Stochastic evolution equations with a spatially homogeneous Wiener process

Large deviations for stochastic heat equation with rough dependence in space

“…Because Λ (that can be formally written as Λ(x − y) = H(2H − 1)|x − y| 2H−2 ) is not a locally integrable function, the classical approach developed, among others, by Da Prato and Zabczyk [6], Peszat and Zabczyk [14], Dalang in [7,8] and Dalang and Quer-Sardanyons [9], cannot be applied to such rough covariance. In [1], Balan, Jolis and Quer-Sardanyons proved the existence and uniqueness of a mild solution to equation (1.1) in the particular case σ(u) = au + b, and assuming that the initial condition u 0 is bounded and Hölder continuous of order H. The stochastic integral is understood in the Itô sense.…”

SPDEs with rough noise in space: Hölder continuity of the solution

“…Since the Fourier transform of the measure µ(dξ) = c H |ξ| 1−2H dξ is not given by a locally integrable function, the study of equations with this kind of noise does not fall under the general theory of SPDEs with colored noise initiated by [3] and [8]. Instead, the family X can be seen as a random stationary distribution (see [6,9]), which allows to define stochastic integrals with respect to this type of noise following the ideas of Basse-O'Connor et al [2].…”

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

“…The case of a linear noise coefficient has been treated by Dawson and Salehi [10] and Noble [28]. Amongst others, Kotelenez [23], Peszat and Zabczyk [31,32], Brzeźniak and Peszat [2], Dalang [5], Manthey and Mittmann [25], Tindel and Viens [41] and Sanz-Solé and Sarrà [37] investigate solutions with Lipschitz coefficients. For some results on equations with non-Lipschitz coefficient see Viot [42], DaPrato and Zabczyk [34], Krylov [24] and Kallianpur and Sundar [20].…”