Initial-boundary value problem for stochastic transport equations

Abstract

This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Itô’s formulation for bounded domains. Moreover, the uniqueness is established without the divergence of the drift vector field bounded from below.

Appendix

At this point we fix some notation and material used through of this paper.

Let us fix a stochastic basis with a d-dimensional Brownian motion

$$ \big ( \Omega , {\mathcal {F}}, \{ {\mathcal {F}}_t: t \in [0,T] \}, {\mathbb {P}}, (B_{t}) \big ). $$

Then, we recall to help the intuition, the following definitions

$$ \begin{aligned} \text {It}{\hat{\text {o}}}:&\ \int _{0}^{t} X_s dB_s= \lim _{n \rightarrow \infty } \sum _{t_i\in \pi _n, t_i\le t} X_{t_i} (B_{t_{i+1} \wedge t} - B_{t_i}), \\ \text {Stratonovich:}&\ \int _{0}^{t} X_s \circ dB_s= \lim _{n \rightarrow \infty } \sum _{t_i\in \pi _n, t_i\le t} \frac{ (X_{t_{i+1} \wedge t } + X_{t_i} ) }{2} (B_{t_{i+1} \wedge t} - B_{t_i}), \\ \text {Covariation:}&\ [X, Y ]_t = \lim _{n \rightarrow \infty } \sum _{t_i\in \pi _n, t_i\le t} (X_{t_{i+1} \wedge t} - X_{t_i} ) (Y_{t_{i+1} \wedge t} - Y_{t_i}), \end{aligned} $$

where $\pi _n$ is a sequence of finite partitions of [0, T] with size $|\pi _n| \rightarrow 0$ and elements $0 = t_0< t_1 < \ldots$. The limits are in the sense of probability, and uniformly in time on compact intervals. Details about these facts can be found in Kunita [18]. Also we address from that book, Itô’s formula, the chain rule for the stochastic integral, for any continuous d-dimensional semimartingale $X = (X_1,X_2,\ldots ,X_d)$, and twice continuously differentiable and real valued function f on ${\mathbb {R}}^{d}$.

Let $U \subset {\mathbb {R}}^{n}$ be an open set, and $\Gamma$ its boundary. A map $\Psi :[0,1] \times \Gamma \rightarrow {\overline{U}}$ is said an admissible deformation, when satisfies the following conditions:

(1):

For all $r \in \Gamma$, $\Psi (0,r)=r$.

(2):

The derivative of the map $[0,1] \ni \tau \mapsto \Psi (\tau , r)$ at $\tau = 0$ is not orthogonal to ${\mathbf {n}}(r)$, for each $r \in \Gamma$.

Moreover, for each $\tau \in [0,1]$, we denote: $\Psi _\tau$ the mapping from $\Gamma$ to ${\overline{U}}$, given by $\Psi _\tau (x):=\Psi (\tau ,x)$; $\Gamma ^\tau = \Psi _\tau (\Gamma )$; ${\mathbf {n}}^\tau$ the unit outward normal field in $\Gamma ^\tau$. In particular, ${\mathbf {n}}^0(r)= {\mathbf {n}}(r)$ is the unit outward normal field in $\Gamma$.

Now, we define a level set function h associated with the deformation $\Psi _\tau$. For $\delta > 0$ sufficiently small we define

$$ h(x):= \left\{ \begin{aligned} \min \{\tau ,\delta \},&\quad \text {if} \, x \in U, \\ - \min \{\tau ,\delta \},&\quad \text {if} \, x \in {\mathbb {R}}^n \setminus U. \end{aligned}\right. $$

The function h(x) is Lipschitz continuous in ${\mathbb {R}}^n$, and $C^2$ on the closure of $\left\{ x \in {\mathbb {R}}^n: |h(x)|< \delta \right\}$, see Gilbarg, Trudinger [16], p. 355.

Given a function $f \in L^1(U)$, we recall the global approximation by smooth functions, that is, $f_\varepsilon \in L^1(U) \cap C^\infty ({\overline{U}})$, such that, $f_\varepsilon \rightarrow f$ in $L^1$, see Evans, Gariepy [9] Chapter 4.2, Theorem 1 and Theorem 3. In fact, this result follows from a convenient modification of the standard mollification of f by a standard (symmetric) mollifier $\rho$, that is a positive radial and regular function with compact support in ${\mathbb {R}}^d$, such that $\int \rho (x) dx= 1$. For each $\varepsilon > 0$, we define $\rho _\varepsilon (x):= \varepsilon ^{-d} \rho (\frac{x}{\varepsilon })$. For convenience, that is to fix the notation, let us give the main idea. For any $\varepsilon > 0$ fixed, $0 \le \delta \le \varepsilon$, and $y \in {\overline{U}}$, we define

$$ y^\varepsilon := y + \lambda \, \varepsilon \, \nabla h(y), $$

for $\lambda > 0$ sufficiently large. Then, we take a standard mollifier $\rho _{\varepsilon }$, and for any $u \in L_{\text {loc}}^{1}(U_T)$, we define the following (space) global approximation

$$ u_{\varepsilon }(t,y) \equiv (u *_{{\mathbf {n}}} \rho _\varepsilon ) (t,y):= \int _{U} u(t,z) \rho _{\varepsilon }(y^\varepsilon -z) \ dz. $$

Therefore, $u_\varepsilon \in L^1_\text {loc}([0,T]; C^\infty ({\overline{U}}))$ and converges to u in $L^1_\text {loc}$.