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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?

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  • $\begingroup$ Can you please remove Stranonovich's circle from the last term in (4)? $\endgroup$
    – Nawaf Bou-Rabee
    Commented Sep 1, 2016 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
    Commented Oct 3, 2016 at 18:03

1 Answer 1

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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).

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  • $\begingroup$ 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
    Commented Sep 22, 2016 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
    Commented Sep 22, 2016 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
    Commented Sep 22, 2016 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
    Commented Sep 22, 2016 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
    Commented Sep 22, 2016 at 15:53

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