A cumulant expansion for stochastic linear differential equations. I

Abstract A linear differential equation whose coefficients are stochastic functions, together with a fixed initial condition, defines a stochastic process. A systematic expansion is given for the expectation value of this process. The expansion parameter is the product of the magnitude of the fluctuations and their auto-correlation time. Secular terms are eluded by expanding the logarithm rather than the expectation value itself, so that the expansion is not confined to short times. The coefficients in the expansion are the cumulants of the coefficients in the differential equation. The result can be expressed as a differntial equation for the expectation value. The expansion is given explicitly to fourth order, and applied to two examples. Finally the method is compared with those of Bourret and of Keller.

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