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It is well-known that a stochastic differential equation (sde) on a Euclideanspace driven by a (possibly infinite-dimensional) Brownian motion withLipschitz coefficients generates a stochastic flow of homeomorphisms. If theLipschitz condition is replaced by an appropriate one-sided Lipschitz condition(sometimes called monotonicity condition) and the number of driving Brownianmotions is finite, then existence and uniqueness of global solutions for eachfixed initial condition is also well-known. In this paper we show that under aslightly stronger one-sided Lipschitz condition the solutions still generate astochastic semiflow which is jointly continuous in all variables (but which isgenerally neither one-to-one nor onto). We also address the question of strong$\Delta$-completeness which means that there exists a modification of thesolution which if restricted to any set of dimension $\Delta$ is almost surelycontinuous with respect to the initial condition.

Strong completeness and semi-flows for stochastic differential equations with monotone drift