Abstract
Let X 1, . . . ,X N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field
First we demonstrate that for all nonrandom Borel sets \({F\subseteq{{\bf R}^d}}\) , the Minkowski sum \({\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}\) , of the range \({\mathfrak{X}({{\bf R}^{N}_{+}})}\) of \({\mathfrak{X}}\) with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.
Article PDF
Similar content being viewed by others
References
Aizenman M.: The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys. 97(1-2), 91–110 (1985)
Albeverio S., Zhou X.Y.: Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45(2), 195–237 (1996)
Berg C., Forst G.: Potential Theory on Locally Compact Abelian Groups. Springer, New York (1975)
Bertoin, J.: Subordinators: Examples and applications, In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717. Springer, Berlin (1999a)
Bertoin J.: Intersection of independent regenerative sets. Probab. Theory Relat. Fields 114(1), 97–121 (1999b)
Bertoin J.: Lévy Processes. Cambridge University Press, Cambridge (1996)
Blumenthal R.M., Getoor R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)
Csiszár I.: A note on limiting distributions on topological groups, English, with Russian summary. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9, 595–599 (1965)
Dellacherie C., Meyer P.-A.: Probabilities and Potential, vol. 29. North-Holland, Amsterdam (1978)
Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin, Reprint of the 1984 edn (2001)
Dynkin E.B.: Self-intersection local times, occupation fields, and stochastic integrals. Adv. Math. 65(3), 254–271 (1987)
Dynkin E.B.: Generalized random fields related to self-intersections of the Brownian motion. Proc. Nat. Acad. Sci. U.S.A. 83(11), 3575–3576 (1986)
Dynkin E.B.: Random fields associated with multiple points of the Brownian motion. J. Funct. Anal. 62(3), 397–434 (1985)
Dynkin, E.B.: Local times and quantum fields, In: Seminar on Stochastic Processes, 1983 (Gainesville, FL, 1983), Progr. Probab. Statist., vol. 7, pp. 69– 83. Birkhäuser Boston, Boston, MA, (1984)
Dynkinm E.B.: Polynomials of the occupation field and related random fields. J. Funct. Anal. 58(1), 20–52 (1984)
Dynkin E.B.: Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55(3), 344–376 (1984)
Dynkin, E.B.: Gaussian random fields and Gaussian evolutions, In: Theory and Application of Random Fields (Bangalore, 1982), Lecture Notes in Control and Inform. Sci., vol. 49, pp. 28–39. Springer, Berlin (1983)
Dynkin E.B.: Markov processes as a tool in field theory. J. Funct. Anal. 50(2), 167–187 (1983)
Dynkin E.B.: Markov processes, random fields and Dirichlet spaces. Phys. Rep. 77(3), 239–247 (1981)
Dynkin E.B.: Markov processes and random fields. Bull. Am. Math. Soc. (N.S.) 3(3), 975–999 (1980)
Dvoretzky A., Erdős P., Kakutani S.: Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3, 364–371 (1954)
Dvoretzky A., Erdős P., Kakutani S.: Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged 12, 75–81 (1950)
Dvoretzky A., Erdős P., Kakutani S., Taylor S.J.: Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 53, 856–862 (1957)
Evans S.N. Potential theory for a family of several Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 23(3), 499–530 (English, with French summary (1987a))
Evans SN.: Multiple points in the sample paths of a Lévy process. Probab. Theory Relat. Fields 76(3), 359–367 (1987b)
Farkas W., Jacob N., Schilling R.L.: Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces. Dissertationes Math. (Rozprawy Mat.) 393, 62 (2001)
Farkas W., Leopold H.-G.: Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl. (4) 185(1), 1–62 (2006)
