Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$

  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $U$ be a separable Hilbert space
  • $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace
  • $(W_t)_{t\ge0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$
  • $H:=\mathbb R^d$ for some $d\in\mathbb N$ and $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,H)$, where $U_0:=Q^{1/2}U$.

Suppose we’re concerned with an SPDE $${\rm d}u_t\left(\Phi_t\left(x\right)\right)=f_t\left(\Phi_t\left(x\right)\right){\rm d}t+\nabla u_t\left(\Phi_t\left(x\right)\right)\cdot\xi_t\left(\Phi_t\left(x\right)\right){\rm d}W_t\;\;\;\text{for all }t\ge 0\text{ and }x\in H\;,\tag 1$$ where

  • $u:\Omega\times[0,\infty)\times H\to\mathbb R$
  • $\Phi:\Omega\times[0,\infty)\times H\to H$
  • $f:\Omega\times[0,\infty)\times H\to\mathbb R$

are suitable.

Can we recast $(1)$ into an equation in $\tilde H:=L^2(\mathbb R^d;\mathbb R^d)$? I want to get rid of the second parameter of $u$, i.e. I want to turn the finite-dimensional multiparameter SDE $(1)$ indexed by time and space into an infinite dimensional single parameter SDE indexed by time only.


$^1$ Let $\mathfrak L(A,B)$ be the set of bounded, linear operators from $A$ to $B$. Moreover, Let $\operatorname{HS}(A,B)$ be the set of Hilbert-Schmidt operators from $A$ to $B$.

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