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I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have anything to do with SDE. My reasons for thinking so are the following.

  • I've read on SDE many times and never encountered a single mention of PDE/ODE
  • My master's programme offers almost nothing on PDE.
  • Searching for both tags on math.SE
  • I encountered PDE literally only once in my life, while dealing with continuous time Markov processes.

Recently, I've been drifting towards biology and started encountering PDEs more and more. This is not surprising, as they are arguably much more useful in that area than SDEs. It makes me wonder, however, whether I should perhaps devote time to ODE/PDE. This leads me to the following questions:

  1. Are PDE really so rarely relevant when it comes to SDE? Or possibly stochastic analysis in general?
  2. What could a stochastics student take away from studying ODE/PDE? What areas should he/she focus on (if any)? (e.g. very basics of ODE, at least)
  3. Since I am going into biology and thus might regret not knowing more PDE, how much sense would it make to make them a serious (secondary) area of study? Could sometimes PDE and SDE be seen as two approaches to the same problem, or be somehow analogous? Could they compliment each other, or would I just be doomed to be studying two mostly unrelated fields?

Thank you.

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  • $\begingroup$ ? math.stackexchange.com/questions/1124771/… $\endgroup$ – BCLC Mar 3 '16 at 20:13
  • $\begingroup$ @BCLC: This one and math.stackexchange.com/questions/988098/… are two questions that are relevant, but seem restricted (i.e. asking about particular problem, not the two fields) and not very accessible to someone who knows next-to-nothing about PDE (and not much about SDE either) $\endgroup$ – Dahn Mar 3 '16 at 20:15
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    $\begingroup$ There is a link between stochastic analysis and PDEs, see this question: math.stackexchange.com/a/697412/36150 Roughly speaking, if you are considering a PDE of the form $$\partial_t u(t,x) + Au(t,x) = 0$$ for a "nice" operator $A$, then this solution can be expressed in terms of a certain stochastic process. $\endgroup$ – saz Mar 4 '16 at 8:03
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    $\begingroup$ @BCLC No, not exactly. $\endgroup$ – saz Jul 24 '16 at 19:37
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    $\begingroup$ @saz I believe that comment was aimed at me. $\endgroup$ – Dahn Jul 25 '16 at 10:14
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Just my thoughts:

  1. I think SDEs and PDEs are rather deeply intertwined. For instance, the Kolmogorov backward equation and Fokker-Planck equations (see the link in the comments above). Indeed, as expected intuitively, Ito diffusion processes and the classical diffusion (heat) PDE are deeply related: on a Riemannian Manifold $(M,g)$, if $X$ is Brownian motion on $M$ (i.e. an Ito diffusion process with infinitesimal generator $\Delta_g/2$) with some transition density function $p(x,y,t)$, then $p$ solves the following: $$ \frac{\partial p}{\partial t} = \frac{1}{2}\Delta_g\, p,\,\;\;\; \lim_{t\rightarrow 0} p(t,x,y) = \delta_x(y) $$ which is of course a heat equation (see e.g. Hsu, Heat Equations on Riemannian Manifolds and Bismut's formula). In the simple case of Euclidean space, we get $\Delta=\Delta_g$ and $$ p(t,x,y) = \frac{1}{ (2\pi t)^{n/2} } \exp\left( \frac{-||x-y||^2}{2t} \right) $$ which is the Gaussian heat kernel of the IVP above (see e.g.: Morters & Peres, Brownian Motion).

  2. Well, beyond PDE/ODE theory itself, which is much more commonly applied (at least for biological models), you get the outlooks from (1) above. :)
    As for what to study, it depends on your goal, but I would look at dynamical systems.

  3. Dynamical systems, ODEs, and PDEs are important tools for ecological, physiological and cellular (since they are often physically based, more in the biomedical engineering realm), and molecular systems modelling (see "systems biology"). But one area I know with heavy attention paid to stochastic partial differential equations (SPDEs) in particular is in modelling neurons (e.g. see Tucker, Stochastic partial differential equations in Neurobiology: linear and nonlinear models for spiking neurons) because the ion channels are noisy and membrane voltage is a function of both space and time.

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