Existence of a solution to an infinite dimensional Stratonovich SDE Let


*

*$U,H$ be separable $\mathbb R$-Hilbert spaces

*$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace

*$U_0:=Q^{1/2}U$

*$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ be a filtered probability space

*$(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$

*$u\in C^{1,\:2}([0,\infty)\times H,H)$

*$G:[0,\infty)\times H\to\operatorname{HS}(U_0,H)$ with $G(t,\;\cdot\;)$ being Fréchet differentiable for all $t\ge 0$

*$X_0$ be a $\mathcal F_0$-measurable random variable on $(\Omega,\mathcal A,\operatorname P)$ and $$X_t:=X_0+\int_0^tu(s,X_s)\:{\rm d}s+\int_0^tG(s,X_s)\circ{\rm d}W_s\;\;\;\text{for }t>0\tag 1$$


By definition of $X$, we obtain $${\rm d}u(t,X_t)=\left[\frac{\partial u}{\partial t}(t,X_t)+{\rm D}u(t,X_t)u(t,X_t)\right]{\rm d}t+\underbrace{{\rm D}u(t,X_t)G(t,X_t)}_{=:\:\Phi(t,X_t)}\circ{\rm d}W_t\tag 2$$ for all $t\ge 0$. If we interpret $t\mapsto X_t$ as being the trajectory of a fluid particle perturbed in the sense of $(1)$ and $u(t,X_t)$ as being the velocity of this particle at time $t$, then it's not far from being reasonable to assume $${\rm d}u(t,X_t)=\left[-\frac1\rho\nabla p(t,X_t)+\nu\Delta u(t,X_t)\right]{\rm d}t\tag 3$$ for all $t\ge 0$ by conservation of momentum for some $\rho,\mu>0$ and $p:[0,\infty)\times H\to\mathbb R$ with $p(t,\;\cdot\;)\in C^1(\Lambda)$ for all $t\ge 0$ for some bounded and open $\Lambda\subseteq\mathbb R^d$ for all $t\ge 0$ in the case $H=\mathbb R^d$ for some $d\in\mathbb N$. By equating $(2)$ and $(3)$, we obtain a SDE for $u$. Actually, I want to solve that SDE numerically. However, I want to ensure that a solution (in some suitable sense) is guaranteed by well-known theory (e.g. the results presented by Da Prato). $(2)$ is understood in the Stratonovich sense. Since I'm not aware of any textbook which establishes existence theory for Stratonovich SDEs, I suppose that we need to convert $(2)$ into its equivalent Itō equation $${\rm d}u(t,X_t)=\left[\frac{\partial u}{\partial t}(t,X_t)+\underbrace{{\rm D}u(t,X_t)u(t,X_t)}_{=:\:C_1(t,X_t)}+\frac12\underbrace{\sum_{n\in\mathbb N}\sqrt{\lambda_n}\left({\rm D}\Phi(t,X_t)\left(\Phi(t,X_t)e_n\right)\right)e_n}_{=:\:C_2(t,X_t)}\right]{\rm d}t+\Phi(t,X_t){\rm d}W_t\tag 4$$ for all $t\ge 0$, where $(e_n)_{n\in\mathbb N}$ is an orthonormal basis of $U$ with $$Qe_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 5$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq[0,\infty)$.

My problem is that I'm unsure whether the nonlinear part of the drift in $(4)$ satisfies the usual growth and Lipschitz conditions as, for example, presented in Da Prato's book in (7.25) and (7.26). Namely: $$\left\|C(t,x)-C(t,y)\right\|_H\le K\left\|x-y\right\|_H\;\;\;\text{for all }t\ge 0\text{ and }x,y\in H\tag 6$$ and $$\left\|C(t,x)\right\|_H\le K(1+\left\|x\right\|_H)\;\;\;\text{for all }t\ge 0\text{ and }x\in H\tag 7$$ for some $K>0$, where $C:=C_1+\frac12C_2$. So, the question is: Does the drift satisfy these assumptions such that we can use the results presented in the book? And if not, can the existence of a solution be obtained in another way?

 A: In order for (2) to hold, it seems that Ito's formula needs to hold for the stochastic process $X(s)$, $0 \le s \le t$.  This already requires that: (i) this process is well-defined; and (b) the Frechet partial derivatives $\partial_t u$, $\partial_x u$, and $\partial_{xx} u$ are continuous and locally bounded.  To check that (1) is well-defined, please write it in Ito form - which is doable since $W$ is a $Q$-Wiener process.  The Ito form of (1) is then: 
$$
dX_t = \underbrace{\left(u(t,X_t) + \frac{1}{2} \sum_{n \in \mathbb{N}} \sqrt{\lambda_n} ( \partial_x G(t,X_t) (G(t,X_t) e_n)) e_n\right)}_{=\tilde u(t,X_t)}  dt + G(t,X_t) dW_t 
$$
If one insists on using existence/uniqueness theory based on Lipschitz nonlinearities, then one has to check that $\tilde u(t,x)$ is globally Lipschitz.  One can do this by verifying that its Frechet partial derivative $\partial_x \tilde u(t,x)$ is bounded uniformly in $x$ for any $t$.  To obtain this bound it suffices that: for any $t>0$ and for all $x \in H$, there exists a real constant $C>0$ such that
$$
\max\left\{ \| \partial_x u(t,x) \|_H,  \| \partial_{xx} G(t,x) \|, \| \partial_x G(t,x) \|, \| G(t,x) \|  \right\} \le C \;.
$$ The linear growth condition follows immediately from this assumption. One can analogously obtain sufficient conditions for (4).
