Abstract
We derive the heat trace asymptotics of the generator of subordinate Brownian motion on Euclidean space for a class of Laplace exponents. The terms in the asymptotic expansion can be computed to arbitrary order and depend both on the geometry of Euclidean space and the short-time behaviour of the process. If the Blumenthal-Getoor index of the process is rational, then the asymptotics may contain logarithmic terms. The key assumption is the existence of a suitable density for the Lévy measure of the subordinator. The analysis is highly explicit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albeverio, S., Rüdiger, B., Wu, J.-L.: Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory. In: Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.) Lévy Processes: Theory and Applications, pp 187–224, Birkhäuser, Boston (2001)
Applebaum, D.: Infinitely divisible central probability measures on compact Lie groups – regularity, semigroups and transition kernels. Ann. Prob. 39(6), 2474–2496 (2011)
Applebaum, D.: Pseudo-differential operators and Markov semigroups on compact Lie groups. J. Math. Anal. Appl. 384, 331–348 (2011)
Applebaum, D.: Probability on Compact Lie Groups, volume 70 of Probability Theory and Stochastic Modelling. Springer-Verlag, Cham (2014)
Bañuelos, R., Baudoin, F.: Trace asymptotics for subordinate semigroups. arXiv:1308.4944 (2013)
Bañuelos, R., Kulczycki, T.: Trace estimates for stable processes. Prob. Theory Relat. Fields 142, 313–338 (2008)
Bañuelos, R., Mijena, J.B., Nane, E.: Two-term trace estimates for relativistic stable processes. J. Math. Anal. Appl. 410(2), 837–846 (2014)
Barndorff-Nielsen, O.E.: Processes of normal inverse gaussian type. Finance Stoch. 2(1), 41–68 (1997)
Bertoin, J.: Lévy Processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998)
Bleistein, N., Handelsman, R.A.: Asymptotic Expansions of Integrals. Dover, New York (1986)
Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10(3), 493–516 (1961)
Boggiatto, P., Buzano, E., Rodino, L.: Global Hypoellipticity and Spectral Theory, volume 92 of Mathematical Research. Akademie Verlag, Berlin (1996)
Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)
Cordes, H.O.: The Technique of Pseudodifferential Operators, volume 202 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1995)
Egorov, Yu.V., Schulze, B.-W.: Pseudo-differential Operators, Singularities, Applications, volume 93 of Operator Theory Advances and Applications, Birkhäuser, Basel (1997)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, volume 194 of Graduate Texts in Mathematics. Springer, New York (2000)
Grubb, G., Seeley, R.T.: Zeta and eta functions for Atiyah-Patodi-Singer operators. J. Geom. Anal. 6(1), 31–77 (1996)
Jacob, N.: Pseudo-differential Operators & Markov Processes: Fourier Analysis and Semigroups, vol. 1. Imperial College Press, London (2001)
Jacob, N.: Pseudo-Differential Operators & Markov Processes: Generators and Their Potential Theory, vol. 2. Imperial College Press, London (2002)
Jacob, N.: Pseudo-Differential Operators & Markov Processes: Markov Processes and Applications, vol. 3. Imperial College Press, London (2005)
Kontsevich, M., Vishik, S.: Determinants of elliptic pseudo-differential operators. arXiv:hep-th/9404046 (1994)
Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. In: Functional Analysis on the Eve of the 21st Century, volume 131 of Progress in Mathematics, Birkhäuser (1995)
Maniccia, L., Schrohe, E., Seiler, J.: Complex powers of classical SG-pseudodifferential operators. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52(2), 353–369 (2006)
Maniccia, L., Schrohe, E., Seiler, J.: Determinants of classical SG-pseudodifferential operators. Math. Nachr. 287(7), 782–802 (2014)
Nicola, F., Rodino, L.: Global Pseudo-Differential Calculus on Euclidean Spaces, volume 4 of Pseudodifferential Operators Theory and Applications, Birkhäuser, Basel (2010)
Park, H., Song, R.: Trace estimates for relativistic stable processes. Potential Anal. 41(4), 1273–1291 (2014)
Pruitt, W.E.: The growth of random walks and Lévy processes. Ann. Prob. 9(6), 948–956 (1981)
Schilling, R.L.: Growth and Hölder conditions for the sample paths of Feller processes. Prob. Theory Relat. Fields 112, 565–611 (1998)
Schilling, R.L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications, volume 37 of Studies in Mathematics, 2nd edn. Walter de Gruyter, Berlin (2012)
Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space, volume 265 of Graduate Texts in Mathematics. Springer-Verlag, Dordrecht (2012)
Scott, S.: Traces and Determinants of Pseudodifferential Operators. Oxford Mathematical Monographs. Oxford University Press, Oxford (2010)
Shubin, M.A.: Pseudodifferential Operators and Spectral Theory, 2nd edn. Springer-Verlag, Berlin (2001)
Wodzicki, M.: Noncommutative residue. I. Fundamentals. In: Manin, Y. (ed.) K-Theory, Arithmetic and Geometry, volume 1289 of Lecture Notes in Mathematics, pp 320–399. Springer-Verlag, Berlin (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fahrenwaldt, M.A. Heat Trace Asymptotics of Subordinate Brownian Motion in Euclidean Space. Potential Anal 44, 331–354 (2016). https://doi.org/10.1007/s11118-015-9514-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-015-9514-1