Felder G., Fröhlich J.: Intersection properties of simple random walks: a renormalization group approach. Comm. Math. Phys. 97(1-2), 111–124 (1985)
Fitzsimmons P.J., Salisbury T.S.: Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 25(3), 325–350 (1989) (English, with French summary)
Fristedt, B.: Sample Functions of Stochastic Processes with Stationary, Independent Increments, In: Advances in Probability and Related Topics, vol. 3, pp. 241–396. Dekker, New York (1974)
Fukushima M., Ōshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes, vol. 19. Walter de Gruyter & Co., Berlin (1994)
Getoor R.K.: Excessive Measures. Birkhäuser Boston Inc., Boston, MA (1990)
Hawkes, J.: Some geometric aspects of potential theory, Stochastic analysis and applications (Swansea, 1983), Lecture Notes in Math., 1095, Springer, Berlin, pp 130–154 (1984)
Hawkes J.: Potential theory of Lévy processes. Proc. London Math. Soc. (3) 38(2), 335–352 (1979)
Hawkes J.: Image and intersection sets for subordinators. J. London Math. Soc. (2) 17(3), 567–576 (1978a)
Hawkes J.: Multiple points for symmetric Lévy processes. Math. Proc. Cambridge Philos. Soc. 83(1), 83–90 (1978b)
Hawkes J.: Local properties of some Gaussian processes. Z. Wahrsch. Verw. Gebiete 40(4), 309–315 (1977)
Hawkes J.: Intersections of Markov random sets. Z. Wahrsch. Verw. Gebiete 37(3), 243–251 (1976/77)
Hendricks W.J.: Multiple points for transient symmetric Lévy processes in R d. Z. Wahrsch. Verw. Gebiete 49(1), 13–21 (1979)
Hendricks W.J.: Multiple points for a process in R 2 with stable components. Z. Wahrsche. Verw. Gebiete 28, 113–128 (1973/74)
Hendricks, W.J., Taylor, S.J.: Concerning some problems about polar sets for processes with stationary independent increments, (1979), unpublished manuscript
Hirsch F.: Potential theory related to some multiparameter processes. Potential Anal. 4(3), 245–267 (1995)
Hirsch, F., Song S.: Multiparameter Markov processes and capacity, In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996), Progr. Probab., vol. 45, pp. 189–200, Birkhäuser, Basel (1999)
Hirsch F., Song S.: Inequalities for Bochner’s subordinates of two-parameter symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 32(5), 589–600 (1996) (English, with English and French summaries)
Hirsch F., Song S.: Markov properties of multiparameter processes and capacities. Probab. Theory Relat. Fields 103(1), 45–71 (1995a)
Hirsch F., Song S.: Symmetric Skorohod topology on n-variable functions and hierarchical Markov properties of n-parameter processes. Probab. Theory Relat. Fields 103(1), 25–43 (1995b)
Hirsch F., Song S.: Une inégalité maximale pour certains processus de Markov à plusieurs paramètres. II. C. R. Acad. Sci. Paris Sér. I Math. 320(7), 867–870 (1995) (French)
Hirsch F., Song S.: Une inégalité maximale pour certains processus de Markov à plusieurs paramètres. I. C. R. Acad. Sci. Paris Sér. I Math. 320(6), 719–722 (1995) (French)
Hirsch F., Song S.: Propriétés de Markov des processus à plusieurs paramètres et capacités. C. R. Acad. Sci. Paris Sér. I Math. 319(5), 483–488 (1994) (French)
Hunt G.A.: Markoff processes and potentials. III. Illinois J. Math. 2, 151–213 (1958)
Hunt G.A.: Markoff processes and potentials. I, II. Illinois J. Math. 1, 316–369 (1957)
Hunt G.A.: Markoff processes and potentials. I, II. Illinois J. Math. 1, 44–93 (1957)
Hunt G.A.: Markoff processes and potentials. Proc. Nat. Acad. Sci. U.S.A. 42, 414–418 (1956)
Jacob N.: Pseudo Differential Operators and Markov Processes, vol. III. Imperial College Press, London (2005)
