We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.

We study generalized Hopf–Cole transformations motivated by the Schrödinger bridge problem, which can be seen as a boundary value Hamiltonian system on the Wasserstein space. We prove that generalized Hopf–Cole transformations are symplectic submersions in the Wasserstein symplectic geometry. Many examples, including a Hopf–Cole transformation for the shallow water equations, are given. Based on this transformation, energy splitting inequalities are provided.

Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain where $\mathbb H$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $\HH$. Given a Jordan arc $\gamma\subset D$ starting at $\partial {\mathbb H},$ let $g_t$ be the unique conformal map from $D\setminus \gamma[0,t]$ onto a standard slit domain $D_t$ satisfying the hydrodynamic normalization. We prove that $g_t$ satisfies an ODE with the kernel on its righthand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$, generalizing the chordal Loewner equation for the simply connected domain $D={\mathbb H}.$ Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in $t$. We establish the differentiability of $g_t$ in $t$ to make the equation a genuine ODE. To this end, we first derive the continuity of $g_t(z)$ in $t$ with a certain uniformity in $z$ from a probabilistic expression of $\Im g_t(z)$ in terms of the BMD for $D$, which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.

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