Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

Abstract

I n this paper we develop a new weak convergence and compact embedding method to study the existence and uniqueness of theferential equations with p-growth coefficients. Then we establish the probabilistic representation of the weak solution of PDEs with p-growth coefficients via corresponding BSDEs.

STATIONARY STOCHASTIC VISCOSITY SOLUTIONS OF SPDEs

“…Zhang and Zhao in [20] proved that under the Lipschitz and monotone conditions, the L 2 ρ (R d ; R 1 ) L 2 ρ (R d ; R d )-valued solution of an infinite horizon BDSDE exists and gives the stationary weak solution of the corresponding parabolic SPDE, and then further considered this problem under the linear growth and monotone conditions in [21]. We believe that this method can even be extended to studying the stationary weak solutions of SPDEs under polynomial growth conditions and our recent work [22] is an intermediate step towards this goal.…”

A study of backward stochastic differential equation on a Riemannian manifold

“…initial point x ∈ T m . This kind of solution has been introduced in [1,28,41,42] to investigate the connection between FBSDE and weak solution of a quasi-linear parabolic system. The motivation of Definition 3.5 is to study the global existence of a solution to N -valued BSDE for more general N , especially for that without any convexity condition.…”

“…Suppose h ∈ C 1 (T m ; N ), then for any T > 0, there exists a solution (Y, Z) of the L 2 (T m ; N )-valued BSDE (3.5) in time interval t ∈ [0, T ]. Intuitively, a global solution of the L 2 (T m ; N )-valued BSDE (3.5) always exists for any compact Riemannian manifold N (without any other convexity condition) since the collection Ξ 0 := {x ∈ T m ; |Z x t | = +∞ for some t ∈ [0, T ]} is a Lebesgue-null set in T m (which could be seen in the proof of Theorem 3.6).Meanwhile, due to the lack of monotone condition on the generator (see the corresponding monotone conditions in[1,28,41,42]), it seems difficult to prove the uniqueness for the solution of the L 2 (T m ; N )-valued BSDE (3.5).EJP 26 (2021), paper 85.…”

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

“…t (1.4) establishes the connection between the classical and viscosity solutions of PDEs and the solutions of BSDEs (or FBSDEs), and provides a new insight into studying non-linear PDEs. Probabilistic representation of weak solutions of semi-linear PDEs in a Sobolev apace was studied by Barles-Lesigne [6], Bally-Matoussi [5] and Zhang-Zhao [31,32,33,34], and for Hamilton-Jacobi-Bellman equations by Wei-Wu-Zhao [27]. For the quasi-linear case, there are a few results about the viscosity solutions (Pardoux-Tang [23], Wu-Yu [28]).…”