Regularity of the sample paths of a class of second-order spde's

Abstract

We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (Electron. J. Probab. 8 (2003) 1), we prove that the solutions belong to a fractional L 2 -Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller. For spatial covariances given by Riesz kernels, we show that our results are optimal.

Sample Path Properties of Anisotropic Gaussian Random Fields

“…Under the latter condition, u = {u(t, x) : t ≥ 0, x ∈ R 2 } can be represented as 25) where S(t, x) = 1 2π (t 2 − |x| 2 ) −1/2 1l {|x|<t} . Sample path regularity of the solution {u(t, x) : t ≥ 0, x ∈ R 2 } has been investigated by Dalang and Frangos (1998), and Dalang and Sanz-Solé (2005).…”

“…Kahane (1985), p.8], one can verify that there is a subsequence of {ν n , n ≥ 1} that converges weakly to a finite measure ν which is positive with positive probability [depending on c 7,12 and c 7,13 only] and ν also satisfies (7.22). Since ν is supported on X −1 (F ) ∩ I, we use the Paley-Zygmund inequality again to derive 27) where c 7,16 = c 2 7,12 /c 7,13 . This implies the lower bound in (7.20).…”

Analytically weak solutions to linear SPDEs with unbounded time-dependent differential operators and an application

“…For complementary results on stochastic beam equations (with operators not being time-dependent and without algebraic constraint) we refer to [5], [6] and [1] and references therein.…”

Theory and application of stability for stochastic reaction diffusion systems

“…As to fractional stochastic partial differential equations, the definitions of mild solutions and variational solutions were proposed and accordingly the conditions for the existence and uniqueness of solutions [15][16][17][18][19] were provided. Further, the continuity and positive principle [18][19][20] of sample trajectory were also investigated; furthermore [21,22] , studied the stability of stochastic differential systems with delays and provided some corresponding effective criteria; for research content, besides some basic property including the existence and uniqueness and the continuity of solutions, there were invariant measure and stochastic viscosity solutions. In ref.…”