A. M. Davie
We consider a d-dimensional stochastic differential equation with additive noise and a drift coefficient which is assumed only to be a bounded Borel function. We show that, for almost all choices of the driving Brownian path, the equation has a unique solution.
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We study existence and uniqueness of solutions for second order ordinary stochastic differential equations with Dirichlet boundary conditions on a given interval. In the first part of the paper we provide sufficient conditions to ensure pathwise uniqueness, extending some known results. In the second part we show sufficient conditions to have the weaker concept of uniqueness in law and provide a significant example. Such conditions involve a linearized equation and are of different type with respect to the ones which are usually imposed to study pathwise uniqueness. This seems to be the first …
This paper considers some the existence and uniqueness of strong solutions of stochastic neutral functional differential equations. The conditions on the neutral functional relax those commonly used to establish the existence and uniqueness of solutions of NSFDEs for many important classes of functional, and parallel the conditions used to ensure existence and uniqueness of solutions of related deterministic neutral equations. Exponential estimates on the almost sure and p--th mean rate of growth of solutions under the weaker existence conditions are also given.
Sufficient conditions are given for existence and uniqueness in Smoluchowski's coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski's equation.
Path-by-path uniqueness for stochastic differential equations under Krylov-Röckner condition
本文证明了在Krylov-Röckner条件下，由布朗运动驱动的随机微分方程具有逐路径唯一性，从而得到强完备性并构成随机动力系统，并讨论了其在边值问题中的应用。
We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate stochastic partial differential equations plays a key role.
We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.
We improve on the strong uniqueness results of [GLM+17], which deal with the following system of SDE. \begin{align*} dX_t&=Y_tdt \\ dY_t&=|X_{t}|^αdB_t \end{align*} and $X_0=x_0,Y_0=y_0$. For $(x_0,y_0)\ne(0,0)$, we show that short-time uniqueness holds for $α>-1/2$.
Based on the weak existence and weak uniqueness, we study the pathwise uniqueness of the solutions for a class of one-dimensional stochastic differential equations driven by pure jump processes. By using Tanaka's formula and the local time technique, we show that there is no gap between the strong uniqueness and weak uniqueness when the coefficients of the Poisson random measures satisfy a suitable condition
In the present paper, we give some examples of stochastic differential equations which have delicateness in the Markov and strong Markov properties, the uniqueness locally in time and globally in time, and initial conditions. Moreover, we show that such stochastic differential equations appear in the limits of stochastic differential equations which have the existence and pathwise uniqueness of solutions. These examples are constructed in motivation to singular stochastic partial differential equations. We also give some examples of shifted equations whose sum of the solutions depends on the c…
The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.
We prove pathwise uniqueness for solutions of parabolic stochastic pde's with multiplicative white noise if the coefficient is Hölder continuous of index $γ>3/4$. The method of proof is an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations.
We introduce a stochastic system of interacting particles which is expected to furnish as the number of particles goes to infinity a stochastic approach of the 2-D Keller-Segel model. In this note, we prove existence and some uniqueness for the stochastic model for the parabolic-elliptic Keller-Segel equation, for all regimes under the critical mass. Prior results for existence and weak uniqueness have been very recently obtained by N. Fournier and B. Jourdain [6].
A practical global existence and uniqueness result for stochastic differential equations on Riemannian manifolds of bounded geometry
本文在有界几何黎曼流形上，针对带随机漂移的正则非齐次张量系数，证明了随机微分方程的全局存在唯一性，并给出了随机流估计。
We prove an existence and uniqueness result for quasilinear Stochastic PDEs with Obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.
In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz conditions on their coefficients.
In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H > \frac{1}{2} and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.
In the article, Zakai and Kushner-Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system are considered. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner-Stratonovich equation is unique in joint law.
We prove the existence and uniqueness of a strong solution for an SDE on a semi-axis with singularities at the point 0. The result obtained yields, for example, the strong uniqueness of non-negative solutions to SDEs governing Bessel processes.
One proves that the stochastic porous media equation in 3-D has a unique nonnegative solution for nonnegative initial data in $H^{-1}(\mathcal O)$ if the nonlinearity is monotone and has polynomial growth.
