Skip to main content
SpringerLink
Log in

Convolution-Like Structures on Multidimensional Spaces

  • Chapter
  • First Online:
Convolution-like Structures, Differential Operators and Diffusion Processes

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2315))

  • 354 Accesses

Abstract

After having constructed convolution-like operators associated with the Shiryaev process and with a family of one-dimensional diffusion processes generated by Sturm–Liouville operators, we devote this final chapter to a more general discussion of the problem formulated in the Introduction: the construction of a generalized convolution associated with a given Feller process on a generally multidimensional state space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Applebaum, Probability on compact Lie groups (Springer, Cham, 2014)

    Book  MATH  Google Scholar 

  2. F.V. Atkinson, A.B. Mingarelli, Multiparameter Eigenvalue Problems—Sturm–Liouville Theory (CRC Press, Boca Raton, 2011)

    MATH  Google Scholar 

  3. F. Avram, N.N. Leonenko, N. Šuvak, On spectral analysis of heavy-tailed Kolmogorov-Pearson diffusions. Markov Process. Related Fields 19(2), 249–298 (2013)

    MathSciNet  MATH  Google Scholar 

  4. R. Bañuelos, K. Burdzy, On the “hot spots” conjecture of J. Rauch. J. Funct. Anal. 164(1), 1–33 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. M.T. Barlow, R.F. Bass, Z.-Q. Chen, M. Kassmann, Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361, 1963–1999 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. R.F. Bass, P. Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19(2), 486–508 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y.M. Berezansky, A.A. Kalyuzhnyi, Harmonic Analysis in Hypercomplex Systems (Kluwer Academic Publishers, Dordrecht, 1998)

    Book  MATH  Google Scholar 

  8. W.R. Bloom, H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups (Walter de Gruyter, Berlin, 1994)

    MATH  Google Scholar 

  9. V. Bobkov, On exact Pleijel’s constant for some domains. Doc. Math. 23, 799–813 (2018)

    MathSciNet  MATH  Google Scholar 

  10. U. Boscain, D. Prandi, Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces. J. Differ. Equ. 260, 3234–3269 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. P.J. Browne, A singular multi-parameter eigenvalue problem in second order ordinary differential equations. J. Differ. Equ. 12, 81–94 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. P.J. Browne, Abstract multiparameter theory I. J. Math. Anal. Appl. 60(1), 259-273 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Burdzy, The hot spots problem in planar domains with one hole. Duke Math. J. 129(3), 481–502 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, in Geometry of the Laplace Operator, ed. by R. Osserman, A. Weinstein. Proc. Sympos. Pure Math. vol. XXXVI (American Mathematical Society, Providence, 1980), pp. 91–146

    Google Scholar 

  15. Z.-Q. Chen, W.-T. (L.) Fan, Functional central limit theorem for Brownian particles in domains with Robin boundary condition. J. Funct. Anal. 269(12), 3765–3811 (2015)

    Google Scholar 

  16. Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change and Boundary Theory (Princeton University Press, Princeton, 2011)

    Book  MATH  Google Scholar 

  17. Z.-Q. Chen, T. Zhang, A probabilistic approach to mixed boundary value problems for elliptic operators with singular coefficients. Proc. Am. Math. Soc. 142(6), 2135–2149 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Choulli, L. Kayser, E. M. Ouhabaz, Observations on Gaussian upper bounds for Neumann heat kernels. Bull. Aust. Math. Soc. 92, 429–439 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Courant, Methods of Mathematical Physics—Vol. II: Partial Differential Equations (Wiley, New York, 1962)

    Google Scholar 

  20. W. Craig, S. Weinstein, On determinism and well-posedness in multiple time dimensions. Proc. R. Soc. A 465, 3023–3046 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. E.B. Davies, One-parameter Semigroups (Academic Press, London, 1980)

