## Electronic Journal of Probability

### An Itô type formula for the additive stochastic heat equation

Carlo Bellingeri

#### 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
Accepted: 15 December 2019
First available in Project Euclid: 7 January 2020

https://projecteuclid.org/euclid.ejp/1578366206

Digital Object Identifier
doi:10.1214/19-EJP404

#### 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

#### 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.