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Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its Green's function $g(t,x)\rightarrow 0$ as $x\rightarrow\infty$ for any fixed time $t$? If it is not in general:

  1. What would be a counterexample?

  2. What are reasonable but general conditions on the diffusion coefficients for the Green's function to possess that limiting property?

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  • $\begingroup$ I would suggest to use the probabilistic interpretation through the stochastic differential equation $dX_t=\sigma(X_t)\cdot dB_t$, the density $u$ of $X_t$ satisfying $\partial_t u-\frac12 |\sigma(x)|^2\Delta u=0$. Problems arise if $\sigma$ is not Lipschitz. $\endgroup$
    – Jean Duchon
    Apr 3, 2018 at 10:26
  • $\begingroup$ @JeanDuchon: Good point. It would give the boundedness of $\int_{R^n}u^2$. However, it is not enough to produce the vanishing of $u$ and possibly its derivatives as $x\rightarrow\infty$. Perhaps we need more regularity condition on $\sigma$ than just Lipschitz. $\endgroup$
    – Hans
    Apr 3, 2018 at 19:07

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