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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!

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1 Answer 1

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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 $\frac{d}{2}+\frac{2}{q}<1$.

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  • $\begingroup$ The mentioned exercise deals with the case $d=1$. Are you aware of any result when $d>1$? $\endgroup$
    – Akira
    Commented Jan 1 at 22:24
  • $\begingroup$ @Akira I added some references. $\endgroup$ Commented Jan 1 at 22:40
  • $\begingroup$ 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? $\endgroup$
    – Akira
    Commented Jan 1 at 23:45
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    $\begingroup$ By the definition of the space $L_q \_ L_p$, I guess you meant the superscript $q/p$ rather than $q$ in the integral. $\endgroup$
    – Akira
    Commented Jan 1 at 23:57
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    $\begingroup$ @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. $\endgroup$ Commented Jan 2 at 0:09

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