Enveloppes convexes des processus gaussiens. (Convex envelopes of Gaussian processes) (Q1863424)

For a measurable real-valued stochastic process \({\mathbf X} = \{X(t)\}_{0\leq t\leq 1}\) let \[ V_{{\mathbf X}}=\text{Conv}\{(t,x)\mid 0\leq t\leq 1,\;x=X(t)\} \] denote the convex hull of its paths. Consider also the set \(\text{ext}(V_{{\mathbf X}})\) of extremal points of \(V_{{\mathbf X}}\). It is proved that if \(\mathbf X\) is a Gaussian process with stationary increments, then: (i) \(\text{ext}(V_{{\mathbf X}})\) is a.s. negligible when \(\mathbf X\) is non-differentiable; (ii) \(\text{ext}(V_{{\mathbf X}})\) is a.s. negligible (moreover, a Cantor set) when \(\mathbf X\) is absolutely continuous and its derivative \(\mathbf X'\) is continuous and nondifferentiable. It is also proved that these properties are stable under a transformation of the form \(Y(t) = f(X(t))\), \(0\leq t\leq 1\), if \(f\) is a sufficiently smooth function.