    MATH  Google Scholar 

  22. E.B. Davies, Heat Kernels and Spectral Theory (Cambridge University Press, Cambridge, 1989)

    Book  MATH  Google Scholar 

  23. N. Dunford, J.T. Schwartz, Linear Operators—Part II: Spectral Theory (Wiley, New York, 1963)

    MATH  Google Scholar 

  24. H. Dym, H.P. McKean, Fourier Series and Integrals (Academic Press, New York, 1972)

    MATH  Google Scholar 

  25. M.S.P. Eastham, The Asymptotic Solution of Linear Differential Systems—Applications of the Levinson Theorem (Oxford University Press, Oxford, 1989)

    MATH  Google Scholar 

  26. M. Faierman, The completeness and expansion theorems associated with the multi-parameter eigenvalue problem in ordinary differential equations. J. Differ. Equ. 5, 197–213 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. G.B. Folland, Real Analysis: Modern Techniques and Their Applications (Wiley, New York, 1999)

    MATH  Google Scholar 

  28. G.B. Folland, A Course in Abstract Harmonic Analysis, 2nd edn. (CRC Press, Boca Raton, 2016)

    Book  MATH  Google Scholar 

  29. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes (Walter De Gruyter, Berlin, 2011)

    MATH  Google Scholar 

  30. L. Gårding, Application of the Theory of Direct Integrals of Hilbert Spaces to some Integral and Differential Operators. Lect. Ser. Inst. Fluid Dynam. Appl. Math., vol. 11 (1954)

    Google Scholar 

  31. G. Gasper, Banach algebras for Jacobi series and positivity of a kernel. Ann. Math. 95(2), 261–280 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. R.K. Getoor, Markov operators and their associated semi-groups. Pac. J. Math. 9, 449–472 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  33. H.-J. Glaeske, A.P. Prudnikov, K.A. Skòrnik, Operational Calculus and Related Topics (CRC Press, Boca Raton, 2006)

    MATH  Google Scholar 

  34. D.S. Grebenkov, B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions. SIAM Rev. 55(4), 601–667 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Grigor’yan, Heat Kernel and Analysis on Manifolds (American Mathematical Society, Providence; International Press, Boston, 2009)

    Google Scholar 

  36. A. Grigor’yan, A. Telcs, Two-sided estimates of heat kernels on metric measure spaces. Ann. Probab. 40(3), 1212–1284 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. B. Helffer, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, N. Nadirashvili, Spectral theory for the dihedral group. Geom. Funct. Anal. 12(5), 989–1017 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. B. Helffer, M.P. Sundqvist, On nodal domains in Euclidean balls. Proc. Am. Math. Soc. 144(11), 4777–4791 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations (Cambridge University Press, Cambridge, 2005)

    Book  MATH  Google Scholar 

  40. E.P. Hsu, Estimates of derivatives of the heat kernel on a compact Riemannian manifold. Proc. Am. Math. Soc. 127(12), 3739–3744 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. S.R. Jain, R. Samajdar, Nodal portraits of quantum billiards: domains, lines and statistics. Rev. Mod. Phys. 89, 045005 (2017)

    Article  MathSciNet  Google Scholar 

  42. C. Judge, S. Mondal, Euclidean triangles have no hot spots. Ann. Math. 191(2), 167–211 (2020)

    MathSciNet  MATH  Google Scholar 

  43. A. Klenke, Probability Theory—A Comprehensive Course, 3rd edn. (Springer, London, 2020)

    Book  MATH  Google Scholar 

  44. V. Komornik, P. Loreti, Fourier Series in Control Theory (Springer, New York, 2005)

    Book  MATH  Google Scholar 

  45. S. Kotani, On a generalized Sturm–Liouville operator with a singular boundary. J. Math. Kyoto Univ. 15(2), 423–454 (1975)

    MathSciNet  MATH  Google Scholar 

  46. S. Kotani, Krein’s strings with singular left boundary. Rep. Math. Phys. 59(3), 305–316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  47. V. Linetsky, The spectral decomposition of the option value. Int. J. Theor. Appl. Financ 7(3), 337–384 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. C. Markett, Norm estimates for generalized translation operators associated with a singular differential operator. Indag. Math. 46, 299–313 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  49. M.A.M. Marrocos, A.L. Pereira, Eigenvalues of the Neumann Laplacian in symmetric regions. J. Math. Phys. 56, 111502 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. I. McGillivray, A recurrence condition for some subordinated strongly local Dirichlet forms. Forum Math. 9(2), 229–246 (1997)

