Preservation of convexity of solutions to parabolic equations
This paper is a continuation of the authors' earlier paper in Ann. Appl. Probab. 13, 890--913 (2003; Zbl 1061.91028), where only the case of one space variables was considered in connection with applications to finance. It deals with the Cauchy problem \[ M F(x,t) = h(x,t), \qquad F(x,0) = f(x), \qquad x \in \mathbb R^n, \;0 \leq t \leq T \] with \[ M = \frac{\partial}{\partial t} - L, \] \(L\) being a linear elliptic operator of second order with coefficients depending on \(x, t\). The aim of the paper is to find conditions on \(M, h\) assuring that for every fixed \(t\) the unique solution \(F(x,t)\) is convex in space variables if \(f(x)\) it does. Several examples of such operators \(M\) are given.
scientific article; zbMATH DE number 2116222