SPDEs with coloured noise : analytic and stochastic approaches
ESAIM: Probability and Statistics, Volume 10 (2006), pp. 380-405.

We study strictly parabolic stochastic partial differential equations on ${ℝ}^{d}$, $d\ge 1$, driven by a gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving path properties of the solution to a parabolic Cauchy problem in evolution form.

DOI: 10.1051/ps:2006016
Classification: 60H15,  60H25,  35R60
Keywords: stochastic partial differential equations, mild and weak solutions, random noise
@article{PS_2006__10__380_0,
author = {Ferrante, Marco and Sanz-Sol\'e, Marta},
title = {SPDEs with coloured noise : analytic and stochastic approaches},
journal = {ESAIM: Probability and Statistics},
pages = {380--405},
publisher = {EDP-Sciences},
volume = {10},
year = {2006},
doi = {10.1051/ps:2006016},
mrnumber = {2263072},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2006016/}
}
TY  - JOUR
AU  - Ferrante, Marco
AU  - Sanz-Solé, Marta
TI  - SPDEs with coloured noise : analytic and stochastic approaches
JO  - ESAIM: Probability and Statistics
PY  - 2006
DA  - 2006///
SP  - 380
EP  - 405
VL  - 10
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2006016/
UR  - https://www.ams.org/mathscinet-getitem?mr=2263072
UR  - https://doi.org/10.1051/ps:2006016
DO  - 10.1051/ps:2006016
LA  - en
ID  - PS_2006__10__380_0
ER  - 
%0 Journal Article
%A Ferrante, Marco
%A Sanz-Solé, Marta
%T SPDEs with coloured noise : analytic and stochastic approaches
%J ESAIM: Probability and Statistics
%D 2006
%P 380-405
%V 10
%I EDP-Sciences
%U https://doi.org/10.1051/ps:2006016
%R 10.1051/ps:2006016
%G en
%F PS_2006__10__380_0
Ferrante, Marco; Sanz-Solé, Marta. SPDEs with coloured noise : analytic and stochastic approaches. ESAIM: Probability and Statistics, Volume 10 (2006), pp. 380-405. doi : 10.1051/ps:2006016. http://www.numdam.org/articles/10.1051/ps:2006016/

[1] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions. National Bureau of Standards (1964).

[2] R.A. Adams, Sobolev spaces. Academic Press, New York-London (1975). | MR | Zbl

[3] E. Alòs, J.A. León and D. Nualart, Stochastic heat equation with random coefficients. Probab. Theory Related Fields 115 (1999) 41-94. | Zbl

[4] R.C. Dalang and N.E. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187-212. | Zbl

[5] R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999) 1-29. | EuDML | Zbl

[6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd Edition. Cambridge University Press (1998). | MR | Zbl

[7] W.F. Donoghue, Distributions and Fourier transforms. Academic Press, New York (1969). | Zbl

[8] S.D. Eidelman and N.V. Zhitarashu, Parabolic Boundary Value Problems. Birkhäuser Verlag, Basel (1998). | MR | Zbl

[9] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964). | MR | Zbl

[10] I.M. Gel'Fand and N.Ya. Vilenkin, Generalized functions. Vol. 4: Applications of harmonic analysis. Academic Press, New York (1964). | Zbl

[11] M.A Krasnoselskii, E.I. Pustylnik, P.E. Sobolevski and P.P. Zabrejko, Integral operators in spaces of summable functions. Noordhoff International Publishing, Leyden (1976). | MR | Zbl

[12] A.A. Kirillov and A.D. Gvishiani, Theorems and problems in functional analysis. Springer-Verlag, New York-Berlin (1982). | MR | Zbl

[13] N.V. Krylov and B.L. Rozovsky, Stochastic evolution systems. Russian Math. Surveys 37 (1982) 81-105. | Zbl

[14] N.V. Krylov, On ${L}_{p}$-theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27 (1996) 313-340. | Zbl

[15] N.V. Krylov, An analytic approach to SPDEs, in Stochastic partial differential equations: six perspectives, Math. Surveys Monogr. 64, American Mathematical Society, Providence (1999) 185-242. | Zbl

[16] N.V. Krylov and V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line. SIAM J. Math. Anal. 30 (1998) 298-325. | Zbl

[17] O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural'Ceva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23, American Mathematical Society (1968). | Zbl

[18] O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse 2452, Lausanne, EPFL (2001).

[19] R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28 (2000) 74-103. | Zbl

[20] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 (1979) 127-167. | Zbl

[21] B.L. Rozovsky, Stochastic evolution equations. Linear theory and applications to non-linear filtering. Kluwer (1990). | Zbl

[22] L. Schwartz, Théorie des distributions. Hermann, Paris (1966). | MR | Zbl

[23] M. Sanz-Solé and M. Sarrà, Path properties of a class of Gaussian processes with applications to spde's. Canadian Mathematical Society Conference Proceedings 28 (2000) 303-316. | Zbl

[24] M. Sanz-Solé and M. Sarrà, Hölder Continuity for the stochastic heat equation with spatially correlated noise, in Progress in Probability 52, Birkhäuser Verlag (2002) 259-268.

[25] M. Sanz-Solé and P.-A. Vuillermot, Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 703-742. | EuDML | Numdam | Zbl

[26] N. Shimakura, Partial differential operators of elliptic type. American Mathematical Society, Providence (1992). | MR | Zbl

[27] H. Triebel, Theory of function spaces. II. Monographs in Mathematics 84, Birkhäuser Verlag, Basel (1992). | MR | Zbl

[28] J.B. Walsh, An Introduction to Stochastic Partial Differential Equations, in École d'été de Probabilités de Saint-Flour XIV (1984). Lect. Notes Math. 1180 (1986) 265-439. | Zbl

Cited by Sources: