Journal of Theoretical Probability最新文献

By investigating the regularity of the nonlinear semigroup (P_t^*) associated with the distribution dependent second-order stochastic differential equations, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions being (beta )-Hölder continuous (with (beta > frac{2}{3})) on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in (L^2)-Wasserstein distance.

In this paper, we consider stochastic reaction–diffusion equations with superlinear drift on the real line (mathbb {R}) driven by space–time white noise. A Freidlin–Wentzell large deviation principle is established by a modified weak convergence method on the space (C([0,T], C_textrm{tem}(mathbb {R}))), where (C_textrm{tem}(mathbb {R}):={fin C(mathbb {R}): sup _{xin mathbb {R}} left( |f(x)|e^{-lambda |x|}right) <infty text { for any } lambda >0}). Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space–time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the specially designed family of norms on the Fréchet space (C([0,T], C_textrm{tem}(mathbb {R}))), one-order moment estimates of the stochastic convolution, and two nonlinear Gronwall-type inequalities play an important role.

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive ((d-6))-dimensional capacity, the solution visits this set almost surely.

We study the density fluctuations at equilibrium of the multi-species stirring process, a natural multi-type generalization of the symmetric (partial) exclusion process. In the diffusive scaling limit, the resulting process is a system of infinite-dimensional Ornstein–Uhlenbeck processes that are coupled in the noise terms. This shows that at the level of equilibrium fluctuations the species start to interact, even though at the level of the hydrodynamic limit each species diffuses separately. We consider also a generalization to a multi-species stirring process with a linear reaction term arising from species mutation. The general techniques used in the proof are based on the Dynkin martingale approach, combined with duality for the computation of the covariances.

Here, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.

For any fixed real (a > 0) and (x in {mathbb {R}}^d, d ge 1), we consider the real-valued random process ((S_n)_{n ge 0}) defined by ( S_0= a, S_n= a+ln vert g_ncdots g_1xvert , n ge 1), where the (g_k, k ge 1, ) are i.i.d. nonnegative random matrices. By using the strategy initiated by Denisov and Wachtel to control fluctuations in cones of d-dimensional random walks, we obtain an asymptotic estimate and bounds on the probability that the process ((S_n)_{n ge 0}) remains nonnegative up to time n and simultaneously belongs to some compact set ([b, b+ell ]subset {mathbb {R}}^+_*) at time n.

This paper gives a new property for stochastic processes, called square-mean (mu -)pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces

We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by G-Brownian motion (G-BSDEs) with diagonal generators. Two methods, including the penalization method and the Picard iteration argument, are provided to prove the existence and uniqueness of the solutions. We also study its connection with the obstacle problem of a system of fully nonlinear PDEs.

The present article describes the precise structure of the (L^{p})-spaces of projective limit measures by introducing a category-theoretical perspective. This analysis is applied to measures on vector spaces and in particular to Gaussian measures on nuclear topological vector spaces. A simple application to constructive quantum field theory (QFT) is given through the Osterwalder–Schrader axioms.

In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift b and the weak gradient of Sobolev diffusion (sigma ) are supposed to satisfy (left| left| bright| cdot mathbbm {1}_{B(R)}right| _{p_1}le O((log R)^{{(p_1-d)^2}/{2p^2_1}})) and (left| left| nabla sigma right| cdot mathbbm {1}_{B(R)}right| _{p_1}le O((log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})), respectively. The main tools for these results are the decomposition of global two-point motions in Fang et al. (Ann Probab 35(1):180–205, 2007), Krylov’s estimate, Khasminskii’s estimate, Zvonkin’s transformation and the characterization for Sobolev differentiability of random fields in Xie and Zhang (Ann Probab 44(6):3661–3687, 2016).