# Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

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?

• 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) – Martingalo May 11 '16 at 10:42
• @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? – 0xbadf00d May 11 '16 at 10:52
• 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. – Martingalo May 11 '16 at 10:59
• @Martingalo $\Phi_t\Phi_s$ is composition or multiplication? – 0xbadf00d May 11 '16 at 10:59
• @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. – 0xbadf00d May 11 '16 at 11:02