Strong uniqueness for an SPDE via backward doubly stochastic differential equations

Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations

“…This can be regarded as a stochastic version of the wellknown Feynman-Kac formula which gives a probabilistic interpretation for second-order SPDEs of parabolic types. Thereafter this subject has attracted many mathematicians; refer to Bally and Matoussi [2], Gomez et al [3], Hu and Ren [4], Ren et al [5]; see also Zhang and Zhao [6][7][8].…”

“…Due to their important significance to SPDEs, the researches for BDSDEs have been in the ascendant (cf. [2][3][4][5][6][7][8] and their references). Peng and Shi [9] introduced a type of time-symmetric forward-backward stochastic differential equations, that is, the so-called fully coupled forward-backward doubly stochastic differential equations (FBDSDEs):…”

A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

“…We introduce the motivations of the problem. In recent decades, a second-order SPDE with a multiplicative space-time white noise has been considered as one of the main problems in the theory of SPDEs; see [2,3,10,14,15,20,[22][23][24]26,27]. For example, consider a nonlinear SPDE driven by a space-time white noiseẆ ; u t (ω, t, x) = u(ω, t, x) + f (ω, t, x, u) + |u(ω, t, x)| γẆ ; u(ω, 0, •) = u 0 (•), (1.2) where γ > 0.…”

“…where γ > 0, Ẇ is the space-time white noise, and O is either R or (0, 1); see [17,18,19,20,21,22,23,24,25,26,27,28]. Especially, [18,19,20,21,27,28] contain the results related to the SPDEs with super-linear diffusion coefficients (e.g.…”

Backward Doubly SDEs with weak Monotonicity and General Growth Generators

“…Was firstly initiated by Pardoux and Peng [1] to give probabilistic interpretation for the solutions of a class of semilinear SPDEs where the coefficients are smooth enough, the idea is to connect the following BDSDEs system After what's realised by Pardoux and Peng [1] numerous authors show the connections between BDSDEs and solutions of stochastic partial differential equations. Bally and Matoussi [5], and [6], [7], studied the solutions of quasilinear SPDEs in Sobolev spaces in terms of BDSDEs with Lipschitz coefficients, in Bahlali et all [8], and [9], they have prove the existence and uniqueness of probabilistic solutions to some semilinear stochastic partial differential equations (SPDEs) with superlinear growth gernerator, Zhang and Zhao [10] considered BDSDEs under Lipschitz conditions in spatial integral form on infinite horizon and related their solutions with the stationary solutions of certain SPDEs. And then [11], [12], [13] studied the same BDSDE but under linear growth and monotonicity conditions.…”