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Stochastic evolution equations with a spatially homogeneous Wiener process. (English) Zbl 0943.60048
A Cauchy problem for a stochastic parabolic equation \[ \frac {\partial X}{\partial t} = AX + f(t,x,X) + b(t,x,X) \dot W, \quad X(0)=\zeta , \tag{1} \] in \(\mathbb{R}^{d}\) is considered. It is assumed that \(A\) is a uniformly elliptic \(2m\)th order differential operator with bounded smooth coefficients, and the functions \(f(t,x,\cdot)\), \(b(t,x,\cdot)\) satisfy the Lipschitz and linear growth conditions uniformly in \((t,x)\). \(W\) denotes a Wiener process in the space \(\mathcal S'\) of tempered distributions on \(\mathbb{R}^{d}\) which is supposed to be spatially homogeneous, that is, the law of \(W(t)\) is invariant with respect to all translations of \(\mathbb{R}^{d}\). It is known that the covariance form of such a process is given by a distribution \(\Gamma \in \mathcal S'\) which is a Fourier transform of a Borel positive and symmetric measure \(\mu \) on \(\mathbb{R}^{d}\). The equation (1) is dealt with by means of the abstract semigroup approach to stochastic partial differential equations. Sufficient conditions on the spectral measure \(\mu \) are found under which (1) has a unique mild solution in a suitable weighted \(L^2\)-space on \(\mathbb{R} ^{d}\). Moreover, (1) is shown to define a Feller Markov process in this space. Further, regularity of solutions in the space variable is investigated. Finally, it is shown that solutions to (1) with a diffusion coefficient \(\varepsilon b\), \(\varepsilon >0\), obey a large deviations principle.
Notwithstanding that equations analogous to (1) have been already treated several times starting with the paper by D. A. Dawson and H. Salehi [J. Multivariate Anal. 10, 141-180 (1980; Zbl 0439.60051)], the paper under review provides both finer and more general results and new methods that are likely to be very useful in related problems.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G60 Random fields
Citations:
Zbl 0439.60051
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[1] Bojdecki, T.; Jakubowski, J., Ito stochastic integral in the dual of a nuclear space, J. multivariate anal., 31, 40-58, (1989) · Zbl 0692.60042
[2] Brzeźniak, Z.; Peszat, S., Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process, (1997), School of Math., University of Hull, Preprint
[3] Brzeźniak, Z., Peszat, S., Stochastic two dimensional Euler equations, in preparation.
[4] Capiński, M., Peszat, S., On the existence of a solution to stochastic Navier-Stokes equations, in preparation.
[5] Carmona, R.; Molchanov, S.A., Parabolic Anderson problem and intermittency, Memoirs amer. math. soc., 108, 1-125, (1994) · Zbl 0925.35074
[6] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge Univ. Press Cambridge · Zbl 0761.60052
[7] Da Prato, G.; Zabczyk, J., Convergence to equilibrium for classical and quantum spin systems, Probab. theory related fields, 103, 529-552, (1995) · Zbl 0839.60083
[8] Da Prato, G.; Zabczyk, J., Ergodicity for infinite dimensional systems, (1996), Cambridge Univ. Press Cambridge · Zbl 0849.60052
[9] Davies, E.B., Heat kernels and spectral theory, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006
[10] Dawson, D.A., Salehi, H., Spatially homogeneous random evolutions. J. Multivariate Anal. 10, 141-180. · Zbl 0439.60051
[11] Eidel’man, S.D., Parabolic systems, (1969), North-Holland Groningen, Amsterdam · Zbl 0181.37403
[12] Funaki, T., Regularity properties for stochastic partial differential equations of parabolic type, Osaka J. math., 28, 495-516, (1991) · Zbl 0770.60062
[13] Gel’fand, I.M.; Vilenkin, N.Ya., Generalized functions. vol. IV: applications of harmonic analysis, (1964), Academic Press New York · Zbl 0136.11201
[14] Handa, K., On stochastic PDE related to Burgers’ equation with noise, collection: nonlinear stochastic PDES, Minneapolis, MN, 1994, (), 147-156
[15] Hida, T.; Streit, L., On quantum theory in terms of white noise, Nagoya math. J., 68, 21-34, (1977) · Zbl 0388.60041
[16] Holley, R.A.; Stroock, D.W., Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motion, Publ. RIMS Kyoto univ., 14, 741-788, (1978) · Zbl 0412.60065
[17] Itô, K., Foundations of stochastic differential equations in infinite dimensional spaces, (1984), SIAM Philadelphia · Zbl 0547.60064
[18] Kifer, Yu., The Burgers equation with a random force and a general model for directed polymers in random environments, (1995), Preprint
[19] Kotelenez, P., Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations, Stochastics stochastics rep., 41, 177-199, (1992) · Zbl 0766.60078
[20] Metivier, M., Semimartingales: A course on stochastic processes, (1982), de Gruyer Berlin · Zbl 0503.60054
[21] Metivier, M., Stochastic partial differential equations in infinite dimensional spaces, (1988), Scuola Normale Superiore Pisa · Zbl 0664.60062
[22] Nobel, J., Evolution equation with Gaussian potential, Nonlinear analysis: theory methods appl., 28, 103-135, (1997) · Zbl 0861.35149
[23] Peszat, S., Large deviation principle for stochastic evolution equations, Probab. theory related fields, 98, 113-136, (1994) · Zbl 0792.60057
[24] Peszat, S., Existence and uniqueness of the solution for stochastic equations on Banach spaces, Stochastics stochastics rep., 55, 167-193, (1995) · Zbl 0886.60064
[25] Peszat, S.; Zabczyk, J., Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. probab., 23, 157-172, (1995) · Zbl 0831.60083
[26] Peszat, S.; Zabczyk, J., Stochastic evolution equations with a spatially homogeneous Wiener process, () · Zbl 0943.60048
[27] Simon, B., The P(φ)2 Euclidean (quantum) field theory, (1974), Princeton Univ. Press Princeton
[28] Tessitore, G.; Zabczyk, J., Invariant measures for stochastic heat equations, () · Zbl 0987.47034
[29] Tessitore, G.; Zabczyk, J., Strong positivity for stochastic heat equations, () · Zbl 0987.47034
[30] Walsh, J.B., An introduction to stochastic partial differential equations, (), 265-439
[31] Zabczyk, J., The fractional calculus and stochastic evolution equations, (), 222-234 · Zbl 0787.60073
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