We prove that the law of the minimum \({m := {\rm min}_{t\in[0,1]}\xi(t)}\) of the solution \({\xi}\) to a one-dimensional stochastic differential equation with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets \({\{x \in C([0,1]) : {\rm inf} x \geq r\}}\) have finite perimeter with respect to the law \({\nu}\) of the solution \({\xi({\cdot})}\) in \({L^{2}(0,2)}\).

中文翻译：

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关于一维SDE中的最小值定律

我们证明了最小的法律\（{M：在= {\ RM分钟} _ {吨\ [0,1]} \ XI（T）} \）的溶液的\（{\ XI} \）到相对于Lebesgue测度，具有良好非线性的一维随机微分方程具有连续的密度。作为该过程的副产品，我们证明集合\（{\ {x \ in C（[0,1]）：{\ rm inf} x \ geq r \}} \\）相对于法\（{\ NU} \）的溶液的\（{\ XI（{\ CDOT}）} \）在\（{L ^ {2}（0,2）} \）。