Phase-Space Realization of Stochastic Multivalued Dynamics with Gyroscopic Subdifferential Coupling
We introduce a new framework for stochastic variational inequalities with oblique reflections generated by non-symmetric matrix coefficients. The loss of symmetry fundamentally changes the structure of the reflected dynamics: besides the dissipative component, it produces a genuinely gyroscopic component describing rotational effects that are invisible to the associated energy forms. To capture this hidden component, we consider planar obstacle problems and we lift the dynamics to an extended phase-space system, where the dissipative and gyroscopic mechanisms are naturally separated. This leads to a new notion of solution, for which we establish existence and uniqueness. The resulting framework allows for genuinely rotated reflections, which can lead to angle-controlled rotations.