Derivative of the Donsker delta functionals

Herry Pribawanto Suryawan

Mathematica Bohemica (2019)

  • Volume: 144, Issue: 2, page 161-176
  • ISSN: 0862-7959

Abstract

top
We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.

How to cite

top

Suryawan, Herry Pribawanto. "Derivative of the Donsker delta functionals." Mathematica Bohemica 144.2 (2019): 161-176. <http://eudml.org/doc/294432>.

@article{Suryawan2019,
abstract = {We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.},
author = {Suryawan, Herry Pribawanto},
journal = {Mathematica Bohemica},
keywords = {Donsker delta functional; white noise analysis; distributional derivative},
language = {eng},
number = {2},
pages = {161-176},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Derivative of the Donsker delta functionals},
url = {http://eudml.org/doc/294432},
volume = {144},
year = {2019},
}

TY - JOUR
AU - Suryawan, Herry Pribawanto
TI - Derivative of the Donsker delta functionals
JO - Mathematica Bohemica
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 144
IS - 2
SP - 161
EP - 176
AB - We prove that derivatives of any finite order of Donsker's delta functionals are well-defined elements in the space of Hida distributions. We also show the convergence to the derivative of Donsker's delta functionals of two different approximations. Finally, we present an existence result of finite product and infinite series of the derivative of the Donsker delta functionals.
LA - eng
KW - Donsker delta functional; white noise analysis; distributional derivative
UR - http://eudml.org/doc/294432
ER -

References

top
  1. Aase, K., Ø{k}sendal, B., Ubø{e}, J., 10.1023/A:1011259820029, Potential Anal. 14 (2001), 351-374. (2001) Zbl0993.91022MR1825691DOI10.1023/A:1011259820029
  2. Benth, F. E., Ng, S.-A., 10.1112/S0024610702003319, J. Lond. Math. Soc., II. Ser. 66 (2002), 1-13. (2002) Zbl1013.60050MR1911216DOI10.1112/S0024610702003319
  3. Bock, W., Silva, J. L. da, Suryawan, H. P., 10.1142/S0219025716500260, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 19 (2016), Article ID 1650026, 16 pages. (2016) Zbl06676046MR3584626DOI10.1142/S0219025716500260
  4. Cesarano, C., 10.15672/HJMS.20154610029, Hacet. J. Math. Stat. 44 (2015), 535-546. (2015) Zbl1336.33021MR3410856DOI10.15672/HJMS.20154610029
  5. Draouil, O., Ø{k}sendal, B., 10.1007/s40304-015-0065-y, Commun. Math. Stat. 3 (2015), 365-421 erratum ibid. 3 2015 535-540. (2015) Zbl1341.49029MR3397146DOI10.1007/s40304-015-0065-y
  6. Gradshteyn, I. S., Ryzhik, I. M., 10.1016/c2010-0-64839-5, Elsevier/Academic Press, Amsterdam (2015). (2015) Zbl1300.65001MR3307944DOI10.1016/c2010-0-64839-5
  7. Grothaus, M., Riemann, F., Suryawan, H. P., 10.1063/1.4862744, J. Math. Phys. 55 (2014), Article ID 013507, 16 pages. (2014) Zbl1291.82128MR3390439DOI10.1063/1.4862744
  8. Hida, T., Kuo, H.-H., Potthoff, J., Streit, L., 10.1007/978-94-017-3680-0, An Infinite-Dimensional Calculus. Mathematics and Its Applications 253. Kluwer Academic Publishers, Dordrecht (1993). (1993) Zbl0771.60048MR1244577DOI10.1007/978-94-017-3680-0
  9. Hu, Y., Watanabe, S., 10.1215/kjm/1250518506, J. Math. Kyoto Univ. 36 (1996), 499-518. (1996) Zbl0883.60056MR1417823DOI10.1215/kjm/1250518506
  10. Hytönen, T., Neerven, J. van, Veraar, M., Weis, L., 10.1007/978-3-319-48520-1, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 63. Springer, Cham (2016). (2016) Zbl1366.46001MR3617205DOI10.1007/978-3-319-48520-1
  11. Kondratiev, Y. G., Leukert, P., Potthoff, J., Streit, L., Westerkamp, W., 10.1006/jfan.1996.0130, J. Funct. Anal. 141 (1996), 301-318. (1996) Zbl0871.60033MR1418508DOI10.1006/jfan.1996.0130
  12. Kuo, H.-H., 10.1007/BFb0044690, Theory and Application of Random Fields. Proc. IFIP-WG 7/1 Working Conf., Bangalore, 1982 Lect. Notes Control Inf. Sci. 49. Springer, Berlin (1983), 167-178. (1983) Zbl0522.60071MR0799941DOI10.1007/BFb0044690
  13. Kuo, H.-H., White Noise Distribution Theory, Probability and Stochastics Series. CRC Press, Boca Raton (1996). (1996) Zbl0853.60001MR1387829
  14. Lascheck, A., Leukert, P., Streit, L., Westerkamp, W., More about Donsker's delta function, Soochow J. Math. 20 (1994), 401-418. (1994) Zbl0803.46043MR1292245
  15. Lee, Y.-J., Shih, H.-H., 10.1023/A:1010734611017, Acta Appl. Math. 63 (2000), 219-231. (2000) Zbl0982.60067MR1834220DOI10.1023/A:1010734611017
  16. Obata, N., 10.1007/BFb0073952, Lecture Notes in Mathematics 1577. Springer, Berlin (1994). (1994) Zbl0814.60058MR1301775DOI10.1007/BFb0073952
  17. Rosen, J., 10.1007/978-3-540-31449-3_18, 38th seminar on probability. Lecture Notes in Math. 1857 Springer, Berlin (2005), 263-281 M. Émery et al. (2005) Zbl1063.60110MR2126979DOI10.1007/978-3-540-31449-3_18
  18. Stein, E. M., Shakarchi, R., Complex Analysis, Princeton Lectures in Analysis 2. Princeton University Press, Princeton (2003). (2003) Zbl1020.30001MR1976398
  19. Suryawan, H. P., 10.22342/jims.20.2.136.111-124, J. Indones. Math. Soc. 20 (2014), 111-124. (2014) Zbl1334.60135MR3274070DOI10.22342/jims.20.2.136.111-124
  20. Watanabe, S., 10.1016/B978-0-12-481005-1.50032-6, Stochastic Analysis. Proc. Conf., Haifa, 1991 Academic Press, Boston (1991), 495-502 E. Mayer-Wolf et al. (1991) Zbl0748.60047MR1119846DOI10.1016/B978-0-12-481005-1.50032-6
  21. Watanabe, S., Some refinements of Donsker's delta functions, Stochastic Analysis on Infinite Dimensional Spaces. Proc. U.S.-Japan bilateral seminar, Baton Rouge, 1994 Pitman Res. Notes Math. Ser. 310. Longman Scientific Technical, Harlow (1994), 308-324 H. Kunita et al. (1994) Zbl0820.60036MR1415679

NotesEmbed ?

top

You must be logged in to post comments.