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Feynman Kac representation for nonlinear heat equation

Consider the following Cauchy problem \begin{align} \begin{cases} \partial_t u=\sigma(t)\partial_{xx} u+ b(u),\; (t,x)\in[0,T]\times \mathbb R\\ u(0,x)=u_0(x)=Ce^{-x^2/2}, \end{cases} \end{align} wher …
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