The authors prove estimates for the transition semigroup and the transition density (heat kernel) of a class of Feller processes which are dominated by rotationally invariant \(\alpha\)-stable Lévy processes. Let \(Y_t\) be a Lévy process (a process with independent and stationary increments and càdlàg paths). It is well known that an \(\alpha\)-stable Lévy process has the characteristic function \(e^{-t|\xi|^\alpha}\) \((\alpha\in (0,2))\) and that the transition semigroup admits a density satisfying \[ p(t,x,y) = p(t,x-y) \asymp t^{-d/\alpha}\wedge \frac{t}{|x-y|^{\alpha+d}} \] (we write \(\asymp\) if the two sides can be estimated against each other with absolute constants). Notice that one of the factors in the minimum, \(y\mapsto |y|^{-\alpha-d}\), is (up to a constant) the density of the Lévy (or jump) measure of \(Y_t\).
Now let \(X_t\) be a Feller process admitting a symbol \(p(x,\xi)\) (we refer to [B. Böttcher et al., Lévy matters III. Lévy-type processes: construction, approximation and sample path properties. Cham: Springer (2013; Zbl 1384.60004)] for details), i.e., for every fixed \(x\), the function \(\xi\mapsto p(x,\xi)\) is the characteristic exponent of a Lévy process: \[ p(x,\xi) = c(x) + i\xi\ell(x) + \sum_{j,k=1}^d q_{jk}(x)\xi_j\xi_k + \int_{y\neq 0}\left(1-e^{iy\xi}+ 1_{(0,1)}(|y|)y\xi\right)\nu(x,dy). \] Here, \((\ell(x),q_{jk}(x),\nu(x,dy))\) is, if \(x\) is fixed, a Lévy triplet. Under certain additional assumptions one can show that such a symbol gives a Feller process. This is assumed throughout. Moreover, it is assumed that

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\(\nu(x,x+dy) = f(x,y)\,dy\) for some continuous \(f:\mathbb R^d\times\mathbb R^d\setminus\{x\neq y\}\to\mathbb R\);

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\(f(x,y)\leq M\phi(|x-y|)|x-y|^{-d-\alpha}\) for some \(C^2\)-function \(\phi\) such that \(\phi|[0,1]\equiv 1\) and \(\phi\) does not oscillate too much;

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symmetric jumps: \(f(x,x+y)=f(x,x-y)\) and reversibility \(f(x,y)=f(y,x)\);

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minimal level of jump activity: \(\int_{|x-y|>\epsilon} f(x,y)/\phi(|x-y|)\,dy\geq c\epsilon^{-\alpha}\) for all \(x\).

Then the semigroup \(P_t\) of the associated Feller process admits a transition density \(p(t,x,y)\) such that \[ p(t,x,y)\leq c'' e^{c't}\left(t^{-d/\alpha}\wedge \frac{t\phi(|x-y|)}{|x-y|^{d+\alpha}}\right). \] The proof uses approximations of Feller processes and their semigropus by compound Poisson processes (obtained by cutting off all small jumps; the assumptions above guarantee that one can control the small-jump activity) and the corresponding semigroups. This part is based on one of the authors’ earlier work [P. Sztonyk, Potential Anal. 33, No. 3, 211–226 (2010; Zbl 1205.60139)], see also [B. Böttcher and R. L. Schilling, Stoch. Dyn. 9, No. 1, 71–80 (2009; Zbl 1168.60359)], where a similar approximation is used]. Several interesting examples are given at the end of the paper.