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\\underline{B}(\\mu,\\nu):=\\inf\\left\\{\\mathbb{E}\\left[\\langle V,X_0\\rangle +\\int_0^T L(t,X,\\beta(t,X))\\,dt\\right]\\middle\\rvert V\\sim\\mu,X_T\\sim \\nu\\right\\}\\\\\\tag{$\\star\\star$}\n  \\overline{B}(\\nu,\\mu):=\\sup\\left\\{\\mathbb{E}\\left[\\langle V,X_T\\rangle -\\int_0^T L(t,X,\\beta(t,X))\\,dt\\right]\\middle\\rvert V\\sim\\mu,X_0\\sim \\nu\\right\\} \\end{align} where $X$ is 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