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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
Zbl 0439.60051
Full Text:
##### References:
 [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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