Stochastic Sobolev spaces and their boundary trace

Yu. A. Rozanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 1, pp. 111-124

Résumé

This is a short self-contained introduction to stochastic Sobolev spaces. The well-known Sobolev spaces $W_2^p(T)$ represent essentially smooth functions which are characterized as being in $L_2 (T)$ with all their generalized derivatives $\partial^ku\in L_2(T)$, $|k|\le p$. $T\subseteq\mathbf R^d$ is the region considered. This kind of characterization of random $u=\xi$ is of no interest, since generalized random functions typically are very chaotic Schwartz distributions, and their primary characterization is supposed to be given in terms of a probability distribution (e.g., by a characteristic functional) or in terms of a covariance $$ \mathbf E(x,\xi)(y,\xi),\quad x,y\in C_0^\infty(T), $$ which itself is determined by the variance $$ \|(x,\xi)\|^2=\mathbf E|(x,\xi)|^2,\quad x\in C_0^\infty(T). $$ What can be common for the deterministic case and the stochastic case is the continuity of $\|(x,\xi)\|$ with respect to the corresponding Sobolev norm $\|x\|_{-p}$. Therefore, we introduce stochastic Sobolev spaces $\mathbf W_2^p(T)$. These spaces contain as the most known representatives random functions such as Brownian motion, Markov free field, Lévy Brownian motion, etc. It is shown, in particular, that the variety of generalized random functions which form a stochastic Sobolev space $\mathbf W_2^p(T)$ in a region $T\subseteq\mathbf R^d$ is represented as a direct product $$ \xi=L^*L\xi\otimes\prod_{k=0}^{p-1}\otimes\xi^{(k)} $$ with arbitrary elements $L^*L\xi\in\mathbf W_2^{-p}(T)$, $\xi^{(k)}\in\mathbf W_2^{p-k-1/2}(\Gamma)$, $k=0,\dots,p-1$, where $\xi^{(k)}$, $k=0,\dots,p-1$, serve as the $k$th generalized derivatives of the corresponding $\xi\in\mathbf W_2^p(T)$ along a non-tangent vector field on the boundary $\Gamma=\partial T$. The differential operator $L$ involved, $$ L=\sum_{|k|\le p}a_k\partial^k, $$ can be any non-degenerate elliptic operator.

Détail

Keywords: generalized random functions, stochastic Sobolev spaces, a generalized boundary trace, embedding theorems.