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.

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 prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

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Multistep cell fate transitions with stepwise changes of transcriptional profiles are common to many developmental, regenerative and pathological processes. The multiple intermediate cell lineage states can serve as differentiation checkpoints or branching points for channeling cells to more than one lineages. However, mechanisms underlying these transitions remain elusive. Here, we explored gene regulatory circuits that can generate multiple intermediate cellular states with stepwise modulations of transcription factors. With unbiased searching in the network topology space, we found a motif family containing a large set of networks can give rise to four attractors with the stepwise regulations of transcription factors, which limit the reversibility of three consecutive steps of the lineage transition. We found that there is an enrichment of these motifs in a transcriptional network controlling the early T cell development, and a mathematical model based on this network recapitulates multistep transitions in the early T cell lineage commitment. By calculating the energy landscape and minimum action paths for the T cell model, we quantified the stochastic dynamics of the critical factors in response to the differentiation signal with fluctuations. These results are in good agreement with experimental observations and they suggest the stable characteristics of the intermediate states in the T cell differentiation. These dynamical features may help to direct the cells to correct lineages during development. Our findings provide general design principles for multistep cell linage transitions and new insights into the early T cell development. The network motifs containing a large family of topologies can be useful for analyzing diverse biological systems with multistep transitions.