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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}

where $\sigma(\cdot)$ is continuous (bounded if needed) and $b(\cdot)$ is bounded and Lipschitz continuous.

I am interested in representation for solutions of the latter equation, I know that if $b=0$ then the differential operator generates a semigroup which is connected to the diffusion $dX_t=\sqrt{2\sigma(t)}dB_t$ where $(B_t)$ is a Brownian Motion and then we can represent the solution y means of the Feynman-Kac formula.

I am now aware if we can still find a Feynman-Kac formula in this case where we have a nonlinear term $b(u)$, is there a representation for the solution?

If not, is it still possible to write this down as

$$u(t,x)=P_t( u_0)(x)+\int_0^t P_{t-s}(b(u))(x)ds$$

where $P_t$ is the Markov semigroup associated with the differential operator?

Last, if you know some reference that treats the heat equation with nonlinear terms please let me know!

Thanks in advance

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  • $\begingroup$ in your last equation, you want to treat $b[u(t,x)]$ as a given source term $B(t,x)$ and then you can just apply the usual path integral solution; but that would not be very helpful, since this source is not actually given. $\endgroup$
    – Carlo Beenakker
    Commented Apr 2, 2021 at 18:58
  • $\begingroup$ @CarloBeenakker indeed, this is just another way to write down the equation, but in my case (there's a lot of context i didn't introduced in the question) this way of writing things down can be useful $\endgroup$
    – Chaos
    Commented Apr 2, 2021 at 20:17
  • $\begingroup$ if it is useful for your purpose, you could certainly do it, but what is then the question? $\endgroup$
    – Carlo Beenakker
    Commented Apr 2, 2021 at 20:30
  • $\begingroup$ Perhaps you may find this reference interesting: Mark Freidlin, Functional integration and partial differential equations, (English) Annals of Mathematics Studies, No. 109. Princeton, New Jersey: Princeton University Press, pp. IX+545 (1985), MR0833742, Zbl 0568.60057. $\endgroup$
    – Daniele Tampieri
    Commented Apr 2, 2021 at 20:32
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    $\begingroup$ @DanieleTampieri Ciao Daniele! I have Freidlin's book but I wasn't able to find a Feynman Kac representation for an equation in which the source was actually nonlinear! The linear case although it could have relevance from a PDE perspective, is not interesting for my purposes $\endgroup$
    – Chaos
    Commented Apr 2, 2021 at 20:37

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