On a class of elliptic and parabolic equations in convex domains without boundary conditions
The authors consider the differential operator \(\mathcal{A}\) defined by \[ \mathcal{A} u = \frac{1}{2} \Delta u - \left<DU,Du\right> \] in a convex open set \(\mathcal{O}\subset \mathbb{R}^N\), where \(U:\mathcal{O}\mapsto \mathbb{R}\) is a convex function such that \[ \lim_{x\to \partial \mathcal{O}, x\in \mathcal{O}} U(x)=+\infty, \quad \lim_{|x|\to \infty, x\in \mathcal{O}} U(x)=+\infty. \] They essentially study the associated Markov semigroup \(\{T(t):t\geq 0\}\), i.e., \(T(t)\varphi\) is the unique solution of the parabolic Cauchy problem defined by \(\mathcal{A}\) (with initial condition \(\varphi\)). Moreover, they prove that \(\{T(t):t\geq 0\}\) coincides with the transition function of the stochastic differential inclusion \[ dX(t)+ \partial U(X(t)) dt \ni dW(t), \quad X(0)=x\in \mathcal{O}, \] where \(W(t)\) is an \(\mathbb{R}^N\)-valued Brownian motion in a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with continuous trajectories, i.e., \(T(t)f(x)= \mathbb{E}(f(X(t))|X_0=x)\).
scientific article; zbMATH DE number 5359501