I am starting to study stochastic partial differential equations, and would like to understand when and why they are used.

It is well known that in many mathematical models for physics PDEs play a central rôle. E.g. the Navier-Stokes equations in continuum mechanics.

Stochastic PDEs often are introduced by considering a physical phenomena with a "random forcing", which then might translate into adding a noise term on the RHS of a given PDE. For example the stochastic Navier-Stokes equations.

In general one can say that the forces acting on a fluid in motion are chaotic since they might be too difficult to describe in a deterministic setting without considering every molecule independently.

Nonetheless I suppose that in most applications most of these chaotic forces can be omitted since, for example, they don't seriously interfere with the movement of the fluid.

  1. So a first question is: where and why do we have to consider random forces? And why are these examples important?

A second question arises from a very naive thought. If the forces in question are random, I would at first expect that the solution to the SPDE does not differ in mean from the solution is the relative PDE.

  1. Are there essential differences between the behaviour of a solution to an SPDE and a PDE?

One answer to the latter question could be that the energy of the system explodes in the stochastic setting, because random forces pump energy into the system.

  • $\begingroup$ You shouldn't expect the effect of the noise to have no effect on the mean unless the equation is completely linear (and the noise has mean zero). Anyway, I'd seek out motivation by just considering particular examples. In my field these examples mainly emerge as "rough continuum limits" of SODEs, where we choose our scaling for the continuum limit such that the noise does not entirely wash out. $\endgroup$ – Ian Nov 10 '16 at 21:45
  • $\begingroup$ Yes, I would very much like to learn about the examples in which spdes occur. By continuum limit I suppose you mean that you have a discrete time setting and send the time interval to zero in some way. But what do you model in the discrete time setting, that brings you to an spde? Of course you do not have to explain it here. You can also just tell me where to "look it up" ;) $\endgroup$ – Kore-N Nov 10 '16 at 22:01
  • $\begingroup$ What I actually mean is a continuum limit in space, so that you wind up with an SPDE rather than an SODE. There are tons of examples for just SODEs, which you might want to look up for some motivation in an applied SDE text (shouldn't be hard to just look up "applied stochastic differential equations" to find some). $\endgroup$ – Ian Nov 10 '16 at 22:47

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