Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions
Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions
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DOI：
10.1016/j.jde.2017.09.030
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发表时间：
2018-01
影响因子：
2.4
通讯作者：
Chunrong Feng;Xince Wang;Huaizhong Zhao
中科院分区：
文献类型：
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作者：
Chunrong Feng;Xince Wang;Huaizhong Zhao
本文研究了索波列夫空间Hρ1(Rd)中拟线性抛物型和椭圆型偏微分方程弱解的存在唯一性及其概率表示。为此，我们首先研究了具有光滑系数的正倒向随机微分方程解的正则性及其与拟线性抛物型随机微分方程解的联系。然后利用逼近方法，证明了它们在Sobolev空间中收敛于空间Lρ2(Rd；Rd)⊗Lρ2(Rd；Rk)⊗Lρ2(Rd；Rk×d)中的非线性偏微分方程解.这与拟线性抛物型偏微分方程解的弱解有关。最后，利用无穷水平上的非线性偏微分方程解，研究了拟线性椭圆型偏微分方程解的唯一性。
In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space H ρ 1 (R d). For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space L ρ 2 (R d; R d)⊗ L ρ 2 (R d; R k)⊗ L ρ 2 (R d; R k× d). This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.