Solving SDEs with singular drifts using PDE techniques
A talk in the Bielefeld Stochastic Afternoon series by
Zimo Hao
| Abstract: | In this talk, we investigate the following stochastic differential equation (SDE) in ${\mathbb R}^d$ driven by Brownian motion
$$
{\textrm d} X_t=b(t,X_t){\textrm d} t+\sqrt{2}{\textrm d} W_t,
$$
where the drift $b$ belongs to the space ${\mathbb L}_T^q \mathbf{H}_p^\alpha$ with $\alpha \in [-1, 0]$ and $p,q\in[2, \infty]$, which is a distribution-valued and divergence-free vector field.
In the subcritical case $\frac dp+\frac 2q<1+\alpha$, we establish the existence and uniqueness of a weak solution;
In the critical and supercritical case $1+\alpha\leq\frac dp+\frac 2q<2+\alpha$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated Markov processes.
Moreover, under the additional assumption that $b=b_1+b_2+\div a$, where $b_1\in {\mathbb L}^\infty_T{\mathbf B}^{-1}_{\infty,2}$, $b_2\in {\mathbb L}^2_TL^2$,
and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence.
To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models. (This part is based on a joint work with Xicheng Zhang)
For the subcritical case $\alpha=0$ and $\frac dp+\frac 2q<1$, we establish the path-by-path uniqueness for the following rough differential equation (RDE):
\begin{align*}
{\textrm d} X_t=b(t,X_t){\textrm d} t+\sigma(X_t){\textrm d} {\mathbf W}_t,
\end{align*}
where ${\mathbf W}_t=(W_t,\int_0^t W_s \, {\mathbf W}_s)$ represents a rough path and $\sigma$ is uniformly elliptic with $\sigma\in {\mathbf C}^3$ (This part is based on a joint work with Khoa L\^e). Within the CRC this talk is associated to the project(s): B1 |