We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are presented.
We study distributions of random vectors whose components are second order polynomials in Gaussian random variables. Assuming that the law of such a vector is not absolutely continuous with respect to Lebesgue measure, we derive some interesting consequences. Our second result gives a characterization of limits in law for sequences of such vectors.
Given the standard Gaussian measure $gamma$ on the countable product of lines $mathbb{R}^{infty}$ and a probability measure $g cdot gamma$ absolutely continuous with respect to $gamma$, we consider the optimal transportation $T(x) = x + abla varphi(x)$ of $g cdot gamma$ to $gamma$. Assume that the function $| abla g|^2/g$ is $gamma$-integrable. We prove that the function $varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {det}_2(I + D^2 varphi) exp bigl(mathcal{L} varphi - 1/2 | abla varphi|^2 bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $varphi$.
Let $A subset mathbb{R}^d$, $dge 2$, be a compact convex set and let $mu = varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ u = varrho_1 dx$ be a probability measure on $B_r := {xcolon |x| le r}$ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping $T$ such that $ u = mu circ T^{-1}$ and $T = phi cdot {rm n}$, where $phicolon A to [0,r]$ is a continuous potential with convex sub-level sets and ${rm n}$ is the Gauss map of the corresponding level sets of $phi$. Moreover, $T$ is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth $phi$ the level sets of $phi$ are driven by the Gauss curvature flow $dot{x}(s) = -s^{d-1} frac{varrho_1(s {rm n})}{varrho_0(x)} K(x) cdot {rm n}(x)$, where $K$ is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.