Jacob N.: Pseudo Differential Operators & Markov Processes, vol. II. Imperial College Press, London (2002)
Jacob N.: Pseudo-Differential Operators and Markov Processes, vol. I. Imperial College Press, London (2001)
Jacob, N., Schilling, R.L.: Function spaces as Dirichlet spaces (about a paper by W. Maz′ya and J. Nagel). Comment on: “On equivalent standardization of anisotropic functional spaces H μ(R n)” (German) [Beiträge Anal. No. 12 (1978), 7–17], Z. Anal. Anwendungen 24(1), 3–28 (2005)
Kahane J.-P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)
Kakutani S: On Brownian motions in n-space. Proc. Imp. Acad. Tokyo 20, 648–652 (1944)
Kakutani S.: Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20, 706–714 (1944)
Kesten, H.: Hitting Probabilities of Single Points for Processes with Stationary Independent Increments, Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence, RI (1969)
Khoshnevisan D.: Intersections of Brownian motions. Expo. Math. 21(2), 97–114 (2003)
Khoshnevisan D.: Brownian sheet images and Bessel–Riesz capacity. Trans. Am. Math. Soc. 351(7), 2607–2622 (1999)
Khoshnevisan D.: Multiparameter Processes. Springer, New York (2002)
Khoshnevisan D., Shieh N.-R., Xiao Y.: Hausdorff dimension of the contours of symmetric additive Lévy processes. Probab. Theory Relat. Fields 140(1), 129–167 (2008)
Khoshnevisan D., Xiao Y.: Level sets of additive Lévy processes. Ann. Probab. 30(2), 62–100 (2002)
Khoshnevisan, D., Xiao, Y.: Additive Lévy processes: capacity and Hausdorff dimension, In: Proc. of Inter. Conf. on Fractal Geometry and Stochastics III., Progress in Probability, vol. 57, pp. 62–100. Birkhäuser, Basel (2004)
Khoshnevisan D., Xiao Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)
Khoshnevisan D., Xiao Y.: Images of the Brownian sheet. Trans. Am. Math. Soc. 359(7), 3125–3151 (2007)
Khoshnevisan D., Xiao Y., Zhong Y.: Measuring the range of an additive Lévy process. Ann. Probab. 31(2), 1097–1141 (2003)
Lawler G.F.: Intersections of random walks with random sets. Israel J. Math. 65(2), 113–132 (1989)
Lawler G.F.: Intersections of random walks in four dimensions. II . Comm. Math. Phys. 97(4), 583–594 (1985)
Lawler G.F.: The probability of intersection of independent random walks in four dimensions. Comm. Math. Phys. 86(4), 539–554 (1982)
Le Gall, J.-F.: Some properties of planar Brownian motion, École d’Été de Probabilités de Saint-Flour XX—1990, Lecture Notes in Math., vol. 1527, pp. 111–235. Springer, Berlin (1992)
Le Gall J.-F.: Le comportement du mouvement brownien entre les deux instants où il passe par un point double. J. Funct. Anal. 71(2), 246–262 (1987)
Le Gall J.-F., Rosen J.S., Shieh N.-R.: Multiple points of Lévy processes. Ann. Probab. 17(2), 503–515 (1989)
Lévy P.: Le mouvement brownien plan (French). Am. J. Math. 62, 487–550 (1940)
Marcus M.B., Rosen J.: Multiple Wick product chaos processes. J. Theor. Probab. 12(2), 489–522 (1999b)
Marcus, M.B., Rosen, J.: Renormalized self-intersection local times and Wick power chaos processes. Mem. Am. Math. Soc. 142(675), (1999a)
Masja W., Nagel J.: Über äquivalente Normierung der anisotropen Funktionalräume H μ(R n). Beiträge Anal. 12, 7–17 (1978) (German)
Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)
Orey, S.: Polar sets for processes with stationary independent increments, In: Markov Processes and Potential Theory (Proc. Sympos. Math. Res. Center, Madison, Wis., 1967). Wiley, New York, (1967)
Pemantle R., Peres Y., Shapiro J.W.: The trace of spatial Brownian motion is capacity-equivalent to the unit square. Probab. Theory Relat. Fields 106(3), 379–399 (1996)
