Sensitivity Analysis of Nonlocal Operators with Applications to Jump Processe

• Applicant: Professor Dr. René Leander Schilling
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426577679

• Final Report Year: 2024

Final Report Abstract

From the general theory of stochastic processes it is well-known that a large class of Feller/Lévytype jump processes are necessarily generated by non-local pseudo-differential operators, whose symbols are non-classical and negative definite. This means that the usual tools from analysis are not available to deal with such symbols. The aims of the present project were (i) to give sufficient conditions on the symbol that guarantee the existence of such processes using (a suitable variation of) the parametrix method, and (ii) to study stochastic properties of thus constructed processes. In total, 26 peer-reviewed research publications were supported by the grant, covering various aspects of (i) and (ii). The joint nature of the grant is also underlined by the rather active dissemination efforts: we organized 6 workshops/conferences, partly with additional external funding. As a by-product, the present grant intensified the links between WUST and TUD, leading to further sustainable (institutional) collaborations and a joint grant application.

Publications

• Construction and heat kernel estimates of generalstable-like Markov processes. Dissertationes Mathematicae, 569, 1-86.
  Knopova, Victoria; Kulik, Alexei & Schilling, René L.
  
• For which functions are are f (xt )−ef (xt ) and g(xt )/eg(xt) martingales?. Theory of Probability and Mathematical Statistics, 105(0), 79-91.
  Kühn, F. & Schilling, R.
  
• Second order elliptic partial differential equations driven by Lévy white noise. Modern Stochastics: Theory and Applications, 179-207.
  Berger, David & Mohamed, Farid
  
• Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups. Studia Mathematica, 262(1), 93-119.
  Kühn, Franziska & Schilling, René L.
  
• Lévy Processes, Generalized Moments and Uniform Integrability. Probability and Mathematical Statistics, 42(1).
  Berger, David; Kühn, Franziska & Schilling, René
  
• Maximum likelihood estimation for discrete exponential families and random graphs. Latin American Journal of Probability and Mathematical Statistics, 19(1), 1045.
  Bogdan, Krzysztof; Bosy, Michał & Skalski, Tomasz
  
• On q-scale functions of spectrally negative Lévy processes. Advances in Applied Probability, 55(1), 56-84.
  Behme, Anita; Oechsler, David & Schilling, René
  
• On the Liouville and strong Liouville properties for a class of non-local operators. MATHEMATICA SCANDINAVICA, 128(2).
  Berger, David & Schilling, René L.
  
• On weak solution of SDE driven by inhomogeneous singular Lévy noise. Transactions of the American Mathematical Society, 375(7), 4567-4618.
  Kulczycki, Tadeusz; Kulik, Alexei & Ryznar, Michał
  
• Singular integrals of subordinators with applications to structural properties of SPDEs. Transactions of the American Mathematical Society.
  Deng, Chang-Song; Schilling, René & Xu, Lihu
  
• Anomalous diffusion limit for a kinetic equation with a thermostatted interface. Probability Theory and Related Fields, 189(1-2), 721-769.
  Bogdan, Krzysztof; Komorowski, Tomasz & Marino, Lorenzo
  
• Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme. Stochastic Processes and their Applications, 163, 136-167.
  Chen, Peng; Deng, Chang-Song; Schilling, René L. & Xu, Lihu
  
• Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials. Communications in Mathematical Physics, 403(2), 795-828.
  Jakubowski, Tomasz; Kaleta, Kamil & Szczypkowski, Karol
  
• Maximal inequalities and some applications. Probability Surveys, 20(none).
  Kühn, Franziska & Schilling, René L.
  
• Quasi-infinite divisibility of a class of distributions with discrete part. Proceedings of the American Mathematical Society.
  Berger, David & Kutlu, Merve
  
• Sharp and plain estimates for Schrödinger perturbation of Gaussian kernel. Journal d'Analyse Mathématique, 152(1), 255-282.
  Jakubowski, Tomasz & Szczypkowski, Karol
  
• Small time chaos approximations for heat kernels of multidimensional diffusions. BIT Numerical Mathematics, 63(1).
  Ivanenko, D.; Kohatsu-Higa, A. & Kulik, A.
  
• The Fractional Laplacian with Reflections. Potential Analysis, 61(2), 317-345.
  Bogdan, Krzysztof & Kunze, Markus
  
• An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions. Journal of Spectral Theory, 14(2), 665-695.
  Berger, David; Schilling, René L.; Shargorodsky, Eugene & Sharia, Teo
  
• Drift reduction method for SDEs driven by heterogeneous singular Lévy noise. Bernoulli, 30(4).
  Kulczycki, Tadeusz; Kulyk, Oleksii & Ryznar, Michał
  
• Relativistic stable operators with critical potentials. Documenta Mathematica, 29(1), 209-245.
  Jakubowski, Tomasz; Kaleta, Kamil & Szczypkowski, Karol
  
• Self-similar solution for fractional Laplacian in cones. Electronic Journal of Probability, 29(none).
  Bogdan, Krzysztof; Knosalla, Piotr; Leżaj, Łukasz & Pilarczyk, Dominika
  
• The Liouville theorem for a class of Fourier multipliers and its connection to coupling. Bulletin of the London Mathematical Society, 56(7), 2374-2394.
  Berger, David; Schilling, René L. & Shargorodsky, Eugene