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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 σ are supposed to satisfy b · 1 B (R) p 1 ≤ O ((log R) (p 1 - d) 2 / 2 p 1 2 ) and ∇ σ · 1 B (R) p 1 ≤ O ((log (R / 3)) (p 1 - d) 2 / 2 p 1 2 ) , 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). [ABSTRACT FROM AUTHOR]

AbstractWe consider a general class of semi-Markov processes on countable state-space, with a differentiable kernel such that the embedded Markov chain may not be homogeneous. Using an SDE representation of the process, we study the associated flow by investigating meetings and merging related events for solutions, starting with different initial conditions. Some sufficient conditions for almost sure meeting and merging are also obtained. Many examples are considered to clarify the intricacies and implications of this study. [ABSTRACT FROM AUTHOR]

We study the existence of the stochastic flow associated to a linear stochastic evolution equation \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A is the generator of a C_0-semigroup, and (B_k) is a sequence of bounded linear operators such that \sum _k\|B_k\|<+\infty. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert–Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well. [ABSTRACT FROM AUTHOR]

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In this paper, we will study an important property on the regularity of the trajectories of the stochastic flow generated by a famous model in finance. More precisely, we prove the differentiability with respect to initial data of the solution of the stochastic differential equation associated with this model based on Gronwall's lemma, Itô's isometry and Burkholder–Davis–Gundy's and Hölder's inequalities. This is the main motivation of our research. [ABSTRACT FROM AUTHOR]

Consider a compact submanifold N without the boundary of a Riemannian manifold M, and a stochastic flow φ t associated with a stochastic differential equation. Let N t = φ t (N) be the random compact submanifold obtained by the action of the stochastic flow. In this work, we present an Itô formula for the volume of the random variable N t and, as a main result, we obtain estimates for its average growth assuming that Ricci curvature is bounded. We first analyse the particular case where the submanifolds are closed curves, thus obtaining estimates for the arc length, and then we study the volume variation of compact submanifolds of dimensions greater than or equal to 2. In addition, we apply our results to the special case where the vector fields of stochastic differential equation are conformal Killing. [ABSTRACT FROM AUTHOR]

Based on the same model stated in [3], we will study the differentiability of the stochastic ow generated by the natural model with respect to the initial data, based on an important idea of H-Kunita, R.M-Dudley and F-Ledrappier. This is the main motivation of our research. [ABSTRACT FROM AUTHOR]

We characterize the revenue-maximizing mechanism for time separable allocation problems in continuous time. The willingness-to-pay of each agent is private information and changes over time.We derive the dynamic revenue-maximizing mechanism, analyze its qualitative structure and frequently derive its closed form solution. In the leading example of repeat sales of a good or service, we establish that commonly observed contract features such as flat rates, free consumption units and two-part tariffs emerge as part of the optimal contract. We investigate in detail the environments in which the type of each agent follows an arithmetic or geometric Brownian motion or a mean-reverting process. We analyze the allocative distortions and show that depending on the nature of the private information the distortion might increase or decrease over time.

In this paper, we will try to study the same result proved in [10]. So, on the same model and with some assumptions, we will study the property of homeomorphism of the stochastic ow generated by the natural model in a one-dimensional case and with some modifications, based on an important theory of Hiroshi Kunita. This is the main motivation of our research. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish the Lipschitz and W 2 , ∞ estimates for a second-order parabolic PDE ∂ t u (t , x) = 1 2 Δ u (t , x) + f (t , x) on R d with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability of strong solutions to a class of SDEs with singular drift coefficients. [ABSTRACT FROM AUTHOR]

This work is devoted to long-time properties of the Arratia flow with drift – a stochastic flow on R whose one-point motions are weak solutions to a stochastic differential equation d X (t) = a (X (t)) d t + d w (t) that move independently before the meeting time and coalesce at the meeting time. We study special modification of such flow that gives rise to a random dynamical system and thus allows to discuss stationary points. Existence of a unique stationary point is proved in the case of a strictly monotone Lipschitz drift by developing a variant of a pullback procedure. Connections between the existence of a stationary point and properties of a dual flow are discussed. [ABSTRACT FROM AUTHOR]

We consider the following stochastic partial differential equation, d Y t = L ∗ Y t dt + A ∗ Y t ⋅ d B t Y 0 = ψ , associated with a stochastic flow {X(t,x)}, for t ≥ 0, x ∈ ℝ d , as in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162, 2008). We show that the strong solutions constructed there are 'locally of compact support'. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space S ′ . We show uniqueness of solutions in the case when ψ is smooth via the 'monotonicity inequality' for (L∗,A∗), which is a known criterion for uniqueness. [ABSTRACT FROM AUTHOR]

We consider a random map x →F(ω; x) and a random variableϴ(ω), and we denote by FN(ω; x) and ϴN(ω) their approximations: We establish a strong convergence result, in Lp-norms, of the compound approximation FN(ω;ϴ(ω)) to the compound variable F(ω; ϴ(ω)), in terms of the approximations of F and ϴ. We then apply this result to the composition of two Stochastic Differential Equations (SDEs) through their initial conditions, which can give a way to solve some Stochastic Partial Dfferential Equations (SPDEs), in particular those from stochastic utilities. [ABSTRACT FROM AUTHOR]

