We study strictly parabolic stochastic partial differential equations on , , 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.
Mots clés : 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 SP - 380 EP - 405 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/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 http://www.numdam.org/articles/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, Tome 10 (2006), pp. 380-405. doi : 10.1051/ps:2006016. http://www.numdam.org/articles/10.1051/ps:2006016/
[1] and, Handbook of mathematical functions. National Bureau of Standards (1964).
[2] , Sobolev spaces. Academic Press, New York-London (1975). | MR | Zbl
[3] , and, Stochastic heat equation with random coefficients. Probab. Theory Related Fields 115 (1999) 41-94. | Zbl
[4] and, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998) 187-212. | Zbl
[5] , 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] and, Stochastic Equations in Infinite Dimensions, 2nd Edition. Cambridge University Press (1998). | MR | Zbl
[7] , Distributions and Fourier transforms. Academic Press, New York (1969). | Zbl
[8] and, Parabolic Boundary Value Problems. Birkhäuser Verlag, Basel (1998). | MR | Zbl
[9] , Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964). | MR | Zbl
[10] and, Generalized functions. Vol. 4: Applications of harmonic analysis. Academic Press, New York (1964). | Zbl
[11] ,, and, Integral operators in spaces of summable functions. Noordhoff International Publishing, Leyden (1976). | MR | Zbl
[12] and, Theorems and problems in functional analysis. Springer-Verlag, New York-Berlin (1982). | MR | Zbl
[13] and, Stochastic evolution systems. Russian Math. Surveys 37 (1982) 81-105. | Zbl
[14] , On -theory of stochastic partial differential equations in the whole space. SIAM J. Math. Anal. 27 (1996) 313-340. | Zbl
[15] , 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] and, A Sobolev space theory of SPDEs with constant coefficients on a half line. SIAM J. Math. Anal. 30 (1998) 298-325. | Zbl
[17] , and, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23, American Mathematical Society (1968). | Zbl
[18] , Hyperbolic stochastic partial differential equations driven by boundary noises. Thèse 2452, Lausanne, EPFL (2001).
[19] , On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28 (2000) 74-103. | Zbl
[20] , Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 (1979) 127-167. | Zbl
[21] , Stochastic evolution equations. Linear theory and applications to non-linear filtering. Kluwer (1990). | Zbl
[22] , Théorie des distributions. Hermann, Paris (1966). | MR | Zbl
[23] and, Path properties of a class of Gaussian processes with applications to spde's. Canadian Mathematical Society Conference Proceedings 28 (2000) 303-316. | Zbl
[24] and, Hölder Continuity for the stochastic heat equation with spatially correlated noise, in Progress in Probability 52, Birkhäuser Verlag (2002) 259-268.
[25] and, 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] , Partial differential operators of elliptic type. American Mathematical Society, Providence (1992). | MR | Zbl
[27] , Theory of function spaces. II. Monographs in Mathematics 84, Birkhäuser Verlag, Basel (1992). | MR | Zbl
[28] , 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
Cité par Sources :
