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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 work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal Burgers equation on a one dimensional bounded domain. The existence of a martingale solution is shown by using a Galerkin approximation, Prokhorov's theorem and Skorokhod's embedding theorem. The same Galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal Burgers equation on a bounded domain. [ABSTRACT FROM AUTHOR]

This paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non-homogeneous boundary condition. We first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general L 1 -data. Our results allow us to give a positive answer to an open problem posed by Bauzet, Vallet and Wittbold in [4] . [ABSTRACT FROM AUTHOR]

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer–Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes where the Meyer–Itô assumptions are weakened even further. We study a version of Itô's formula for multi-dimensional finite variation Lévy processes assuming that the underlying function is continuous and admits weak derivatives. We also discuss some applications of this extension, particularly in finance. [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: In this paper we solve an infinite-horizon linear quadratic control problem for a class of differential equations with countably infinite Markov jumps and multiplicative noise. The global solvability of the associated differential Riccati-type equations is studied under detectability hypotheses. A nonstochastic, operatorial approach is used. Some properties of the linear stochastic systems, such as stability, stabilizability and detectability, are also discussed on the basis of a new solution representation result. A generalized Ito's formula which applies to infinite dimensional stochastic differential equations with countably infinite Markov jumps is also provided. [Copyright &y& Elsevier]

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]

Abstract: In this paper, we present a DI SIR epidemic model with two categories stochastic perturbations. The long time behavior of the two stochastic systems is studied. Mainly, we show how the solution goes around the infection-free equilibrium and the endemic equilibrium of deterministic system under different conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. We show there is a unique positive solution to the system with positive initial value, and mainly investigate the long time behavior of the system. Condition for the system to be extinct is given and persistent condition is established. At last, numerical simulations are carried out to support our results. [Copyright &y& Elsevier]

Abstract: This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69–84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population. [Copyright &y& Elsevier]

Abstract: In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. Numerical simulations conclude the paper. [Copyright &y& Elsevier]

Abstract: We establish a scaling limit theorem for a large class of Dawson–Watanabe superprocesses whose underlying spatial motions are symmetric Hunt processes, where the convergence is in the sense of convergence in probability. When the underling process is a symmetric diffusion with -coefficients or a symmetric Lévy process on whose Lévy exponent is bounded from below by for some and when is large, a stronger almost sure limit theorem is established for the superprocess. Our approach uses the principal eigenvalue and the ground state for some associated Schrödinger operator. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized non-autonomous logistic equation , where is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164–172], the authors show that has a unique positive T-periodic solution provided , and are continuous T-periodic functions, , and . We show that this equation is stochastically permanent and the solution is globally attractive provided , and are continuous T-periodic functions, , and . By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded. [Copyright &y& Elsevier]

Abstract: In this paper we will develop a new stochastic population model under regime switching. Our model takes both white and color environmental noises into account. We will show that the white noise suppresses explosions in population dynamics. Moreover, from the point of population dynamics, our new model has more desired properties than some existing stochastic population models. In particular, we show that our model is stochastically ultimately bounded. [Copyright &y& Elsevier]

Abstract: In this paper we introduce stochasticity into a model of AIDS and condom use via the technique of parameter perturbation which is standard in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as desired in any population dynamics. We also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment. Our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to improve some results obtained by Mao [X. Mao, The LaSalle-type theorems for stochastic functional differential equations, Nonlinear Stud. 7 (2000) 307–328]. Our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above. For example, we need only the local Lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions. To guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation (SFDE) under the weaker conditions imposed in this paper, we establish a generalised existence-and-uniqueness theorem which covers a wider class of nonlinear SFDEs as demonstrated by the examples discussed in this paper. Moreover, from our improved results follow some new criteria on the stochastic asymptotic stability for SFDEs. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô''s- and Tanaka''s-type formulae associated to . [Copyright &y& Elsevier]

Abstract: In this paper we stochastically perturb the delay Lotka–Volterra model into the stochastic delay differential equation (SDDE) The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: [(i)] If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small. [(ii)] When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties. [Copyright &y& Elsevier]

Abstract: This paper discusses a randomized nonautonomous logistic equation where is 1-dimensional standard Brownian motion. We show that has a unique positive T-periodic solution provided , , and are continuous T-periodic functions, , and . [Copyright &y& Elsevier]

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka–Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka–Volterra model x˙(t)=diag(x1(t),…,xn(t))[b+Ax(t−τ)] into the Itoˆ form dx(t)=diag(x1(t),…,xn(t))[(b+Ax(t−τ)) dt+σx(t) dw(t)], and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A. [Copyright &y& Elsevier]

The main aim of this note is to improve some results obtained in the author''s earlier paper (1999, J. Math. Anal. Appl.236, 350–369). From the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness. [Copyright &y& Elsevier]

Using Itô's formula for processes with jumps, we extend the Hardy–Stein identity proved in [7] to non-symmetric pure jump Lévy processes and derive its martingale version. By a symmetrization argument for Littlewood–Paley functions, the L p boundedness of Fourier multipliers arising from non-symmetric pure jump Lévy processes is proved. [ABSTRACT FROM AUTHOR]