Stochastic differential equations give a constructive way to obtain a large number of stochastic processes. They can be used as well to simulate these processes numerically. As a by-product we can construct numerically solutions of the Cauchy problem for some parabolic equations and systems. In this paper we derive systems of stochastic equations for stochastic processes underlying systems of coupled parabolic equations which are particular cases of cross-diffusion systems of PDEs. Stochastic processes satisfying these stochastic systems allow to obtain probabilistic representations of classical and weak solutions of the Cauchy problem for original PDE systems and can be used to simulate the PDE solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the following stochastic differential equation (SDE) in Rd: dXt = dZt + b(t,Xt) dt, X0 = x, where Z is a L'evy process. We show that for a large class of L'evy processes Z and H¨older continuous drifts b, the SDE above has a unique strong solution for every starting point x ∈ Rd. Moreover, these strong solutions form a C1-stochastic flow. As a consequence, we show that, when Z is an α-stable-type L'evy process with α ∈ (0, 2) and b is a bounded β-H¨older continuous function with β ∈ (1 - α/2, 1), the SDE above has a unique strong solution. When α ∈ (0, 1), this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for ∇Exf(Xt) when Z is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with H¨older continuous b and f: ∂tu + Lu + b · ∇u + f = 0, u(1, ·) = 0, where L is the generator of the L'evy process Z. [ABSTRACT FROM AUTHOR]

Existence of random dynamical systems for a class of coalescing stochastic flows on ℝ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered. [ABSTRACT FROM AUTHOR]

Permeability measurements show that unconventional rock is considerably stress sensitive because the matrix permeability is likely controlled by micro cracks and bedding structures in unconventional reservoirs. While there are a number of laboratory studies in the literature on the dependency of core-scale permeability on effective stress for unconventional rocks, how to determine the corresponding large-scale stress-dependency relationship (of more interest to practical applications) from the laboratory measurements has not been systematically investigated. This work proposes a method to estimate such a large-scale relationship from laboratory measurements. Based on the stochastic approach commonly used for parameter upscaling, we derived relationships between the large-scale effective permeability and the stress for the two- and three-dimensional isotropic porous media. The development is based on the empirical observation that at core scale permeability is an exponential function of effective stress. The developed large-scale relationships can be written in terms of the same mathematical form as the local-scale relationship except parameters in the large-scale relationships correspond to effective ones. The effective stress sensitivity parameter (that characterizes the stress-dependency) is simply the expected value of that at the local scale, or the arithmetic average of local values, for the two-dimensional flow problem and a function of effective stress for the three-dimensional problem. Because of its dominant two-dimensional flow along beddings (resulting from the fact that vertical permeability is significantly smaller than the horizontal one), the relationship for the two-dimensional flow case is valid for unconventional rocks. Nevertheless, we demonstrate that for typical local-scale parameter values from unconventional rocks (e.g., Barnett shale and a carbonate source rock), the relationships obtained for two- and three-dimensional problems give the essentially same results. [ABSTRACT FROM AUTHOR]

We study the degenerate Itô SDE on whose drift coefficient only fulfills a mixed Osgood and Sobolev regularity. Under suitable assumptions on the gradient of the diffusion coefficient and on the divergence of the drift coefficient, we prove the existence and uniqueness of generalized stochastic flows associated to such equations. We also prove the uniqueness of solutions to the corresponding Fokker-Planck equation by using the probabilistic method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a probabilistic representation formula for the Navier-Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of ℝ or of T. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham-Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose it as a sum of the square of Lie derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we propose a local model reduction approach for subsurface flow problems in stochastic and highly heterogeneous media. Generally, we apply the mixed generalized multiscale finite element method (MGMsFEM) to solve the single-phase flow equation. As for the two-phase flow, we use mixed finite element method (MFEM) with piecewise constant bases on a fine mesh to deal with the transport equation and a standard Implicit Pressure Explicit Saturation (IMPES) scheme is utilized in the time-marching process. To guarantee the mass conservation, we consider the mixed formulation of the flow problem and aim to solve the problem in a coarse grid to reduce the complexity of a large-scale system. We decompose the entire problem into a training and a testing stage, namely the offline coarse-grid multiscale basis generation stage and online simulation stage with different parameters. In the training stage, a parameter-independent and small-dimensional multiscale basis function space is constructed, which includes the media, source and boundary information. The key part of the basis generation stage is to solve some special local problems. With the parameter-independent basis space, one can efficiently solve the concerned problems corresponding to different samples of permeability field in a coarse grid without repeatedly constructing a multiscale space for each new sample. A rigorous analysis on convergence of the proposed method is presented. In particular, we consider a generalization error, where bases constructed with one source will be used to a different source. In the numerical experiments, we apply the proposed method for both single-phase and two-phase flow problems. Simulation results for both 2D and 3D representative models demonstrate the high accuracy and efficiency of the proposed model reduction techniques. [ABSTRACT FROM AUTHOR]

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDSDEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a system of reflected SPDEs in a domain is given via such RBDSDEs. The solution is expressed as a pair ( u , ν ) where u is a predictable continuous process which takes values in a Sobolev space and ν is a random regular measure. The bounded variation process K , the component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D . This bounded variation process determines the measure ν from a particular relation by using the inverse of the flow associated to the diffusion operator. [ABSTRACT FROM AUTHOR]

We construct a probabilistic representation for the generalized solution of the Cauchy problem for the system of strongly coupled nonlinear parabolic equations describing the growth of cells under contact inhibition. [ABSTRACT FROM AUTHOR]

We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on $${\mathbb {T}}^2$$ driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. $$\alpha >\frac{1}{2}$$ ) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the $$L^p$$ -norm and a boot-strapping argument. The upper semicontinuity of random attractors is also established. Moreover, if the viscosity constant is large enough, the system has a trivial random attractor. [ABSTRACT FROM AUTHOR]

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong Δ-completeness which means that there exists a modification of the solution which if restricted to any set A ⊂ R d of dimension Δ is almost surely continuous with respect to the initial condition. [ABSTRACT FROM AUTHOR]