On Adjoint Additive Processes

• Autorzy: Evans Kristian P. , Jacob Niels

Identyfikatory

DOI

10.37190/0208-4147.40.2.2

• Języki publikacji: EN

Abstrakty

EN

Starting with an additive process (Y_t)_t≥0, it is in certain cases possible to construct an adjoint process (X_t)_t≥0 which is itself additive. Moreover, assuming that the transition densities of (Y_t)_t≥0 are controlled by a natural pair of metrics d_ψ;t and δ_ψ;t, we can prove that the transition densities of (X_t)_t≥0 are controlled by the metrics δ _ψ,1/treplacing d_ψ,t and d_ψ,/treplacing δ_ψ,t.

Słowa kluczowe

EN

additive processes; Lévy processes; adjoint densities; transition functions; metric measure space

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2020
• Tom: Vol. 40, Fasc. 2
• Strony: 205--223
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Evans Kristian P.

• Mathematics Department, Swansea University Bay Campus, Fabian Way, Swansea, SA1 8EN. U.K.

autor

Jacob Niels

• Mathematics Department, Swansea University Bay Campus, Fabian Way, Swansea, SA1 8EN. U.K.

Bibliografia

• [1] H. Bauer, Probability Theory, de Gruyter, Berlin, 1996.
• [2] B. Böttcher, Construction of time inhomogeneous Markov processes via evolution equations using pseudo-differential operators, J. London Math. Soc. 78 (2008), 605-621.
• [3] B. Böttcher, R. L. Schilling and J. Wang, Lévy-type Processes: Construction, Approximation and Sample Path Properties, Lecture Notes in Math. 2099, Springer, Berlin, 2013.
• [4] L. Bray and N. Jacob, Some considerations on the structure of transition densities of symmetric Lévy processes, Comm. Stoch. Anal. 10 (2016), 405-420.
• [5] W. Hoh, A symbolic calculus for pseudo-differential operators generating Feller semigroups, Osaka J. Math. 35 (1998), 798-820.
• [6] N. Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z. 215 (1994), 151-166.
• [7] N. Jacob, V. Knopova, S. Landwehr and R. Schilling, A geometric interpretation of the transition density of a symmetric Lévy process, Sci. China Math. 55 (2012), 1099-1126.
• [8] N. Jacob and E. O. T. Rhind, Aspects of micro-local analysis and geometry in the study of Lévy-type generators, in: Open Quantum Systems, Birkhäuser/Springer, 2019, 77-140.
• [9] V. Knopova and R. Schilling, A note on the existence of transition probability densities for Lévy processes, Forum Math. 25 (2013), 125-149.
• [10] F. Kühn, Lévy-Type Processes: Moments, Construction and Heat Kernel Estimates, Lecture Notes in Math. 2187, Springer, Berlin, 2017.
• [11] G. Laue, M. Riedel and H.-J. Roßberg, Unimodale und positiv definite Dichten, B.G. Teubner, Stuttgart, 1999.
• [12] T. Lewis, Probability functions which are proportional to characteristic functions and the infinite divisibility of the von Mises distribution, in: Perspectives in Probability and Statistics, Papers in honour of M. S. Bartlett on the occasion of his sixty-fifth birthday (ed. by J. Gani), Academic Press, London, 1976, 19-28.
• [13] J. Pitman and M. Yor, Infinitely divisible laws associated with hyperbolic functions, Canad. J. Math. 55 (2003), 292-330.
• [14] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, 1999.
• [15] R. L. Schilling, Growth and H¨older conditions for the sample paths of Feller processes, Probab. Theory Related Fields 112 (1998), 565-611.
• [16] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, 2nd ed., de Gruyter, Berlin, 2012.
• [17] H. Tanabe, Equations of Evolution, Pitman, Boston, MA, 1979.
• [18] R. Zhang, Fundamental solutions of a class of pseudo-differential operators with time-dependent negative definite symbols, PhD thesis, Swansea Univ., Swansea, 2011.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).