hspace{2.5pt}\text{large enough,}\hspace{2.5pt}\]]]>

Assume that the functions a(x) and m(x,u) in (2.7) satisfy the assumptions A2–A4 given below.

Denote by flow and fup the functions of the form (2.10) flow(x):=a1(1−a2‖x‖)+,fup(x):=a3e−a4‖x‖,x∈Rn, where ai>0 0$]]>, 1≤i≤4 , are some constants.

Finally, define q∗(r):=supℓ∈SnqU(rℓ) , r>0 0$]]>. 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 (2.11) ρt:=inf{r:q∗(r)=1/t},t∈(0,1].

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>0 0$]]> is the constant appearing in flow in (2.12)), that is, the minimal N for which (2.13) ‖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 (2.14) 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.

Assumption (2.15) can be relaxed, provided that more information about the initial transition probability density is available. Put (2.17) 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 (2.18) limt→0supx∫0t|ϖ|{y:‖x−y‖≤s}sn+1−αds=0. The following lemma shows that for d>n−α n-\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.

In the proposition below, we state the “compact” upper bound for ptA(x,y) .

In Section 5, we provide examples that illustrate Theorem 3 and Proposition 4.