On weak solution of SDE driven by inhomogeneous singular Lévy noise

Kulczycki, Tadeusz; Kulik, Alexei; Ryznar, Michał

doi:10.1090/tran/8612

On weak solution of SDE driven by inhomogeneous singular Lévy noise
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by Tadeusz Kulczycki, Alexei Kulik and Michał Ryznar PDF

Trans. Amer. Math. Soc. 375 (2022), 4567-4618 Request permission

Abstract:

We study a time-inhomogeneous SDE in $\mathbb {R}^d$ driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit ‘principal part’ and a ‘residual part’, subject to certain $L^\infty (dx)\otimes L^1(dy)$ and $L^\infty (dx)\otimes L^\infty (dy)$-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.

References

Krzysztof Bogdan, Tomasz Grzywny, and MichałRyznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), no. 6, 3543–3571. MR 3165234, DOI 10.1016/j.jfa.2014.01.007
Krzysztof Bogdan, PawełSztonyk, and Victoria Knopova, Heat kernel of anisotropic nonlocal operators, Doc. Math. 25 (2020), 1–54. MR 4077549
Zhen-Qing Chen, Zimo Hao, and Xicheng Zhang, Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes, Electron. J. Probab. 25 (2020), Paper No. 137, 23. MR 4179301, DOI 10.1214/20-ejp542
Zhen-Qing Chen, Eryan Hu, Longjie Xie, and Xicheng Zhang, Heat kernels for non-symmetric diffusion operators with jumps, J. Differential Equations 263 (2017), no. 10, 6576–6634. MR 3693184, DOI 10.1016/j.jde.2017.07.023
Zhen-Qing Chen and Xicheng Zhang, Heat kernels and analyticity of non-symmetric jump diffusion semigroups, Probab. Theory Related Fields 165 (2016), no. 1-2, 267–312. MR 3500272, DOI 10.1007/s00440-015-0631-y
Zhen-Qing Chen and Xicheng Zhang, Heat kernels for time-dependent non-symmetric stable-like operators, J. Math. Anal. Appl. 465 (2018), no. 1, 1–21. MR 3806688, DOI 10.1016/j.jmaa.2018.03.054
F. H. Clarke, On the inverse function theorem, Pacific J. Math. 64 (1976), no. 1, 97–102. MR 425047, DOI 10.2140/pjm.1976.64.97
Arnaud Debussche and Nicolas Fournier, Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients, J. Funct. Anal. 264 (2013), no. 8, 1757–1778. MR 3022725, DOI 10.1016/j.jfa.2013.01.009
Samuil D. Eidelman, Stepan D. Ivasyshen, and Anatoly N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Operator Theory: Advances and Applications, vol. 152, Birkhäuser Verlag, Basel, 2004. MR 2093219, DOI 10.1007/978-3-0348-7844-9
Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
Willy Feller, Zur Theorie der stochastischen Prozesse, Math. Ann. 113 (1937), no. 1, 113–160 (German). MR 1513083, DOI 10.1007/BF01571626
Martin Friesen, Peng Jin, and Barbara Rüdiger, Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 1, 250–271 (English, with English and French summaries). MR 4255174, DOI 10.1214/20-aihp1077
M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, J. Math. Pures Appl. 9 (1913), 305–471; 10 (1914), 105–148.
Tomasz Grzywny, On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes, Potential Anal. 41 (2014), no. 1, 1–29. MR 3225805, DOI 10.1007/s11118-013-9360-y
Tomasz Grzywny and Karol Szczypkowski, Heat kernels of non-symmetric Lévy-type operators, J. Differential Equations 267 (2019), no. 10, 6004–6064. MR 3996792, DOI 10.1016/j.jde.2019.06.013
Tomasz Grzywny and Karol Szczypkowski, Lévy processes: concentration function and heat kernel bounds, Bernoulli 26 (2020), no. 4, 3191–3223. MR 4140542, DOI 10.3150/20-BEJ1220
J. Hadamard, Sur la solution fondamentale des équations aux dérivées partielles du type parabolique, Comptes Rendus de l’Academie des Sciences, Paris 152 (1911) 1148–1149.
Piotr Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), no. 1, 93–101. MR 1201446, DOI 10.4064/cm-64-1-93-101
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
Victoria P. Knopova, Anatoly N. Kochubei, and Alexei M. Kulik, Parametrix methods for equations with fractional Laplacians, Handbook of fractional calculus with applications. Vol. 2, De Gruyter, Berlin, 2019, pp. 267–297. MR 3965398
Victoria Knopova and Alexei Kulik, Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 100–140 (English, with English and French summaries). MR 3765882, DOI 10.1214/16-AIHP796
V. Knopova, A. Kulik, and R. Schilling, Construction and heat kernel estimates of general stable-like Markov processes, arXiv:2005.08491.
Viktorya Knopova and René L. Schilling, Transition density estimates for a class of Lévy and Lévy-type processes, J. Theoret. Probab. 25 (2012), no. 1, 144–170. MR 2886383, DOI 10.1007/s10959-010-0300-0
A. N. Kochubeĭ, Parabolic pseudodifferential equations, hypersingular integrals and Markov processes, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 909–934, 1118 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 2, 233–259. MR 972089, DOI 10.1070/IM1989v033n02ABEH000825
Vassili Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. (3) 80 (2000), no. 3, 725–768. MR 1744782, DOI 10.1112/S0024611500012314
Franziska Kühn, Lévy matters. VI, Lecture Notes in Mathematics, vol. 2187, Springer, Cham, 2017. Lévy-type processes: moments, construction and heat kernel estimates; With a short biography of Paul Lévy by Jean Jacod; Lévy Matters. MR 3701414, DOI 10.1007/978-3-319-60888-4
Franziska Kühn, Transition probabilities of Lévy-type processes: parametrix construction, Math. Nachr. 292 (2019), no. 2, 358–376. MR 3912204, DOI 10.1002/mana.201700441
Tadeusz Kulczycki and MichałRyznar, Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates, Stochastic Process. Appl. 130 (2020), no. 12, 7185–7217. MR 4167204, DOI 10.1016/j.spa.2020.07.011
Tadeusz Kulczycki and MichałRyznar, Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 2, 1335–1375. MR 3877025, DOI 10.30757/alea.v15-50
Tadeusz Kulczycki, MichałRyznar, and PawełSztonyk, Strong feller property for SDEs driven by multiplicative cylindrical stable noise, Potential Anal. 55 (2021), no. 1, 75–126. MR 4261305, DOI 10.1007/s11118-020-09850-8
Alexei Kulik, Approximation in law of locally $\alpha$-stable Lévy-type processes by non-linear regressions, Electron. J. Probab. 24 (2019), Paper No. 83, 45. MR 4003136, DOI 10.1214/19-ejp339
Alexei M. Kulik, On weak uniqueness and distributional properties of a solution to an SDE with $\alpha$-stable noise, Stochastic Process. Appl. 129 (2019), no. 2, 473–506. MR 3907007, DOI 10.1016/j.spa.2018.03.010
E. E. Levi, Sulle equazioni lineari totalmente ellittiche alle derivate parziali, Rend. Circ. Mat. Palermo 24 (1907), 275–317.
Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR 532498
PawełSztonyk, Estimates of densities for Lévy processes with lower intensity of large jumps, Math. Nachr. 290 (2017), no. 1, 120–141. MR 3604626, DOI 10.1002/mana.201500189