How to Evaluate This Stochastic Expectation Value?

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SUMMARY

The discussion centers on evaluating the stochastic expectation value $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$, where the random variables satisfy the conditions $$\mathbb{E}[V_{i}(t)] = 0$$ and $$\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')$$ for a constant $$\gamma$$. Participants emphasize the importance of understanding stochastic processes and the properties of expectation values in this context. The evaluation requires familiarity with concepts such as Gaussian processes and moment-generating functions.

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thatboi
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Hi all,
I am not familiar with stochastic processes, but I would like to know how to evaluate the following expectation value: $$\mathbb{E}[e^{\int_{0}^{t}d\tau(V_{i}(\tau)-V_{j}(\tau))}]$$ where ##\mathbb{E}[V_{i}(t)] = 0,\mathbb{E}[V_{i}(t),V_{j}(t')] = \gamma\delta_{ij}\delta(t-t')## for some constant ##\gamma##.
Any assistance is appreciated.
 

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