[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b170835

A NOTE ON THE GENERALIZED HEAT CONTENT FOR LÉVY PROCESSES

Cygan, Wojciech (Instytut Matematyczny Uniwersytet Wroclawski)
Grzywny, Tomasz (Wydzial Matematyki Politechnika Wroclawska)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.5, 2018 , pp. 1463-1481 More about this Journal

Abstract

Let $X=\{X_t\}_{t{\geq}0}$ be a $L{\acute{e}}vy$ process in ${\mathbb{R}}^d$ and ${\Omega}$ be an open subset of ${\mathbb{R}}^d$ with finite Lebesgue measure. The quantity $H_{\Omega}(t)={\int_{\Omega}}{\mathbb{P}}^x(X_t{\in}{\Omega})$ dx is called the heat content. In this article we consider its generalized version $H^{\mu}_g(t)={\int_{\mathbb{R}^d}}{\mathbb{E}^xg(X_t){\mu}(dx)$ , where g is a bounded function and ${\mu}$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of $L{\acute{e}}vy$ processes.

Keywords

heat content; isotropic $L{\acute{e}}vy$ process; multivariate regular variation;

Citations & Related Records

Reference

1	L. Acuna Valverde, Heat content estimates over sets of finite perimeter, J. Math. Anal. Appl. 441 (2016), no. 1, 104-120. DOI
2	L. Acuna Valverde, On the one dimensional spectral heat content for stable processes, J. Math. Anal. Appl. 441 (2016), no. 1, 11-24. DOI
3	L. Acuna Valverde, Heat content for stable processes in domains of ${\mathbb{R}}^d$ , J. Geom. Anal. 27 (2017), no. 1, 492-524. DOI
4	L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
5	K. Sato, Levy Processes and Infinitely Divisible Distributions, translated from the 1990 Japanese original, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.
6	M. van den Berg, Heat content and Brownian motion for some regions with a fractal boundary, Probab. Theory Related Fields 100 (1994), no. 4, 439-456. DOI
7	M. van den Berg, Heat flow and perimeter in ${\mathbb{R}}^m$ , Potential Anal. 39 (2013), no. 4, 369-387. DOI
8	M. van den Berg, E. B. Dryden, and T. Kappeler, Isospectrality and heat content, Bull. Lond. Math. Soc. 46 (2014), no. 4, 793-808. DOI
9	M. van den Berg and P. Gilkey, Heat flow out of a compact manifold, J. Geom. Anal. 25 (2015), no. 3, 1576-1601. DOI
10	M. van den Berg and K. Gittins, Uniform bounds for the heat content of open sets in Euclidean space, Differential Geom. Appl. 40 (2015), 67-85. DOI
11	M. van den Berg and J.-F. Le Gall, Mean curvature and the heat equation, Math. Z. 215 (1994), no. 3, 437-464. DOI
12	W. Cygan and T. Grzywny, Heat content for convolution semigroups, J. Math. Anal. Appl. 446 (2017), no. 2, 1393-1414. DOI
13	D. Applebaum, Levy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
14	N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987.
15	K. Bogdan, T. Grzywny, and M. Ryznar, Density and tails of unimodal convolution semigroups, J. Funct. Anal. 266 (2014), no. 6, 3543-3571. DOI
16	J. Jacod, Asymptotic properties of power variations of Levy processes, ESAIM Probab. Stat. 11 (2007), 173-196. DOI
17	J. E. Figueroa-Lopez, Small-time asymptotics for Levy processes. , Statistics & Probability Letters 78 (2008), 3355-3365. DOI
18	B. Galerne, Computation of the perimeter of measurable sets via their covariogram: Applications to random sets, Image Analysis & Stereology 30 (2011), no. 1, 39-51. DOI
19	T. Grzywny, On Harnack inequality and Holder regularity for isotropic unimodal Levy processes, Potential Anal. 41 (2014), no. 1, 1-29. DOI
20	V. Knopova and R. L. Schilling, A note on the existence of transition probability densities of Levy processes, Forum Math. 25 (2013), no. 1, 125-149.
21	F. Kuhn and R. L. Schilling, On the domain of fractional Laplacians and related generators of Feller processes, Preprint: arXiv:1610.08197, 2016.
22	W. E. Pruitt, The growth of random walks and Levy processes, Ann. Probab. 9 (1981), no. 6, 948-956. DOI
23	S. I. Resnick, Heavy-Tail Phenomena, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2007.