Abstract
We consider the perturbation of parabolic operators of the form ∂ t + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed.
The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.
Similar content being viewed by others
References
Bal G.: Central limits and homogenization in random media. Multiscale Model. Simul. 7(2), 677–702 (2008)
Bal, G.: Homogenization with large spatial random potential, Submitted, 2009. available at http://www.columbia.edu/~gb2030/PAPERS/large-potential-homogenization.pdf, 2009
Bardina X., Jolis M.: Weak convergence to the multiple Stratonovich integral. Stochastic Processes Appl. 90, 277–300 (2000)
Budhiraja A., Kallianpur G.: Two results on multiple Stratonovich integrals. Statistica Sinica 7, 907–922 (1997)
Carmona R.A., Molchanov S.A.: Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108(518), 125 (1994)
Dalang R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E.’s. Electron. J. Probab. 26, 1–29 (1999)
Delgado R., Sanz M.: The Hu-Meyer formula for non deterministic kernels. Stochastics Stochastics Rep. 38, 149–158 (1992)
Figari R., Orlandi E., Papanicolaou G.: Mean field and Gaussian approximation for partial differential equations with random coefficients. SIAM J. Appl. Math. 42, 1069–1077 (1982)
Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. A modeling, white noise functional approach. Prob. Appl., Boston, MA: Birkhäuser, 1996
Hu Y.: Chaos expansion of heat equations with white noise potentials. Potential Anal. 16, 45–66 (2002)
Hu, Y.Z., Meyer, P.-A.: Chaos de Wiener et intégrale de Feynman. In: Séminaire de Probabilités, XXII, Vol. 1321 of Lecture Notes in Math.; Berlin: Springer, pp. 51–71, 1988
Itô, K.: Foundations of stochastic differential equations in infinite-dimensional spaces. Vol. 47 of CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, PA:SIAM, 1984
Johnson G.W., Kallianpur G.: Homogeneous chaos, p-forms, scaling and the feynman integral. Trans. Amer. Math. Soc. 340, 503–548 (1993)
Jolis M.: On a multiple Stratonovich-type integral for some Gaussian processes. J. Theoret. Probab. 19, 121–133 (2006)
Nualart D., Rozovskii B.: Weighted stochastic sobolev spaces and bilinear spdes driven by space-time white noise. J. Funct. Anal. 149, 200–225 (1997)
Nualart D., Zakai M.: Generalized Brownian functionals and the solution to a stochastic partial differential equation. J. Funct. Anal. 84, 279–296 (1989)
Nualart D., Zakai M.: On the relation between the stratonovich and ogawa integrals. Ann. Probab 17, 1536–1540 (1989)
Pardoux E., Piatnitski A.: Homogenization of a singular random one dimensional PDE. GAKUTO Internat. Ser. Math. Sci. Appl. 24, 291–303 (2006)
Reed M., Simon B.: Methods of modern mathematical physics. I. Functional analysis. 2nd ed. Academic Press, Inc., New York (1980)
Solé J.L., Utzet F.: Stratonovich integral and trace. Stochastics Stochastics Rep. 2, 203–220 (1990)
Taylor M.E.: Partial Differential Equations I. Springer Verlag, New York (1997)
Walsh, J.B.: An introduction to stochastic partial differential equations. École d’été de probabilités de Saint-Flour, XIV—1984, Vol. 1180 of Lecture Notes in Math., Berlin: Springer, pp. 265–439, 1986
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H.-T. Yau
Rights and permissions
About this article
Cite this article
Bal, G. Convergence to SPDEs in Stratonovich Form. Commun. Math. Phys. 292, 457–477 (2009). https://doi.org/10.1007/s00220-009-0898-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-009-0898-x