• $U,H$ 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?

  • $\begingroup$ Can you please remove Stranonovich's circle from the last term in (4)? $\endgroup$ – Nawaf Bou-Rabee Sep 1 '16 at 19:04
  • $\begingroup$ @NawafBou-Rabee Why did you delete your answers of this and the other question? I wasn't able to work at these problems for the last couple of days and just wanted to face them again. I'm sorry if it takes long, but I cannot accept an answer before I understood it. Could you please undelete them? $\endgroup$ – 0xbadf00d Oct 3 '16 at 18:03

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).

| cite | improve this answer | |
  • $\begingroup$ Thank you very much for your effort. Let's be rigorous: (1) You say that $∂u/∂t$, ${\rm D}u$ and ${\rm D}^2u$ need to be continuous and locally bounded. By definition of $u$, that's the case (and note that a continuous function between normed vector spaces is always locally bounded). I guess you think that we need that in order to apply the Itō formula. However, in the formulation of the Itō formula that I know (Google Books link), we need that $∂u/∂t$, ${\rm D}u$ and ${\rm D}^2u$ are uniformly continuous on bounded subsets of $[0,T]\times H$. In the scenario of $\endgroup$ – 0xbadf00d Sep 22 '16 at 15:50
  • $\begingroup$ the question, $u$ should actually be a mapping $[0,T]\times\overlineλ\to\mathbb R^d$ for some $T>0$ and a bounded and open $λ\subseteq\mathbb R^d$ (let's assume that from now on). So, since $∂u/∂t$, ${\rm D}u$ and ${\rm D}^2u$ are continuous and $\overlineλ$ is compact, the former property is satisfied too. (2) You say that its possible to write $(1)$ in an equivalent Itō form, since $W$ is a $Q$-Wiener process. Maybe you can explain why you think that it's important that $W$ is a $Q$-Wiener process. I think that it should work in the case of a cylindrical Wiener process (which is nothing else $\endgroup$ – 0xbadf00d Sep 22 '16 at 15:51
  • $\begingroup$ than a $ιι^*$-Wiener process on another separable Hilbert space $V$ with a Hilbert-Schmidt embedding $ι$ from $U_0$ to $V$) too. (3) I think there is something wrong with your correction term. $G(t,x)$ is a mapping $U_0\to\mathbb R^d$, but your correction term involves an invocation of $G(t,x)$ with parameter $e_n$, which is in $U\supseteq U_0$. Maybe there is something I'm missing. That's likely to be the case, cause I've found the same correction term in a book (Google Books link). However, as I said, $G(t,x)e_n$ isn't defined and I think the correction term $\endgroup$ – 0xbadf00d Sep 22 '16 at 15:52
  • $\begingroup$ should be $\frac12\tilde{\text{tr}}\:Q^{1/2}{\rm D}G(t,x)G(t,x)$ with $$\underbrace{\tilde{\text{tr}}\:Q^{1/2}{\rm D}G(t,x)G(t,x)}_{=:\:F(t,x)}:=\sum_{n\in\mathbb N}\sqrt{λ_n}\left({\rm D}G(t,x)G(t,x)e_n^0\right)e_n^0$$ where $e_n^0:=\sqrt{λ_n}e_n$. $(4)$ Indeed, it's easy to see that $F(t,\;\cdot\;)$ is Lipschitz continuous for all $t\in[0,T]$. Together with the other mentioned properties of the coefficients, we obtain the existence of a mild (Itō-)solution (cf. Theorem 7.2 in Da Prato's book (Google Books link). However, we need a strong solution in order to $\endgroup$ – 0xbadf00d Sep 22 '16 at 15:53
  • $\begingroup$ apply the Itō formula, don't we? What am I missing? $(5)$ In the context of the Navier-Stokes scenario described in the question: Do you know a suitable choice for $G$? [I know these are a lot of question. Thank you, once again, in advance for your effort]. $\endgroup$ – 0xbadf00d Sep 22 '16 at 15:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.