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An Elliptic PDE with Convex Solutions

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  24 January 2018

Jon Warren
Jon Warren*
Affiliation:
Department of Statistics, University of Warwick, Coventry CV4 7AL, UK (j.warren@warwick.ac.uk)
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Abstract

Using a mixture of classical and probabilistic techniques, we investigate the convexity of solutions to the elliptic partial differential equation associated with a certain generalized Ornstein–Uhlenbeck process.

Keywords

convexityelliptic partial differential equationBrownian motion

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 201 - 214
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1. Buraczewski, D., Damek, E. and Mikosch, T., Stochastic models with power-law tails (Springer, 2016).Google Scholar
2. Carmona, P., Petit, F. and Yor, M., Exponential functionals of Lévy Processes, In Lévy processes (ed. Barndorff-Nielson, O. E., Mikosch, T. and Resnick, S. I.), pp. 41–55 (Birkhäuser, Boston, 2001).Google Scholar
3. Dufresne, D., The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Actuar. J. 1990(1) (1990), 39–79.Google Scholar
4. Gawȩdzki, K. and Horvai, P., Sticky behavior of fluid particles in the compressible Kraichnan model, J. Stat. Phys. 116(5–6) (2004), 1247–1300.CrossRefGoogle Scholar
5. Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order (Springer, 2001).Google Scholar
6. Janson, S. and Tysk, J., Preservation of convexity of solutions to parabolic equations, J. Differential Equations 206 (2004), 182–226.CrossRefGoogle Scholar
7. Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics , Volume 1150 (Springer, 1985).Google Scholar
8. Korevaar, N. J., Convexity properties of solutions to elliptic PDEs, In Variational methods for free surface interfaces (ed. Concus, P. and Finn, R.), pp. 115–121 (Springer, New York, 1987).Google Scholar
9. Lions, P. L. and Musliela, M., Convexity of solutions to parabolic equations, C. R. Acad. Sci. Paris, Ser. I 342 (2006), 215–921.Google Scholar
10. Matsumoto, H. and Yor, M., Exponential functionals of Brownian motion, I: Probability laws at fixed time, Probab. Surv. 2 (2005), 312–347.Google Scholar
11. Müller, C., Spherical harmonics, Lecture Notes in Mathematics , Volume 17 (Springer, 1966).Google Scholar
12. Pinsky, R. G., Positive harmonic functions and diffusion (Cambridge University Press, 1995)CrossRefGoogle Scholar
13. Revuz, D. and Yor, M., Continuous martingales and Brownian motion (Springer, 1999).Google Scholar
14. Warren, J., Sticky particles and stochastic flows, In Memoriam Marc Yor – Séminaire de probabilités XLVII, (ed. Donati-Martin, C., Lejay, A. and Rouault, A.) Lecture Notes in Mathematics, Volume 2137, pp. 17–35 (Springer, 2015).Google Scholar
15. Yor, M., Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity (ed. Barndorff-Nielson, O. E., Mikosch, T. and Resnick, S. I.), In Lévy processes, pp. 361–375 (Birkhäuser, Boston, 2001).Google Scholar