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Probabilistic representation of heat equation of convolution type

In this paper we give a probabilistic representation for the solution of the heat equation of convolution type with a generalized function ƒ as initial condition. The method uses a combination between convolution calculus and the generalized stochastic calculus, namely Itô's formula for generalized functions. Finally, generalization to the stochastic heat equation with a gradient term and generalized coefficients is presented.