Fully nonlinear stochastic partial differential equations: Non-smooth equations and applications. (English. Abridged French version) Zbl 0924.35203

Lions, Pierre-Louis; Souganidis, Panagiotis E.

C. R. Acad. Sci., Paris, Sér. I, Math. 327, No. 8, 735-741 (1998).

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 \(u\in \text{BUC} (\mathbb{R}^N \times[0,T])\).

Reviewer: S.Wedrychowicz (Rzeszów)

MSC:

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

Keywords:

uniform convergence; almost everywhere convergence; bounded uniformly continuous; Brownian motion; white noise; degenerate elliptic