Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line
- PDF / 492,635 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 102 Downloads / 185 Views
Support theorem for an SPDE with multiplicative noise driven by a cylindrical Wiener process on the real line Timur Yastrzhembskiy1 Received: 22 February 2019 / Revised: 5 July 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We prove a Stroock–Varadhan’s type support theorem for a stochastic partial differential equation on the real line with a noise term driven by a cylindrical Wiener process on L 2 (R). The main ingredients of the proof are V. Mackeviˇcius’s approach to support theorem for diffusion processes and N.V. Krylov’s L p -theory of SPDEs. Keywords SPDE · Stroock–Varadhan’s support theorem · Wong–Zakai approximation · Krylov’s L p -theory of SPDEs Mathematics Subject Classification 35R60 · 60H15
1 Introduction Let (Ω, F, P) be a complete probability space, and let (Ft , t ≥ 0) be an increasing filtration of σ -fields Ft ⊂ F containing all P-null sets of Ω. By P we denote the predictable σ -field generated by (Ft , t ≥ 0). Let N = {1, 2, . . .}, R be the real line, and R+ = [0, ∞). Denote Dx =
∂ ∂ , ∂t = . ∂x ∂t
For a function u : R+ × R → R, the temporal argument is denoted by t (or ·), and the spatial argument – by x (or ). For a function f : R → R, we denote D f (x) =
B 1
df (x). dx
Timur Yastrzhembskiy [email protected] University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA
123
Stoch PDE: Anal Comp
Let W (t), t ∈ R+ be an Ft -adapted cylindrical Wiener process on L 2 (R) on the probability space (Ω, F, P) (see Sect. 2 for the definition). We consider the following SPDE: du(t, x) = [a(t, x)Dx2 u(t, x) + b(t, x)Dx u(t, x) + f (u, t, x)]dt + u(t, x)dW (t), u(0, x) = u 0 (x), x ∈ R.
(1.1)
Here, a and b are some Hölder space-valued functions, a is bounded from below by a positive constant, and f (u, ·, ) is a ‘zero-order’ term. We point out that we do not assume continuity in the temporal variable for a, b and f . In this paper we adopt N.V. Krylov’s approach to parabolic SPDEs (see [7]), which allows us to treat parabolic SPDEs with relatively mild smoothness assumptions on the coefficients and the initial data. Under certain conditions the Eq. (1.1) has a unique 1/2−κ solution u that belongs to some stochastic Banach space H p (T ), p > 2, κ ∈ (0, 1/2] (see Sect. 2), which is a generalization of the parabolic counterpart of the γ space of Bessel potentials H p (T ). The other approaches to the regularity theory of SPDEs can be found in [3,14,18]. Our goal is to characterize the topological support of the distribution of u in the space C γ ([0, T ], H ps (R)), for some γ , s ∈ (0, 1/2), p > 2, where H ps (R) is the space of Bessel potentials. Let H(T ) be the set of Borel functions h : [0, T ] × R → R such that ∂t h ∈ B([0, T ] × R) ∩ L 2 ([0, T ] × R), where B([0, T ] × R) is the space of bounded Borel functions. In Theorem 2.1 we prove that the support of u coincides with the closure in the aforementioned Hölder–Bessel space of the set R = {Rh : h ∈ 1/2−κ (T ) of the following PDE H(T )}, where Rh is the unique solution of class H p (see De
Data Loading...
