Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle $X_0\in\mathcal V_0$ over time. Assuming that the motion of $X_0$ is perturbed by space-time white noise, we may assume that $${\rm d}X_t=v_t{\rm d}t+\xi_t{\rm d}B_t\;\;\;\text{for all }t\ge 0\tag 2$$ for a smooth resolved velocity $v$, a random unresolved component $\xi=(\xi^1,\xi^2,\xi^3)$ and a $\mathbb R^3$-valued Brownian motion $(B_t)_{t\ge 0}$.

Letting $\Phi_t(X_0):=X_t$ and $u_t(X_t):=v_t$, we obtain $${\rm d}u=\left(\partial_tu+(u\cdot\nabla)u+\frac 12\langle C,C\rangle u\right){\rm d}t+\nabla u\cdot\xi_t{\rm d}B_t\;\;\;\text{for }t\ge 0\tag 3$$ with $C:=\sum_{i=1}^3\xi^i\partial_i$ and $u=u_t(X_t)$ by a well-known Itō formula.

Instead of considering the multiparameter SDE $(3)$ for the $\mathbb R^3$-valued process $u$ indexed by time and space, I would like to write down an infinite dimensional SDE for a process indexed by time only and taking values in a infinite dimensional Hilbert space $H$ of functions in space.

$H$ will be some Sobolev space, e.g. $H=H_0^1(\mathcal V\to\mathbb R^3)$ where $\mathcal V$ is the union of the $\mathcal V_t$ up to some finite time $T\ge 0$ (which is assumed to be bounded and open).

What do I need to do? I tried to perturb the fluid flow map $\Phi$ by a $H$-valued cylindrical Brownian motion $(W_t)_{0\le t\le T}$, i.e. I've tried to assume that $${\rm d}\Phi_t=u_t(\Phi_t){\rm d}t+\xi_t(\Phi_t){\rm d}W_t\;\;\;\text{for all }t\in [0,T]\;,\tag 4$$ but I don't know how I need to proceed. I think I need to apply some kind of Itō formula in order to find an expression for $u_t(\Phi_t)$ as I did with $(2)$ to obtain $(3)$.

  • $\begingroup$ @oxbadfood : Maybe you could have a look at le LNM by Filipovic "Consistency Problems for Heath-Jarrow-Morton Interest Rate Models" "link.springer.com/book/10.1007/b76888" best regards. $\endgroup$ – TheBridge Apr 7 '16 at 17:03
  • $\begingroup$ @TheBridge Could you please explain why you think I should take a look at this book? $\endgroup$ – 0xbadf00d Apr 7 '16 at 17:35
  • $\begingroup$ @oxbadfood : If I remember well the authors is studying infinite dimensional stochastic calculus with variables taking value in Hilbert space, and is deriving such things as Itô's formula in the context. Best regards $\endgroup$ – TheBridge Apr 7 '16 at 20:46
  • $\begingroup$ @TheBridge Thanks for your effort, but there are many books on infinite dimensional SDEs and I'm aware of a suitable Itō formula for them. The problem is, that they are only applicable for Itō processes, but here (in $(4)$) the integrands on the rhs depend on $\Phi_t$. $\endgroup$ – 0xbadf00d Apr 7 '16 at 21:38

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