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I am currently studying the path integral approach to stochastic processes. Recently I was reading Functional integral approach for multiplicative stochastic processes about the path integral formulation in multiplicative stochastic processes.

My question is, in some point of the paper, they find a determinant of a single derivative, that is

$\det\left(\frac{d}{dt}\right)$.

What I want to understand is how to solve this determinant? the paper doesn't make clear how.

I tried to solve searching for the eigenvalues using Dirichelet boundary conditions, however this approach doesn't give me any result. It appears that is not possible to find a closed relation for the eigenvalues, something like $\lambda_n\sim n$.

It is possible to find an exact solution? what is this solution? Moreover, if we have a multiplicative constant, in the det like $a\frac{d}{dt}$ what will change? (this will be relevant in the context of stochastic thermodynamics systems).

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