In this Note, we present a Calderón-type uniqueness theorem on the Cauchy problem of stochastic partial differential equations. To this aim, we introduce the concept of stochastic pseudo-differential operators, and establish their boundedness and other fundamental properties. The proof of our uniqueness theorem is based on a new Carleman-type estimate.
We study a class of stochastic differential equations with non-Lipschitzian coefficients.A unique strong solution is obtained and a large deviation principle of Freidln-Wentzell type has been established.
The aim of this work is to prove existence and uniqueness of $L^{2}-$solutions of stochastic fractional partial differential equations in one spatial dimension. We prove also the equivalence between several notions of $L^{2}-$solutions. The Fourier transform is used to give meaning to SFPDEs. This method is valid also when the diffusion coefficient is random.
Pathwise uniqueness for stochastic heat and damped equations with Hölder continuous drift
本文在不要求结构条件成立的前提下，证明了无穷维随机微分方程的逐路径唯一性，并覆盖最高到三维的随机热方程、一维随机阻尼波动方程及最高到三维的随机Euler-Bernoulli阻尼梁方程。
The existence-uniqueness theory for solutions to stochastic dynamic systems is always a significant theme and has received a huge attention. The objective of this article is to study the mentioned theory for stochastic functional differential equations (SFDEs) driven by G-Lévy process. The existence-uniqueness theorem for solutions to SFDEs driven by G-Lévy process has been determined. The error estimation between the exact solution and Picard approximate solutions has been shown. In addition, the exponential estimate has been derived.
The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense that initial and terminal values depend on solutions, and are not just fixed random variables. We establish under some monotonicity conditions, the existence and uniqueness of strong solutions of such equations by using a continuation method.
This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an $m$-dimensional Brownian motion and a $d$-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in modelling hybrid systems, where the phenomena are simultaneously subjected to two kinds of uncertainties: randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coefficients are proved.
We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on IR^d for d greater or equal to 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hoelder continuity of the solutions near zero is established.
We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled and don't need any approximation approach. Similar equations without jumps were studied in the same context by \cite{Le Gall}, \cite{Ouknine} and others authors. As an application we get a new condition on the pathwise uniqueness for the solutions to stochastic differential equations driven by a symmetric stable Lévy processes.
One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier-Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hold for solutions to linear evolution equations in Hilbert spaces with self-adjoint principal part.
Consider the stochastic differential equation $\mathrm dX_t = -A X_t \,\mathrm dt + f(t, X_t) \,\mathrm dt + \mathrm dB_t$ in a (possibly infinite-dimensional) separable Hilbert space, where $B$ is a cylindrical Brownian motion and $f$ is a just measurable, bounded function. If the components of $f$ decay to 0 in a faster than exponential way we establish path-by-path uniqueness for mild solutions of this stochastic differential equation. This extends A. M. Davie's result from $\mathbb R^d$ to Hilbert space-valued stochastic differential equations.
Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the $L^p$-setting is discussed. As a direct application essential self--adjointness and strong uniqueness in $L^p$ is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume $Λ\subset \R^2$.
Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.
We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b,σ$. When $α\in (1,2)$, we investigate pathwise uniqueness for this equation. When $α\in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α\in (0,1)$ or $α\in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $σ…
In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.
In this paper we are concerned with the stochastic partial differential equations of super-fast diffusion processes describing behavior of plasma dX(t)-Δln(X(t)+1)dt=\surd(Q)dW(t), in (0,T)\timesO, where O is a bounded open subset of R. We define a strong solution adequate to the properties of the natural logarithm and we prove the corresponding existence and uniqueness result.
Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients
本文研究了一维广义随机微分方程中存在性和唯一性问题，特别是在分布漂移和退化、可能不连续的扩散系数下的情况。提出路径唯一性条件和粘贴方法，为广义随机微分方程提供了首个显式粘贴定理。
Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde $$\frac{\partial X}{\partial t}=\fracΔ{2}X+X^p\dot W+ψ,\ X_0\equiv 0$$ where $\dot W$ is a white noise, $ψ\ge 0$ is smooth, compactly supported and non-trivial, and $0<p<1/2$. We further show that any solution spends positive time at the 0 function.
Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
研究具有赫尔德连续系数的退化随机微分方程的路径唯一性问题，利用马利瓦因紧致性准则构造强解，通过对偶山田-渡边论证及弱唯一性得出路径唯一性。
We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $α$-stable Lévy processes, $α\in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space $\mathcal{C}^β$ for $β>1/2-α/2$.
