Electronic Journal of Probability

An Itô type formula for the additive stochastic heat equation

Carlo Bellingeri

Full-text: Open access

Abstract

We use the theory of regularity structures to develop an Itô formula for $u$, the solution of the one-dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular, for any smooth enough function $\varphi $ we can express the random distribution $(\partial _{t}-\partial _{xx})\varphi (u)$ and the random field $\varphi (u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of Malliavin calculus.

Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 2, 52 pp.

Dates
Received: 7 March 2018
Accepted: 15 December 2019
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1578366206

Digital Object Identifier
doi:10.1214/19-EJP404

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
stochastic partial differential equations Itô formula Malliavin calculus regularity structures

Rights
Creative Commons Attribution 4.0 International License.

Citation

Bellingeri, Carlo. An Itô type formula for the additive stochastic heat equation. Electron. J. Probab. 25 (2020), paper no. 2, 52 pp. doi:10.1214/19-EJP404. https://projecteuclid.org/euclid.ejp/1578366206


Export citation

References

  • [1] Christian Bayer, Peter K. Friz, Paul Gassiat, Jorg Martin, and Benjamin Stemper, A regularity structure for rough volatility, Mathematical Finance (2019), 1–51.
  • [2] Y. Bruned, M. Hairer, and L. Zambotti, Algebraic renormalisation of regularity structures, Inventiones mathematicae 215 (2019), no. 3, 1039–1156.
  • [3] Yvain Bruned, Singular KPZ type equations, Theses, Université Pierre et Marie Curie - Paris VI, 2015.
  • [4] Yvain Bruned, Recursive formulae in regularity structures, Stochastic Partial Differential Equations. Analysis and Computations 6 (2018), no. 4, 525–564.
  • [5] Yvain Bruned, Ajay Chandra, Ilya Chevyrev, and Martin Hairer, Renormalising SPDEs in regularity structures, arXiv e-prints (2017), 1–85.
  • [6] Yvain Bruned, Franck Gabriel, Martin Hairer, and Lorenzo Zambotti, Geometric stochastic heat equations, arXiv e-prints (2019), 1–83.
  • [7] R. Cairoli and John B. Walsh, Stochastic integrals in the plane, Acta Mathematica 134 (1975), 111–183.
  • [8] Ajay Chandra and Martin Hairer, An analytic BPHZ theorem for regularity structures, arXiv e-prints (2016), 1–129.
  • [9] Giuseppe Da Prato and Jerzy Zabczyk, Stochastic Equations in Infinite Dimensions, 2 ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2014.
  • [10] Franco Flandoli and Francesco Russo, Generalized integration and stochastic ODEs, Annals of Probability 30 (2002), no. 1, 270–292.
  • [11] Tadahisa Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Mathematical Journal 89 (1983), 129–193.
  • [12] Donald Geman and Joseph Horowitz, Occupation densities, Annals of Probability 8 (1980), no. 1, 1–67.
  • [13] Máté Gerencsér and Martin Hairer, Singular SPDEs in domains with boundaries, Probability Theory and Related Fields 173 (2019), no. 3, 697–758.
  • [14] Mihai Gradinaru, Ivan Nourdin, and Samy Tindel, Ito’s- and Tanaka’s-type formulae for the stochastic heat equation: The linear case, Journal of Functional Analysis 228 (2005), 114–143.
  • [15] M. Hairer, A theory of regularity structures, Inventiones Mathematicae 198 (2014), no. 2, 269–504.
  • [16] Martin Hairer, Introduction to regularity structures, Brazilian Journal of Probability and Statistics 29 (2015), no. 2, 175–210.
  • [17] Martin Hairer, Regularity structures and the dynamical $\phi ^{4}_{3}$ model, arXiv e-prints (2015), 1–46.
  • [18] Martin Hairer, The motion of a random string, arXiv e-prints (2016), 1–20.
  • [19] Martin Hairer and Étienne Pardoux, A Wong-Zakai theorem for stochastic PDEs, Journal of the Mathematical Society of Japan 67 (2015), no. 4, 1551–1604.
  • [20] N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder spaces, Graduate Studies in Mathematics, American Mathematical Society, 1996.
  • [21] Alberto Lanconelli, On a new version of the Itô’s formula for the stochastic heat equation, Communications on Stochastic Analysis 1 (2007), no. 2, 311–320.
  • [22] C. Mueller and R. Tribe, Hitting properties of a random string, Electronic Journal of Probability 7 (2002), no. 10, 29.
  • [23] Ivan Nourdin and David Nualart, Central limit theorems for multiple Skorokhod integrals, Journal of Theoretical Probability 23 (2010), no. 1, 39–64.
  • [24] David Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, Springer-Verlag, 1995.
  • [25] David Nualart and Moshe Zakai, Generalized multiple stochastic integrals and the representation of Wiener functionals, Stochastics 23 (1998), no. 3, 311–330.
  • [26] Giovanni Peccati and Murad Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams, Springer, 2011.
  • [27] Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion, Grundlehren der mathematischen Wissenschaften, Springer Berlin Heidelberg, 2004.
  • [28] Francesco Russo and Pierre Vallois, Forward, backward and symmetric stochastic integration, Probability Theory and Related Fields 97 (1993), no. 3, 403–421.
  • [29] Leon Simon, Schauder estimates by scaling, Calculus of Variations and Partial Differential Equations 5 (1997), no. 5, 391–407.
  • [30] Jason Swanson, Variations of the solution to a stochastic heat equation, Annals of Probability 35 (2007), no. 6, 2122–2159.
  • [31] John B. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1984.
  • [32] Lorenzo Zambotti, Itô-Tanaka’s formula for stochastic partial differential equations driven by additive space-time white noise, Stochastic Partial Differential Equations and Applications - VII 245 (2006), 337–347.