Stochastic PDEs via convex minimization

Abstract

We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.

Existence of variational solutions to doubly nonlinear nonlocal evolution equations via minimizing movements

“…These methods were greatly refined and expaned upon by the group of Bögelein, Duzaar, Marcellini and coauthors in a series of papers such as [7,20,8,9,10,11,12,62]. The group of Stefanelli has also conducted extensive work on variational methods in the existence theory of evolution equations starting from his early work on De Giorgi conjecture [64] and subsequent works [2,59,60].…”

The Energy-Dissipation Principle for stochastic parabolic equations

“…In particular, the minimization of the Weighted-Energy-Dissipation functional corresponds to a noncausal differential problem. As ε → 0 one can prove [53] that u ε converge to the solution to the Cauchy problem (1). In particular, causality is restored in the limit.…”

“…In the additive case, a different global variational approach to (1) is in [53], where the Weighted-Energy-Dissipation functional…”

Existence of variational solutions to nonlocal evolution equations via convex minimization

“…Some forms of these conjectures were proved in [44,42]. Variational notions are also useful in the realm of stochastic partial differential equations, for example, see [35,40].…”