Spatial Quadratic Variation for Stochastic Heat Equations Driven by Multiplicative Noise with Piecewise Constant Coefficients

Abstract

In this paper, we study a stochastic partial differential equation (SPDE) with piecewise constant coefficients, driven by a nonlinear Gaussian space-time white noise. We investigate the asymptotic behavior of the spatial quadratic variation and, as an application, propose an estimator derived from the limiting results.

Keywords

Quadratic variations / stochastic partial differential equations / parameter estimation / discontinuity of coefficients / space-time white noise / 60H15 / 60G15 / 60H05 / 60H30 / 35A08

References

[1] Agram N., Turpin I., Zougar E., Spatially controlled evolution of composite materials via stochastic partial differential equations. 2025, arXiv:2502.15351

[2] Chen Z-Q, Zili M. One-dimensional heat equation with discontinuous conductance. Sci. China Math., 2015, 58(1): 97-108

[3] Dalang R., Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab., 1999, 4: Paper No. 6

[4] Hildebrandt F, Trabs M. Parameter estimation for SPDEs based on discrete observations in time and space. Electron. J. Stat., 2021, 15(1): 2716-2776

[5] Li Y., Shu H., Yan L., Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients. 2024, arXiv:2410.22750

[6] Mishura Y., Ralchenko K., Zili M., On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity. Statist. Probab. Lett., 2020, 159: Paper No. 108682, 9 pp.

[7] Mishura Y., Ralchenko K., Zili M., Zougar E., Fractional stochastic heat equation with piecewise constant coefficients. Stoch. Dyn., 2021, 21(1): Paper No. 2150002, 39 pp.

[8] Pospíšil J, Tribe R. Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise. Stoch. Anal. Appl., 2007, 25(3): 593-611

[9] Walsh J. An Introduction to Stochastic Partial Differential Equations, 1986, Berlin, Springer1180

[10] Zili M. Développement asymptotique en temps petits de la solution d’une équation aux dérivées partielles de type parabolique. C. R. Acad. Sci. Paris Sér. I Math., 1995, 321(8): 1049-1052

[11] Zili M. Construction d’une solution fondamentale d’une équation aux dérivées partielles à coefficients constants par morceaux. Bull. Sci. Math., 1999, 123(2): 115-155

[12] Zili M, Zougar E. One-dimensional stochastic heat equation with discontinuous conductance. Appl. Anal., 2019, 98(12): 2178-2191

[13] Zili M, Zougar E. Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation. Teor. Ĭmovīr. Mat. Stat., 2019, 100: 75-101

[14] Zili M, Zougar E. Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator. Mod. Stoch. Theory Appl., 2019, 6(3): 345-375

[15] Zili M, Zougar E. Stochastic heat equation with piecewise constant coefficients and generalized fractional type noise. Theory Probab. Math. Statist., 2021, 2021(104): 123-144

RIGHTS & PERMISSIONS

Peking University