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The Feynman-Kac formula (or Kolmogorov backward equation) describes a link between SDEs and PDEs. This answer introduces Ricci flows on Riemann manifolds which also seems to provide a mapping between PDEs to SDEs.

Are there other non-trivial examples besides the above mentioned where a mapping between SDEs and PDEs is possible?

Jurek
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    For example, the book Brownian Motion and Stochastic Calculus by Karatzas & Shreve contains a whole chapter devoted to this subject. – Kurt G. Dec 10 '22 at 07:42
  • Thank you, Kurt, for the reference! I had a quick look at the book: The connections presented are related to the Feynman-Kac formula (as far as I can tell) and consider general cases of elliptical and parabolic PDEs. I wonder if a connection had been found beyond this, e.g. hyperbolic PDEs. – Jurek Dec 11 '22 at 17:46
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    Hyperbolic PDEs can be ruled out as they typically describe wave phenomena while elliptic/parabolic are very closely related to convection of heat and diffusion of many particles. If not K&S, then maybe the good book by Olver might explain this a bit more. – Kurt G. Dec 11 '22 at 17:50
  • That's interesting and I am not sure I understand this: From a physics perspective I can follow the reasoning, but is there a mathematical reason why a mapping for hyperbolic PDEs to SDEs can be ruled out? Do you mean the Introduction to Partial Differential Equations by P. Olver?

    Thanks again, Kurt, for your replies and the book recommendations!

     – Jurek Dec 11 '22 at 20:40
  • Yes. That book. I am sure you will find a better non-physical explanation that I can produce here in those comments. Just a hint: look at the wave equation $\partial_t^2f-\partial_x^2f=0$ and flip the sign of time $t$. Symmetry! Processes from SDEs don't have that symmetry. Look up what the Generator $A$ of a Markov process is and how the transition probabilities satisfy the parabolic (never hyperbolic) equation $\frac{d}{dt}f=Af$. – Kurt G. Dec 12 '22 at 08:45
  • Kurt, thanks so much for the explanation and the hint -- that was very helpful! :) – Jurek Dec 12 '22 at 11:20
  • You may also want to contemplate about the reason why all PDEs related to SDEs are linear of order two in the space variables. I don't have a quick answer for this but there must be something in a good reference. I think it has something to do with a conservation law - but that's physics again :). – Kurt G. Dec 13 '22 at 09:39
  • Actually, I was thinking about the hint you gave and did have a few more questions regarding it: One can find solutions to the Poisson equation $\partial_x^2 \phi(x) = f(x)$ with Dirichlet boundary conditions via the mapping to a SDE and stochastic processes (see e.g. here). What I wonder: Couldn't one do similarly by taking $\partial_t^2$ as a space-variable (and maybe do some mapping to to the complex plane to get rid of the minus sign)? If you have any thoughts on that, I'd be happy to hear them. :) – Jurek Dec 13 '22 at 13:31
  • No. I have no thoughts on that. Sometimes younger people have to explore such ideas without the burden of that old folklore and all that literature. Only this generates innovations. See Feynman and his path integral. – Kurt G. Dec 13 '22 at 13:52
  • Yeah, I might update the question or ask a new one. In any way, thanks a lot for your help and the interesting discussion with you! :) – Jurek Dec 13 '22 at 14:02

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As mentioned in the comments, this not possible eg. symmetry. And also because of the finite-propagation. People have found ways around those issues using Poisson measures, for some references see here

Zhang–Yu–Mascagni, Revisiting Kac’s method: A Monte Carlo algorithm for solving the Telegrapher’s equations.

Dalang–Mueller–Tribe, A Feynman-Kac-type formula for the deterministic and stochastic wave equations, arXiv:0710.2861.

Thomas Kojar
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