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A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation

We prove pathwise uniqueness property for a class of one-dimensional stochastic differential equation involving the local time of the unknown process and with a sojourn time on the boundary. We first formulate the associated nonlinear martingale's problem to prove weak uniqueness of solutions and then, by using local time's technics, we show that the extremas of two solutions are also solutions.

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