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  "body": "\n## Summary\n\nA random integral equation is an [[integral equation|R3SG1SN7JQ]] whose kernel, forcing term or both depend on a random parameter.\nFor each outcome of the randomness one gets an ordinary integral equation, and the solution is a [[stochastic process|0QH5ENKZM1]].\nThe basic questions are existence, uniqueness and measurability of such random solutions, and the laws of the solutions.\nSuch equations model systems with uncertain coefficients, such as materials with random properties.\nThey differ from stochastic integral equations with Itô integrals, where the integral against a [[Wiener process|22735FGJV6]] is not defined path by path.\n\n## Explanations\n\n### L1 human\n\nSuppose that the coefficients of an integral equation are measured with uncertainty and are modelled as random variables.\nFor each possible value of the coefficients the equation has a solution, so the solution itself is random.\nFor example, with a random number $Y$ between $0$ and $1$, the equation $x(t) = 1 + \\frac{Y}{2} \\int_0^1 x(s) \\, ds$ has the solution $x(t) = 2/(2 - Y)$.\nThe solution is a random variable, and one can ask for its mean or its distribution.\n",
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