## 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)].

### MSC:

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

### Citations:

Zbl 1281.60060; Zbl 1323.35230
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### References:

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