For a (non-symmetric) strong Markov process

Let

Feynman–Kac semigroup is well studied in the case of a Brownian motion (see [

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

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 [

Up to the author’s knowledge, in general, the results on two-sided estimates of

The paper is organized as follows. In Section

For functions

Let

For fixed

Recall some notions on the Kato class of measures and related continuous additive functionals.

We say that a functional

A positive Borel measure

Let

To show that the semigroup

Denote by

The next proposition is essentially contained in [

When

Proposition

Let

There exists

Assume that the functions

The functions

There exist constants

In the case

Denote by

Finally, define

The constructed process is a

Let us show that, under the conditions of Theorem

Finally, for a signed Borel measure

The following theorem is the main result of the paper. Let

In general,

Assumption (

In the proposition below, we state the “compact” upper bound for

a) For

b) In view of Lemma

In Section

a) By the upper bound in (

Statement b) is already contained in Proposition

For the proof of Theorem

Second, we show that on

Finally, observe that the kernel

Before we prove that (

Rewrite the upper bound in (

By induction we can get

In the next subsection, we handle the general case, in particular,

We give the generic calculation, which allows us to estimate the convolution

We estimate the convolutions

We change the estimation procedure after

The change of the estimation procedure could be unnecessary if we would know that

In the case when

For

Take

Take a sequence

Finally, define inductively the sequence of measures

We use induction. Rewrite the upper estimate on

Take

As we observed in the proof, the estimation procedure depends on condition H1, which guarantees the existence of the number

From (

Let us show that the integral equation (

Suppose that there are two solutions

Estimating series (

For the lower bound, observe that by (

Since the proof of the proposition follows with minor changes from the proof of the upper estimate in [

Let us briefly discuss the crucial difference between the proofs of Theorem

As one might observe, the scope of applicability of Theorem

a) Consider a “discretized version” of an

Take some functions

Let

Thus, by Theorem

b) Consider now the one-dimensional situation. In this case, the Lévy measure

To end this example, we remark that it is still possible to construct the upper bound for such

Consider the Lévy measure

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.

We follow the idea of the proof of [

The converse is straightforward. □