# Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?

Consider the $$d$$-dimensional SDE, $$d > 1$$,

$$dX_t = b(X_t) \, dt + \sqrt 2 \, dW_t$$

where

• $$b$$ is locally Lipschitz such that $$|b(x)| \le C |x|^2$$ for $$x \in \mathbb R^d$$.
• $$W$$ is a standard $$d$$-dimensional Brownian motion.

Do you know some references about the weak uniqueness of above SDE?

Thank you so much for your elaboration!

• mathoverflow.net/questions/288996/… Commented Jan 1 at 16:49

For d=1, this is covered in Karatzas-Shreve in the section on Feller-tests. In particular, in exercise 5.38, they have that if $$\int_{x-\epsilon}^{x+\epsilon}b^{2}(y)dy<\infty,$$

then $$dX_{t}=b(X_{t})dt+dW_{t}$$ has a weak solution up to explosion time $$S$$ and it is unique in probability law. The main idea is using Girsanov theorem (hence that square-integrability condition).

For $$d\geq 1$$, see "Strong solutions of stochastic equations with singular time dependent drift" and "strong solutions of stochastic differential equations with square integrable drift", where again they require integrability: there exists open set $$Q\subset\mathbb{R}^{d+1}$$

$$\int \left(\int |b(t,x)1_{Q}(t,x)|^{p} dx\right)^{q/p}dt<\infty,$$

for some $$p\geq 2, q>2$$ with $$\frac{d}{2}+\frac{2}{q}<1$$.

• The mentioned exercise deals with the case $d=1$. Are you aware of any result when $d>1$? Commented Jan 1 at 22:24
• @Akira I added some references. Commented Jan 1 at 22:40
• It seems from this screenshot (of the first paper) that the following integrability condition is sufficient, i.e., $$\int_0^n \left(\int_{B(0, n)} |b(t,x)|^{p} dx\right)^{q}dt<\infty, \quad \forall n \in \mathbb N^*.$$ Here $B(0, n)$ is the open ball centered at $0$ with radius $n$. Could you confirm if my understanding is correct? Commented Jan 1 at 23:45
• By the definition of the space $L_q \_ L_p$, I guess you meant the superscript $q/p$ rather than $q$ in the integral. Commented Jan 1 at 23:57
• @Akira but to be clear, these solutions can blow up in finite time eg. $dX_{t}=X^2_{t}dt+dB_{t}$ blows up in finite time. So all the above results are only valid up to the explosion time. Commented Jan 2 at 0:09