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Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications. (English. Abridged French version) Zbl 0924.35203
The authors consider equations of the form $du=F(D^2u,Du)dt+ \sum^M_{i+1} H_i(Du) \circ dW_i\quad \text{in }\mathbb{R}^N \times (0, \infty),\;u=u_0 \text{ on }\mathbb{R}^N \times\{0\}.$ Under the assumption that $$u_0\in\text{BUC}(\mathbb{R}^N)$$, the space of bounded uniformly continuous functions on $$\mathbb{R}^N$$; $$W=(W_1, \dots,W_M)$$ is the standard $$M$$-dimensional Brownian motion in time and, hence, $$dW=(dW_1, \dots,dW_M)$$ is the usual “$$M$$-dimensional white noise” in time and $$F$$ is continuous and degenerate elliptic, i.e., it satisfies, for all $$X,Y\in S^N$$, the space of $$N\times N$$ symmetric matrices, and $$p\in\mathbb{R}^N$$, $\text{if }X\leq Y,\text{ then } F(X,p) \leq F(Y,p). \tag{1}$
$u^\varepsilon_t= F(D^2u^\varepsilon, Du^\varepsilon)+ \sum^M_{i=1} H_i(Du^ \varepsilon)\zeta^\varepsilon_i(t)\quad\text{ in }\mathbb{R}^N \times (0,\infty),\quad u^\varepsilon= u^\varepsilon_0 \text{ on }\mathbb{R}^N \times \{0\},\tag{2}$ where $$u_0^\varepsilon\in\text{BUC} (\mathbb{R}^N)$$ and the smooth functions $$\zeta^\varepsilon= (\zeta^\varepsilon_1, \dots, \zeta^\varepsilon_M)$$: $$[0,\infty] \times\Omega \to \mathbb{R}^M$$ are such that, as $$\varepsilon\to 0$$ and for all $$T>0$$,
(3) $$\zeta^\varepsilon\to W$$ uniformly and a.e. in $$(0,T)$$, and $$u^\varepsilon_0\to u_0$$.
The authors assume the following condition (4): There exists $$G\in C(S^{2N}\times\mathbb{R}^N)$$, degenerate elliptic such that, for all $$p\in\mathbb{R}^N$$
(i) $$G\left(\begin{matrix} \lambda X & -\lambda X\\ -\lambda X & \lambda X\end{matrix}, p \right) =0$$, for all $$\lambda\in\mathbb{R}$$ and all $$X\in S^N$$, and
(ii) $$F(X,p)-F(Y, p)\leq G(Z,p)$$, for all $$X,Y\in S^N$$ and such that $$\left(\begin{matrix} X & 0\\ 0 & Y \end{matrix}\right)\leq Z$$.
The authors prove the following main theorem: Let $$(\zeta^\varepsilon)_{\varepsilon>0}$$ and $$(\xi^r)_{r>0}$$ satisfy (3). Assume that $$F$$ satisfies (1) and (4) and $$H$$ satisfies the condition: $$H\in C^{0,1} (\mathbb{R}^N, \mathbb{R}^M)$$, and for each $$i\in\{1,\dots,M\}$$, $$H_i$$ can be written as the difference of two convex functions.
If $$\| u^\varepsilon_0-v^\eta_0 \| \to 0$$ as $$\varepsilon,\eta\to 0$$, then $\lim_{\varepsilon,\eta\to 0} \| u^\varepsilon- v^\eta\|_{C (\mathbb{R}^N \times[0,T])} =0$ for every $$T>0$$ and a.e. in $$\omega$$, where $$u^\varepsilon$$ and $$v^\varepsilon$$ solve (2) with initial data $$u_0^\varepsilon$$ and $$v_0^\eta$$ respectively. In particular, each family $$(u^\varepsilon)_{\varepsilon>0}$$ is Cauchy in $$\mathbb{R}^N \times [0,T]$$ a.e. in $$\omega$$, and hence, it converges, uniformly in $$(x,t)$$ and a.e. in $$\omega$$, to a unique $$u\in \text{BUC} (\mathbb{R}^N \times[0,T])$$.

##### MSC:
 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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