    MathSciNet  MATH  Google Scholar 

  51. Y. Miyamoto, The hot spots conjecture for a certain class of planar convex domains. J. Math. Phys. 50(10), 103530 (2009)

    Google Scholar 

  52. M.A. Naimark, Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space (Frederick Ungar Publishing, New York, 1968)

    Google Scholar 

  53. F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010)

    MATH  Google Scholar 

  54. M.N. Pascu, Scaling coupling of reflecting Brownian motions and the hot spots problem. Trans. Am. Math. Soc. 354(11), 4681–4702 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  55. I.G. Petrovsky, Lectures on Partial Differential Equations (Interscience Publishers, 1954)

    Google Scholar 

  56. M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, 2nd edn. (Springer, Berlin, 2004)

    MATH  Google Scholar 

  57. M.Rösler, Convolution algebras which are not necessarily positivity-preserving, in Applications of Hypergroups and Related Measure Algebras—A Joint Summer Research Conference, July 31-August 6, 1993, Seattle WA, ed. by W.C. Connett, M.-O. Gebuhrer, A.L. Schwartz (American Mathematical Society, Providence, 1995), pp. 299–318

    Google Scholar 

  58. K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space (Springer, Dordrecht, 2012)

    Book  MATH  Google Scholar 

  59. G.F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, New York, 1963)

    MATH  Google Scholar 

  60. B. Simon, Operator Theory—A Comprehensive Course in Analysis, Part 4 (American Mathematical Society, Providence, 2015)

    MATH  Google Scholar 

  61. B.D. Sleeman, Multiparameter spectral theory and separation of variables. J. Phys. A 41(1), 015209 (2008)

    Google Scholar 

  62. R. Sousa, M. Guerra, S. Yakubovich, Product formulas and convolutions for Laplace–Beltrami operators on product spaces: beyond the trivial case (2020). Preprint, arXiv:2006.14522. [Submitted for publication on a peer-reviewed journal]

    Google Scholar 

  63. K.-T. Sturm, How to construct diffusion processes on metric spaces. Potential Anal. 8(2), 149–161 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  64. K.-T. Sturm, Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26(1), 1–55 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  65. G. Teschl, Mathematical methods in quantum mechanics. With applications to Schrödinger operators, 2nd edn. (American Mathematical Society, Providence, 2014)

    Google Scholar 

  66. K. Uhlenbeck, Generic properties of eigenfunctions. Am. J. Math. 98(4), 1059–1078 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  67. V.E. Volkovich, Infinitely divisible distributions in algebras with stochastic convolution. J. Sov. Math. 40(4), 459–467 (1988)

    Article  MathSciNet  Google Scholar 

  68. V.E. Volkovich, Quasiregular stochastic convolutions. J. Sov. Math. 47(5), 2685–2699 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  69. F.-Y. Wang, L. Yan, Gradient estimate on convex domains and applications. Proc. Am. Math. Soc. 141(3), 1067–1081 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  70. G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edn. (Cambridge University Press, Cambridge, 1944)

    MATH  Google Scholar 

  71. J. Wloka, Partial Differential Equations (Cambridge University Press, Cambridge, 1987)

    Book  MATH  Google Scholar 

  72. H. Zeuner, Domains of attraction with inner norming on Sturm–Liouville hypergroups. J. Appl. Anal. 1(2), 213–221 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, University of Porto, Porto, Portugal

    Rúben Sousa & Semyon Yakubovich

  2. Department of Mathematics, ISEG, Universidade de Lisboa, Lisbon, Portugal

    Manuel Guerra

Authors
  1. Rúben Sousa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel Guerra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Semyon Yakubovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Sousa, R., Guerra, M., Yakubovich, S. (2022). Convolution-Like Structures on Multidimensional Spaces. In: Convolution-like Structures, Differential Operators and Diffusion Processes. Lecture Notes in Mathematics, vol 2315. Springer, Cham. https://doi.org/10.1007/978-3-031-05296-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation