In this paper we consider the following stochastic partial differential equation (SPDE) in the whole space: \( du (t, x) = [a^{i j} (t, x) D_{i j} u(t, x) + f(u, t, x)]\, dt + \sum _{k = 1}^m g^k (u(t, x)) dw^k (t). \) We prove the convergence of a Wong–Zakai type approximation scheme of the above equation in the space \( C^{\theta } ([0, T], H^{\gamma }_p ({\mathbb {R}}^d)) \) in probability, for some \( \theta \in (0,1/2), \gamma \in (1, 2)\), and \(p > 2\). We also prove a Stroock–Varadhan’s type support theorem. To prove the results we combine V. Mackevičius’s ideas from his papers on Wong–Zakai theorem and the support theorem for diffusion processes with N. V. Krylov’s \(L_p\)-theory of SPDEs.

在本文中，我们考虑整个空间中的以下随机偏微分方程（SPDE）：\（du（t，x）= [a ^ {ij}（t，x）D_ {ij} u（t，x）+ f（u，t，x）] \，dt + \ sum _ {k = 1} ^ mg ^ k（u（t，x））dw ^ k（t）\）我们证明了Wong–的收敛性在空间\（C ^ {\ theta}（[0，T]，H ^ {\ gamma} _p（{\ mathbb {R}} ^ d））\）中的上述方程式的Zakai型逼近方案，对于一些\（\ theta \ in（0,1 / 2），\ gamma \ in（1，2）\）和\（p> 2 \）。我们还证明了Stroock–Varadhan的类型支持定理。为了证明结果，我们结合了V.Mackevičius的Wong-Zakai定理和扩散过程的支持定理的理论，并结合了NV Krylov的SPDE的\（L_p \）-理论。