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Examples of Renormalized SDEs

  • Y. Bruned
  • I. Chevyrev
  • P. K. FrizEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a lead-lag process of discretised fractional Brownian motion with Hurst parameter \(H \in (1/4,1/2)\), in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures.

Keywords

Renormalization Rough paths Gaussian analysis 

Subject classification (MSC 2010)

Primary 60H99 Secondary 60H10 

Notes

Acknowledgements

P.K.F. is partially supported by the European Research Council through CoG-683164 and DFG research unit FOR2402. I.C., affiliated to TU Berlin when this project was commenced, was supported by DFG research unit FOR2402, and is currently supported by a Junior Research Fellowship of St John’s College, Oxford. Y.B. thanks Martin Hairer for financial support from his Leverhulme Trust award.

References

  1. 1.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014).  https://doi.org/10.1007/s00222-014-0505-4
  2. 2.
    Bruned, Y., Hairer, M., Zambotti, L.: Algebraic renormalisation of regularity structures. ArXiv e-prints (2016)Google Scholar
  3. 3.
    Hairer, M.: The motion of a random string. ArXiv e-prints (2016)Google Scholar
  4. 4.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6, 75 (2015).  https://doi.org/10.1017/fmp.2015.2
  5. 5.
    Kupiainen, A.: Renormalization group and stochastic PDEs. Ann. Henri Poincaré 17(3), 497–535 (2016).  https://doi.org/10.1007/s00023-015-0408-y
  6. 6.
    Lyons, T.J.: Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998).  https://doi.org/10.4171/RMI/240
  7. 7.
    Friz, P.K., Victoir, N.B.: Cambridge studies in advanced mathematics. In: Multidimensional stochastic processes as rough paths, vol. 120. Cambridge University Press, Cambridge (2010)Google Scholar
  8. 8.
    Friz, P., Gassiat, P., Lyons, T.: Physical Brownian motion in a magnetic field as a rough path. Trans. Am. Math. Soc. 367(11), 7939–7955 (2015).  https://doi.org/10.1090/S0002-9947-2015-06272-2
  9. 9.
    Flint, G., Hambly, B., Lyons, T.: Discretely sampled signals and the rough Hoff process. Stoch. Process. Appl. 126(9), 2593–2614 (2016).  https://doi.org/10.1016/j.spa.2016.02.011
  10. 10.
    Cannizzaro, G., Friz, P.K., Gassiat, P.: Malliavin calculus for regularity structures: the case of gPAM. J. Funct. Anal. 272(1), 363–419 (2017).  https://doi.org/10.1016/j.jfa.2016.09.024
  11. 11.
    Bruned, Y., Chevyrev, I., Friz, P.K., Preiss, R.: A rough path perspective on renormalization. ArXiv e-prints (2017)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonEngland
  2. 2.Mathematical InstituteUniversity of OxfordOxfordEngland
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany
  4. 4.Weierstraß–Institut für Angewandte Analysis und StochastikBerlinGermany

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