A Wong-Zakai theorem for stochastic PDEs. (English) Zbl 1341.60062

This paper treats the Wong-Zakai type theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise: \[ du = \partial_x^2 u dt + H(u) dt + G(u) d W(t). \tag{1} \] Let \(H\) and \(G\) be of class \(C^2\) and \(C^5\), respectively, both with bounded first derivatives. Let \(u\) denote the solution to (1), and let \(u_{\varepsilon}\) denote the classical solution to the random PDE \[ \partial_t u_{\varepsilon} = \partial_x^2 u_{\varepsilon} + H( u_{\varepsilon} ) - C_{\varepsilon} G'( u_{\varepsilon} ) G( u_{\varepsilon} ) + G( u_{\varepsilon} )\xi_{\varepsilon}, \tag{2} \] with \(C_{\varepsilon} = \varepsilon^{-1} c_{\rho}\) and \(H\) replaced by \[ \bar{H}(u) = H(u) - c_{\rho}^{(1)} G'( u )^3 G(u) - c_{\rho}^{(2)} G''(u) G'(u) G^2(u), \tag{3} \] for some constants \(c_{\rho}^{(i)}\) possibly depending on \(\rho\) but not on \(\varepsilon\). Both solutions are started with the same initial condition \(u_{\varepsilon}(0, \cdot)\) \(=\) \(u(0, \cdot)\) \(\in\) \(C(S^1)\). Then, there exists a choice of \(c_{\rho}^{(i)}\) such that, for any \(T >0\), one has \[ \lim_{ \varepsilon \to 0} \sup_{ (t,x) \in [0, T] \times S^1 } | u(t,x) - u_{\varepsilon} (t,x) | = 0 \tag{4} \] in probability. Moreover, for any \(\alpha \in (0, 1/2)\) and \(t > 0\), the restriction of \(u_{\varepsilon}\) to \([0, T] \times S^1\) converges to \(u\) in probability for the topology of \(C^{\alpha/2, \alpha }\), where \(\xi_{\varepsilon}\) is an \(\varepsilon\)-approximation to the space-time white noise, defined by \[ \xi_{\varepsilon} (t,x) = \int_{-\infty}^{\infty} \langle \rho_{\varepsilon} (t-s, x- \cdot ), d W(s) \rangle \tag{5} \] with the cylindrical Wiener process \(W\) driving (1). For other related works, see, e. g., [M. Hairer, Ann. Math. (2) 178, No. 2, 559–664 (2013; Zbl 1281.60060)] and [M. Hairer et al., Stoch. Partial Differ. Equ., Anal. Comput. 1, No. 4, 571–605 (2013; Zbl 1323.35230)].


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
35R60 PDEs with randomness, stochastic partial differential equations
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