Stochastic differential equations with singular coefficients on the straight line

Consider the following stochastic differential equation (SDE): Xt=x+∫0tb(s,Xs)ds+∫0tσ(s,Xs)dBs,0≤t≤T,x∈R,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$\end{document} where {Bs}0≤s≤T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{B_{s}\}_{0\leq s\leq T}$\end{document} is a 1-dimensional standard Brownian motion on [0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,T]$\end{document}. Suppose that q∈(1,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\in (1,\infty ]$\end{document}, p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in (1,\infty )$\end{document}, b=b1+b2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b=b_{1}+b_{2}$\end{document}, b1∈Lq(0,T;Lp(R))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$\end{document} such that 1/p+2/q<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1/p+2/q<1$\end{document} and b2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{2}$\end{document} is bounded measurable, with σ∈L∞(0,T;Cu(R))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma \in L^{\infty }(0,T;{\mathcal{C}}_{u}(\mathbb{R}))$\end{document} there being a real number δ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\delta >0$\end{document} such that σ2≥δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma ^{2}\geq \delta $\end{document}. Then there exists a weak solution to the above equation. Moreover, (i) if σ∈C([0,T];Cu(R))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$\end{document}, all weak solutions have the same probability law on 1-dimensional classical Wiener space on [0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,T]$\end{document} and there is a density associated with the above SDE; (ii) if b2=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{2}=0$\end{document}, p∈[2,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in [2,\infty )$\end{document} and σ∈L2(0,T;Cb1/2(R))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$\end{document}, the pathwise uniqueness holds.


Introduction and main results
Consider the following stochastic differential equation (SDE) in R d : where T > 0 is a given real number, b : [0, T] × R d − → R d , σ : [0, T] × R d − → R d×k are Borel measurable functions and {B t } 0≤t≤T is a k-dimensional standard Brownian motion defined on a given stochastic basis ( , F, {F t } 0≤t≤T , P). The fundamental theory for (1.1) is developed mainly by Itô and furnishes a very important tool to construct diffusion process. Under the Lipschitz and linear growing conditions, Itô showed the existence and uniqueness of strong solutions.
Later, the result was sharped by a series of authors on the case of bounded measurable coefficients. In [1], Skorokhod proved that (1.1) had a solution under the condition that b and σ are only continuous (also see [2]), and then the problem of the uniqueness of solutions becomes important. When b is bounded measurable, σ is bounded continuous and σ σ is strictly elliptic, Strook-Varadhan [3,4] showed the uniqueness in the probability laws. This uniqueness result is then strengthened by Veretennikov [5] for strong uniqueness if b is only bounded measurable but σ (t, ·) is Lipschitz continuous uniformly in t ∈ [0, T].
When the coefficients are not bounded but only integrable, the existence and uniqueness for solutions is more difficult. A breathtaking work in this direction has been established by  for σ = I d×d and This result was then extended by Fedrizzi-Flandoli [7,8]. Later, Zhang [9] generalized their results to the non-constant diffusion coefficients: σ (t, ·) is uniformly continuous uniformly in t ∈ [0, T], σ σ is uniformly elliptic and |∇ x σ | ∈ L q (0, T; L p (R d )) with p, q ∈ (1, ∞) and 2/q + d/p < 1. For more details in this direction, we refer to [10][11][12][13]. For some extensions and applications, we refer to [14][15][16][17][18] and the references cited therein.
Since b is only integrable in [6], the non-degenerate assumption on σ σ T is needed. When the diffusion coefficients are degenerate, we should assume b more regular. When where is a positive increasing concave function, ρ is positive and increasing, Yamada-Watanabe [19] proved the pathwise uniqueness. Recently, Fang-Zhang [20] generalized this result to d ≥ 1. By assuming that there is a small enough constant c 0 such that when |x -y| ≤ c 0 , (|x -y|) = |x -y|r(|x -y|) and ρ(|x -y|) = |x -y|r(|x -y|) (r ∈ C 1 (R + )), they derived the pathwise uniqueness. Set the space L q (0, T; L p (R d )), 2/q + d/p < 1 by L. Then all above results for (1.1) can be summed by the scheme in Table 1. From the table, we will ask: if b is in class of L and σ is non-degenerate, does there exist a unique weak/strong solution to (1.1) if σ is continuous or satisfies (1.4)?
To solve the above question, let us consider (1.1) on the straight line, where T > 0 is a given real number, b : [0, T] × R − → R, σ : [0, T] × R − → R are Borel measurable functions. We will give a positive answer for the above question, and initially to denote the space consisted of functions which is bounded and continuous on R, and use C u (R) to denote the space consisted of functions which is bounded and uniformly continuous on R. Our first main result is presented now.
and almost surely, for all t ∈ [0, T], . Therefore, we develop a new and different existence and uniqueness result to (1.5).
If σ is not Hölder continuous in spatial variable but only uniformly continuous, the uniqueness for weak solutions holds true as well if we suppose further that it is continuous in t. It is our second main result.

Theorem 1.2 Let p, q and b 1 be described in Theorem
) and there is a real number δ > 0 such that σ 2 ≥ δ. Then all weak solutions of (1.5) possess the same probabil- (ii) Thanks to [21,Lemma p. 75], the uniqueness in probability law implies the pathwise uniqueness for d = 1, therefore we obtain the existence and uniqueness for strong solutions.

Lemma 2.2 ([9, Theorem 2.2])
SupposeX · ∈ S b,σ . Let p, q ∈ (1, ∞) such that 1/p + 2/q < 1 and b, f ∈ L q (0, T; L p (R)). Then there is a constant C > 0, which depends on p, q, T, b and σ , such that We are now in a position to give the proof details of Theorem 1.1.

Proof of Theorem 1.2
Let (X t , B t ) 0≤t≤T be a weak solution of (1.5) on a probability space ( , F, P) with a reference family {F t } 0≤t≤T , and let (X t ,B t ) 0≤t≤T be another weak solution of (1.5) on a probability space (˜ ,F ,P) with a reference family {F t } 0≤t≤T . We denote the probability laws of by P x = P • X -1 andP x = P •X -1 , respectively. for every t ∈ [0, T] and every f ∈ C b (R).
Let λ > 0, we consider the following Cauchy problem: where a(t, x) = σ 2 (t, x). By virtue of Lemma 2.1, there is a unique solution u of (3.2). Moreover, if we define Y t = (t, X t ) = X t + u(t, X t ), = -1 , then (2.4) is true. In view of Itô's rule and using the same notation as in (2.5), it yields dY t = λu t, (t, Y t ) dt + b 2 t, (t, Y t )