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In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift b and the weak gradient of Sobolev diffusion σ are supposed to satisfy b · 1 B (R) p 1 ≤ O ((log R) (p 1 - d) 2 / 2 p 1 2 ) and ∇ σ · 1 B (R) p 1 ≤ O ((log (R / 3)) (p 1 - d) 2 / 2 p 1 2 ) , respectively. The main tools for these results are the decomposition of global two-point motions in Fang et al. (Ann Probab 35(1):180–205, 2007), Krylov's estimate, Khasminskii's estimate, Zvonkin's transformation and the characterization for Sobolev differentiability of random fields in Xie and Zhang (Ann Probab 44(6):3661–3687, 2016). [ABSTRACT FROM AUTHOR]

AbstractWe consider a general class of semi-Markov processes on countable state-space, with a differentiable kernel such that the embedded Markov chain may not be homogeneous. Using an SDE representation of the process, we study the associated flow by investigating meetings and merging related events for solutions, starting with different initial conditions. Some sufficient conditions for almost sure meeting and merging are also obtained. Many examples are considered to clarify the intricacies and implications of this study. [ABSTRACT FROM AUTHOR]

We study the existence of the stochastic flow associated to a linear stochastic evolution equation \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A is the generator of a C_0-semigroup, and (B_k) is a sequence of bounded linear operators such that \sum _k\|B_k\|<+\infty. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert–Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well. [ABSTRACT FROM AUTHOR]