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Motivated by P.L. Lions' seminal course on Mean Field Games (MFG) and the associated Master equation at Collège de France [19] , we investigate in the case of absence of control mean-field backward doubly stochastic differential equations (BDSDEs), i.e., BDSDEs whose driving coefficients also depend on the joint law of the solution process. This BDSDE can be interpreted as randomly perturbed recursive cost functional which we associate with a forward mean-field SDE. Defining the value function V = V (t , x , μ) through this BDSDE, we prove that V = V (t , x , μ) is the unique classical solution of a backward stochastic PDE over [ 0 , T ] × R d × P 2 (R d). This backward SPDE extends in the non-control case the Master equation studied in earlier works by different authors. The associated MFG can be interpreted as a forward dynamics endowed with a randomly perturbed recursive cost functional, the solution of the MFG system (u (t , x) , m (t)) remains also in this extended case related with V = V (t , x , μ) by the relation u (t , x) = V (t , x , m (t)). Our main objective is to characterise the random field V as classical solution of the backward SPDE. For this we study the regularity of V over [ 0 , T ] × R d × P 2 (R d). However, unlike the pioneering paper on BDSDEs by Pardoux-Peng [25] where V does not depend on the measure, here we have only an L 2 -regularity of V with respect to its variables. Other difficulties to overcome and subtle technical proofs are related with the study of the L 2 -regularity and, hence, that of the solution of the underlying mean-field BDSDE. It adds that our BDSDE needs weaker assumptions on the coefficients than in [25]. Finally, let us point out that for our approach we extend the classical mean-field Itô formula to smooth functions of solutions of mean-field BDSDEs. This formula contains earlier (mean-field) Itô formulas as special cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t (x) = Δ α 2 u t (x) d t + g (t , x) d η t , u 0 = 0 , t ∈ (0 , T ] , x ∈ G , for a random field u : (t , x) ∈ [ 0 , T ] × G ↦ u (t , x) = : u t (x) ∈ R , where Δ α 2 : = − (− Δ) α 2 , α ∈ (0 , 2 ] , is the fractional Laplacian, T ∈ (0 , ∞) is arbitrarily fixed, G ⊂ R d is a bounded domain, g : [ 0 , T ] × G × Ω → R is a joint measurable coefficient, and η t , t ∈ [ 0 , ∞) , is either a Brownian motion or a Lévy process on a given filtered probability space (Ω , F , P ; { F t } t ∈ [ 0 , T ]). To this end, we derive the BMO estimates and Morrey–Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilizing the embedding theory between the Campanato space and the Hölder space, we establish the controllability of the norm of the space C θ , θ / 2 (D ¯) , where θ ≥ 0 , D ¯ = [ 0 , T ] × G ¯. With all these in hand, we are able to show that the q -th order BMO quasi-norm of the α q 0 -order derivative of the solution u is controlled by the norm of g under the condition that η t is a Lévy process. Finally, we derive the Schauder estimate for the p -moments of the solution of the above stochastic fractional heat equations driven by Lévy noise. [ABSTRACT FROM AUTHOR]

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen–Cahn equation does not have finite time singularities and the unique solution exists globally. [ABSTRACT FROM AUTHOR]

Abstract: The paper is concerned with the problem of non-existence of global solutions for a class of stochastic reaction–diffusion equations of Itô type. Under some sufficient conditions on the initial state, the nonlinear term and the multiplicative noise, it is proven that, in a bounded domain , there exist positive solutions whose mean -norm will blow up in finite time for , while, if , the previous result holds in any compact subset of . Two examples are given to illustrate some application of the theorems. [ABSTRACT FROM AUTHOR]