Abstract
We present simple assumptions on the constraints defining a hard core dynamics for the associated reflected stochastic differential equation to have a unique strong solution. Time-reversibility is proven for gradient systems with normal reflection, or oblique reflection with a fixed oblicity matrix. An application is given concerning the clustering at equilibrium of particles around a large attractive sphere.
Highlights
Since the first works of Skorokhod [14] on existence and uniqueness for pathwize solutions of reflected stochastic differential equations, many authors have investigated this type of equation and extended his results on half-spaces to more general domains: convex sets (Tanaka [15]), admissible sets (Lions-Sznitman [8]), domains satisfying only the Uniform Exterior Sphere and the Uniform Normal Cone conditions (Saisho [10]), or some weaker version of these conditions (Dupuis and Ishii [4])
We present simple assumptions on the constraints defining a hard core dynamics for the associated reflected stochastic differential equation to have a unique strong solution
We present in this note a constraint-based assumption to construct pathwise solutions of Skorokhod problems
Summary
Since the first works of Skorokhod [14] on existence and uniqueness for pathwize solutions of reflected stochastic differential equations, many authors have investigated this type of equation and extended his results on half-spaces to more general domains: convex sets (Tanaka [15]), admissible sets (Lions-Sznitman [8]), domains satisfying only the Uniform Exterior Sphere and the Uniform Normal Cone conditions (Saisho [10]), or some weaker version of these conditions (Dupuis and Ishii [4]). The domain in which the process has to live is defined by constraints which are physically natural rather than by its geometrical properties as a subset of some Euclidean space. We present in this note a constraint-based assumption to construct pathwise solutions of Skorokhod problems (even for non-reversible dynamics). In the special case of time-reversible dynamics, Skorokhod problems can be studied using potentiel theory. This Dirichlet form approach allows constructions on relatively non-smooth domains, as done in the seminal article of Chen [3]. The first part (section 2) exhibits a new compatibility criterion for constraints If it is satisfied, the reflected stochastic differential equation admits a unique strong solution. Applications to more realistic models (see e.g. [9] or [1]) are currently investigated
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