Peres, Y.: Probability on trees: an introductory climb, In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, pp. 193– 280. Springer, Berlin, (1999)
Peres Y.: Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phys. Théor. 64(3), 339–347 (1996) (English, with English and French summaries)
Peres Y.: Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177(2), 417–434 (1996b)
Ren J.G.: Topologie p-fine sur l’espace de Wiener et théorème des fonctions implicite. Bull. Sci. Math. 114(2), 99–114 (1990) (French, with English summary)
Röckner, M.: General Theory of Dirichlet Forms and Applications, In: Dirichlet forms, Varenna, 1992, Lecture Notes in Math., vol. 1563, pp. 129–193. Springer, Berlin (1993)
Rogers, L.C.G.: Multiple points of Markov processes in a complete metric space, In: Séminaire de Probabilités XXIII, Lecture Notes in Math., vol. 1372, pp. 186– 197. Springer, Berlin (1989)
Rosen J.: Self-intersections of random fields. Ann. Probab. 12(1), 108–119 (1984)
Rosen J.: A local time approach to the self-intersections of Brownian paths in space. Comm. Math. Phys. 88(3), 327–338 (1983)
Salisbury, T.S.: Energy, and intersections of Markov chains, Random discrete structures, Minneapolis, MN, 1993,IMA Vol. Math. Appl., vol. 76, pp. 213–225. Springer, New York (1996)
Salisbury T.S.: A low intensity maximum principle for bi-Brownian motion. Illinois J. Math. 36(1), 1–14 (1992)
Salisbury, T.S.: Brownian bitransforms, Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987), Progr. Probab. Statist., vol. 15, pp. 249–263. Birkhäuser Boston, Boston, MA (1988)
Sato, Ken-iti.: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge (1999) Translated from the 1990 Japanese original, Revised by the author
Schoenberg I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44, 522–536 (1938)
Slobodeckiĭ, L.N., S.L.: Sobolev’s spaces of fractional order and their application to boundary problems for partial differential equations, Russian, Dokl. Akad. Nauk SSSR (N.S.), 118, 243–246 (1958)
Tongring N.: Which sets contain multiple points of Brownian motion?. Math. Proc. Cambridge Philos. Soc. 103(1), 181–187 (1988)
Ville J.: Sur un problème de géométrie suggéré par l’étude du mouvement brownien. C. R. Acad. Sci Paris 215, 51–52 (1942) (French)
Walsh, J.B.: Martingales with a Multidimensional Parameter and Stochastic Integrals in the Plane, Lectures in Probability and Statistics, Santiago de Chile, 1986, Lecture Notes in Math., vol. 1215, pp. 329–491. Springer, Berlin (1986)
Westwater J.: On Edwards’ model for long polymer chains. Comm. Math. Phys. 72(2), 131–174 (1980)
Westwater J.: On Edwards’ model for polymer chains. II. The self-consistent potential. Comm. Math. Phys. 79(1), 53–73 (1981)
Westwater J.: On Edwards’ model for polymer chains. III. Borel summability. Comm. Math. Phys. 84(4), 459–470 (1982)
Wolpert R.L.: Local time and a particle picture for Euclidean field theory. J. Funct. Anal. 30(3), 341–357 (1978)
Yang M.: Hausdorff dimension of the image of additive processes. Stoch. Process Appl. 118(4), 681–702 (2008)
Yang M.: On a theorem in multi-parameter potential theory. Electron. Comm. Probab. 12, 267–275 (2007) (electronic)
Yang M.: A short proof of the dimension formula for Lévy processes. Electron. Comm. Probab. 11, 217–219 (2006) (electronic)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by a grant from the National Science Foundation (DMS-0706728).
Rights and permissions
About this article
Cite this article
Khoshnevisan, D., Xiao, Y. Harmonic analysis of additive Lévy processes. Probab. Theory Relat. Fields 145, 459–515 (2009). https://doi.org/10.1007/s00440-008-0175-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-008-0175-5