We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.
We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $γ>1-\fracη{2(η+1)}$. Here $η\in(0,1)$ is a constant that defines the spatial regularity of the noise.
We show pathwise uniqueness for a class of degenerate Itô-SDE among all of its weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Consequently, by the Yamada-Watanabe Theorem and a weak existence result, the pathwise unique solutions can be shown to be strong and to exist. The main tools to show pathwise uniqueness are inequalities associated with maximal functions and a Krylov type estimate derived from elliptic regularity and uniqueness in law.
On conditional uniqueness of solutions to stochastic Navier-Stokes Equations
本文延续先前工作，针对随机Navier-Stokes方程，在$\u211d^d$空间中建立了解的条件唯一性和正则性定理。
We provide sufficient conditions on the coefficients of a stochastic functional differential equation with bounded memory driven by Brownian motion which guarantee existence and uniqueness of a maximal local and global strong solution for each initial condition. Our results extend those of previous works. For local existence and uniqueness, we only require the coefficients to be continuous and to satisfy a one-sided local Lipschitz (or monotonicity) condition. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general s…
A stochastic differential equation with infinite memory is considered. The drift coefficient of the equation is a nonlinear functional of the past history of the solution. Sufficient conditions for existence and uniqueness of stationary solution are given.
Uniqueness for a stochastic ideal dyadic MHD model
研究无粘性无电阻磁流体动力学随机二进模型，针对特定随机强迫证明弱唯一性，但解耗散形式上守恒的能量。
In this paper we consider stochastic differential equations with discontinuous diffusion coefficient of varying sign, for which weak existence and uniqueness holds but strong uniqueness fails. We introduce the notion of $φ$-strong solution, and we show that under certain conditions on the diffusion coefficient a $φ$-strong solution exists and it is unique. We also give a construction of a $φ$-strong solution and a weak solution in terms of the notion of \emph{i.i.d. sign choice} introduced in the paper, and we give an explicit representation of the set of all weak solutions, which in particula…
We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^α dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $α\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.
Second-order McKean-Vlasov stochastic evolution equation driven by Poisson jumps: existence, uniqueness and averaging principle
本文研究非Lipschitz条件下泊松跳驱动的二阶McKean-Vlasov随机发展方程，用Carathéodory逼近证明温和解的存在唯一性，并得到其与简化方程解在均方意义下的平均原理。
Pathwise uniqueness for singular stochastic Volterra equations with Hölder coefficients
本文证明了一类由布朗运动驱动、带奇异核和Hölder连续扩散系数的一维随机Volterra方程的道路唯一性，进而得到唯一强解的存在性。
We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along with additional restrictions. For example, consider the forcing $f(u) = u \log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.
We prove a dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.
We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochas…
In order to extend the study of uniqueness property of multi-dimensional systems of stochastic differential equations, in this paper, we look at the following three-dimensional system of equations, of which the two-dimensional case was well-studied before: $dX_t=Y_tdt\quad, dY_t=Z_tdt,\quad dZ_t=|X_t|^αdB_t$. We proved that if $(X_0,Y_0,Z_0)\neq(0,0,0)$, and $\frac{3}{4}<α<1$, then the system of equations has a unique solution in the strong sense.
In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.
Pathwise uniqueness by noise for singular stochastic PDEs
本文通过噪声正则化方法，解决了奇异随机偏微分方程中具有微分形式漂移项的路径唯一性问题，为流体动力学和相分离模型等提供了新的唯一性结果。
In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.
This paper considers the problem of uniqueness of the solutions to a class of Markovian backward stochastic differential equations (BSDEs) which are also connected to certain nonlinear partial differential equation (PDE) through a probabilistic representation. Assuming that there is a solution to the BSDE or to the corresponding PDE, we use the probabilistic interpretation to show the uniqueness of the solutions, and provide an example of a stochastic control application.
Strong existence and uniqueness for a class of quasilinear stochastic evolution equations
本文研究了在有界域上的一类quasilinear随机演算方程的强解存在性和路径唯一性，结合了最近的弱存在性结果和Yamada-Watanabe理论，利用L^1收缩论证证明路径唯一性。
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