Perturbation of an α -stable type stochastic process by a pseudo-gradient
Volume 12, Issue 1 (2025), pp. 1–25
Pub. online: 25 June 2024
Type: Research Article
Open Access
Open Access
Received
25 June 2023
25 June 2023
Revised
18 May 2024
18 May 2024
Accepted
25 May 2024
25 May 2024
Published
25 June 2024
25 June 2024
Abstract
A Markov process defined by some pseudo-differential operator of an order $1\lt \alpha \lt 2$ as the process generator is considered. Using a pseudo-gradient operator, that is, the operator defined by the symbol $i\lambda |\lambda {|^{\beta -1}}$ with some $0\lt \beta \lt 1$, the perturbation of the Markov process under consideration by the pseudo-gradient with a multiplier, which is integrable at some large enough power, is constructed. Such perturbation defines a family of evolution operators, properties of which are investigated. A corresponding Cauchy problem is considered.
References
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional laplacian perturbed by gradient operators. Commun. Math. Phys. 271, 179–198 (2007). MR2283957. https://doi.org/10.1007/s00220-006-0178-y
Boyko, M.V., Osypchuk, M.M.: Perturbation of a rotationally invariant α-stable stochastic process by a pseudo-gradient operator. Precarpathian bulletin of the Shevchenko scientific society (Number) (16(60)), 20–32 (2021). (In Ukrainian) MR4745257. https://doi.org/10.31471/2304-7399-2021-16(60)-20-32
Boyko, M.V., Osypchuk, M.M.: Perturbation of an isotropic α-stable stochastic process by a pseudo-gradient with a generalized coefficient. Carpathian Math. Publ. 16(1), 53–60 (2024). MR4745257. https://doi.org/10.15330/cmp.16.1.53-60
Chen, Z.-Q., Wang, J.-M.: Perturbation by non-local operators. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 54(2), 606–639 (2018). MR3795061. https://doi.org/10.1214/16-AIHP816
Eidelman, S.D., Ivasyshen, S.D., Kochubei, A.N.: Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type. Birkhäuser, Basel (2004). MR2093219. https://doi.org/10.1007/978-3-0348-7844-9
Jakubowski, T.: Fundamental solution of the fractional diffusion equation with a singular drift. J. Math. Sci. (N.Y.) 218(2) (2016). MR3550165. https://doi.org/10.1007/s10958-016-3016-6
Jakubowski, T., Szczypkowski, K.: Time-dependent gradient perturbations of fractional laplacian. J. Evol. Equ. 10(2), 319–339 (2010). MR2643799. https://doi.org/10.1007/s00028-009-0051-5
Loebus, J.-U., Portenko, M.I.: On a class of perturbations of a stable process. Theor. Probability and Math. Statist. 52, 102–111 (1995). (In Ukrainian) MR1445543
Maekawa, Y., Miura, H.: On fundamental solutions for non-local parabolic equations with divergence free drift. Advances in Mathematics 247, 123–191 (2013). MR3096796. https://doi.org/10.1016/j.aim.2013.07.011
Osypchuk, M.M.: On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations. Carpathian Math. Publ. 7(1), 101–107 (2015). MR3427542. https://doi.org/10.15330/cmp.7.1.101-107
Osypchuk, M.M.: On some perturbations of a symmetric stable process and the corresponding Cauchy problems. Theory Stoch. Process. 21(37)(1), 64–72 (2016) MR3571413
Podolynny, S.I., Portenko, N.I.: On multidimentional stable processes with locally unbounded drift. Random Oper. Stoch. Equ. 3(2), 113–124 (1995). MR1341116. https://doi.org/10.1515/rose.1995.3.2.113
Portenko, M.I.: Diffusion processes in media with membranes. In: Proceedings of the Institute of Mathematics of the Ukrainian National Academy of Sciences (1995). (In Ukrainian) MR1356720
Portenko, N.I.: Generalized Diffusion Processes. American Mathematical Society, Providence, Rhode Island (1990) MR1104660. https://doi.org/10.1090/mmono/083
Portenko, N.I.: Some perturbations of drift-type for symmetric stable processes. Random Oper. Stoch. Equ. 2(3), 211–224 (1994). MR1310558. https://doi.org/10.1515/rose.1994.2.3.211
Portenko, N.I.: On some perturbations of symmetric stable processes. In: Probability Theory and Mathematical Statistics. Proceedings of the Seventh Japan-Russia Symposium, Tokyo, Japan, July 26–30, 1995, pp. 414–422. World Scientific, Singapore (1996) MR1467959
Zhang, Q.: Gaussian bounds for the fundamental solutions of $\nabla (a\nabla u)+b\nabla u-{u_{t}}=0$. Manuscripta Mathematica 93(1), 381–390 (2007). MR1457736. https://doi.org/10.1007/BF02677479
