Annals of Probability

Variations of the solution to a stochastic heat equation

Jason Swanson

Full-text: Open access


We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, F(t), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of t, converge weakly to Brownian motion.

Article information

Ann. Probab., Volume 35, Number 6 (2007), 2122-2159.

First available in Project Euclid: 8 October 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 60H05: Stochastic integrals 60H15: Stochastic partial differential equations [See also 35R60]

Quartic variation quadratic variation stochastic partial differential equations stochastic integration long-range dependence iterated Brownian motion fractional Brownian motion self-similar processes


Swanson, Jason. Variations of the solution to a stochastic heat equation. Ann. Probab. 35 (2007), no. 6, 2122--2159. doi:10.1214/009117907000000196.

Export citation


  • Bulinski, A. V. (1996). On the convergence rates in the CLT for positively and negatively dependent random fields. In Probability Theory and Mathematical Statistics (I. A. Ibragimov and A. Yu. Zaitsev, eds.) 3--14. Gordon and Breach, Amsterdam.
  • Cheridito, P. and Nualart, D. (2005). Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter $H\in(0,\frac12)$. Ann. Inst. H. Poincaré Probab. Statist. 41 1049--1081.
  • Decreusefond, L. (2003). Stochastic integration with respect to fractional Brownian motion. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. S. Taqqu, eds.) 203--226. Birkhäuser, Boston.
  • Denis, L. (2004). Solutions of stochastic partial differential equations considered as Dirichlet processes. Bernoulli 10 783--827.
  • Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
  • Gradinaru, M., Nourdin, I., Russo, F. and Vallois, P. (2005). $m$-order integrals and generalized Itô's formula: The case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 781--806.
  • Gradinaru, M., Russo, F. and Vallois, P. (2003). Generalized covariations, local time and Stratonovich Itô's formula for fractional Brownian motion with Hurst index $H\ge \frac14$. Ann. Probab. 31 1772--1820.
  • Herrndorf, N. (1984). A functional central limit theorem for weakly dependent sequences of random variables. Ann. Probab. 12 141--153.
  • Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267--307.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Krylov, N. V. (1999). An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives (R. A. Carmona and B. Rozovskii, eds.) 185--242. Amer. Math. Soc., Providence, RI.
  • Nualart, D. and Ortiz, S. (2006). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Preprint.
  • Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403--421.
  • Russo, F. and Vallois, P. (1995). The generalized covariation process and Itô formula. Stochastic Process. Appl. 59 81--104.