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Let

  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $U$ be a separable Hilbert space
  • $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace
  • $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$

Moreover, let

  • $d\in\left\{2,3\right\}$
  • $\mathcal V_0\subseteq\mathbb R^d$ be a bounded domain
  • $u:[0,\infty)\times\mathbb R^d\to\mathbb R^d$ and $\Phi:\Omega\times[0,\infty)\times\mathbb R^d\to\mathbb R^d$ with $${\rm d}\Phi_t(x_0)=u_t\left(\Phi_t\left(x_0\right)\right){\rm d}t+\xi_t\left(\Phi_t\left(x_0\right)\right){\rm d}W_t\;\;\;\text{for all }t\ge 0\text{ and }x_0\in\mathcal V_0\tag 1$$ for some $\xi:\Omega\times[0,\infty)\times\mathbb R^d\to\operatorname{HS}(U_0,\mathbb R^d)$, where $\operatorname{HS}(U_0,\mathbb R^d)$ is the space of Hilbert-Schmidt operators from $U_0:=Q^{1/2}U$ to $\mathbb R^d$

The idea is that $\Phi_t(x_0)$ is the position at time $t\ge 0$ of a particle which started at position $x_0\in\mathcal V_0$ at time $0$. By $(1)$, the trajectory of each particle is perturbed by a random forcing. Let's assume that the particle system is closed (i.e. no particle is destroyed and no new particle is created). Then, if $\xi=0$, it's clear that each $\Phi_t$ should be a bijection. If $\xi\ne 0$, I somehow want to preserve that property. In particular, I want to be able to talk about the bounded domain $\mathcal V_t\subseteq\mathbb R^d$ occupied by the particle system at time $t\ge 0$. Is this possible?

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  • $\begingroup$ Have you checked the book by Da Prato and Zabczyk (SDEs in infinite dimensions) or Kunita (stochastic flows)? I guess it depends on the regularity of the functions $u_t$ and indeed $\xi_t$. In the classical case when $W$ is a std. Wiener process, the family $\{\Phi_t(x)\}_{t>0}$ is a one.parameter semigroup of diffeomorphisms (at least when the coefficients are regular) $\endgroup$ – Martingalo May 11 '16 at 10:42
  • $\begingroup$ @Martingalo I've checked Da Prato, but that doesn't help. I don't see that $(\Phi_t(x_0))_{t\ge 0}$ has the semigroup property. Can you prove it? $\endgroup$ – 0xbadf00d May 11 '16 at 10:52
  • $\begingroup$ It should be easy if the coefficients are Lipschitz. Just just have to compare $\Phi_t\Phi_s$ and $\Phi_{t+s}$ (use Grönwall). Or just simply show that both satisfy the same SDE and then it follows by uniqueness of solutions. $\endgroup$ – Martingalo May 11 '16 at 10:59
  • $\begingroup$ @Martingalo $\Phi_t\Phi_s$ is composition or multiplication? $\endgroup$ – 0xbadf00d May 11 '16 at 10:59
  • $\begingroup$ @Martingalo Side note: I've applied the Itō formula to $u_t\left(\Phi_t\left(x_0\right)\right)$ and obtained the equation $(1)$ in an other question. This question is somehow related to that question. Maybe you can help there too. $\endgroup$ – 0xbadf00d May 11 '16 at 11:02

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