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Uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs  [PDF]
Richard F. Bass,Edwin A. Perkins
Mathematics , 2011,
Abstract: We prove uniqueness in law for a class of parabolic stochastic partial differential equations in an interval driven by a functional A(u) of the temperature u times a space-time white noise. The functional A(u) is H\"older continuous in u of order greater than 1/2. Our method involves looking at an associated system of infinite-dimensional stochastic differential equations and we obtain a uniqueness result for such systems.
On weak uniqueness for some degenerate SDEs by global $L^p$ estimates  [PDF]
Enrico Priola
Mathematics , 2013,
Abstract: We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the H\"ormander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.
On local equivalence of star-products on Poisson manifolds  [PDF]
Ziemowit Domanski,Maciej Blaszak
Physics , 2013,
Abstract: We present a proof that every star-product defined on a Poisson manifold and written in a given quantum canonical coordinate system is uniquely equivalent with a Moyal product associated with this coordinate system. The equivalence is assumed to satisfy some additional conditions which guarantee its uniqueness. Moreover, the systematic construction of such equivalence is presented and a formula for this equivalence in a case of a particular class of star-products is given, to the fourth order in $\hbar$.
Pathwise uniqueness and continuous dependence for SDEs with nonregular drift  [PDF]
E. Fedrizzi,F. Flandoli
Mathematics , 2010, DOI: 10.1080/17442508.2011.553681
Abstract: A new proof of a pathwise uniqueness result of Krylov and R\"{o}ckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
Yamada-Watanabe Theorem for Stochastic Evolution Equation Driven by Poisson Random Measure  [PDF]
Huiyan Zhao
ISRN Probability and Statistics , 2014, DOI: 10.1155/2014/982190
Abstract: The purpose of this paper is to give a detailed proof of Yamada-Watanabe theorem for stochastic evolution equation driven by pure Poisson random measure. 1. Introduction The main purpose of this paper is to establish the Yamada-Watanabe theory of uniqueness and existence of solutions of stochastic evolution equation driven by pure Poisson random measure in the variational approach. The classical paper [1] has initiated a comprehensive study of relations between different types of uniqueness and existence (e.g., strong solutions, weak solutions, pathwise uniqueness, uniqueness, and joint uniqueness in law) arising in the study of SDEs (see, e.g., [2–4]) and the study is still alive today. New papers are published (see, e.g., [2, 3, 5–7]). In this paper we are concerned with the similar question for stochastic evolution equation driven by Poisson random measure by using the method of Yamada and Watanabe. Yamada and Watanabe's initial work [1] proved that weak existence and pathwise uniqueness imply strong existence and weak uniqueness. For -dimensional case, see [8, 9]. For infinite dimensional stochastic differential equation, Ondreját [6] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process, where the solutions are understood in the mild sense. Lately, R?ckner et al. [7] proved similar result for stochastic evolution equation in Banach space driven by cylindrical Wiener process under the variational framework. On the other hand, Kurtz [2, 3] obtained a pleasant version of Yamada-Watanabe and Engelbert theorem in an abstract form, which covered most of the work mentioned above. However, we will consider the following concrete stochastic evolution equation by using a different method. In this paper, we will consider the following stochastic evolution equation driven by pure Poisson random measure under the variational framework: This type of equations can be applied to many SPDEs, for example, stochastic Burgers equation, stochastic porous media equation, and stochastic Navier-Stokes equation (see, e.g., [9–13]). We will introduce the above equation precisely in Section 2. Our aim is to obtain this jump-case Yamada-Watanabe theorem; that is, weak existence and strong uniqueness (which will be stated in Section 2) imply strong existence and weak uniqueness and vice versa. We note that there are some differences between the jump-case case and the Brownian motion case. It is well known that a Brownian motion can be treated as a canonical map on or (for some Hilbert space ), while for jump-case we have
Uniqueness in Law for the Allen-Cahn SPDE via Change of Measure  [PDF]
Hassan Allouba
Mathematics , 2010, DOI: 10.1016/S0764-4442(00)00190-7
Abstract: We start by first using change of measure to prove the transfer of uniqueness in law among pairs of parabolic SPDEs differing only by a drift function, under an almost sure $L^2$ condition on the drift/diffusion ratio. This is a considerably weaker condition than the usual Novikov one, and it allows us to prove uniqueness in law for the Allen-Cahn SPDE driven by space-time white noise with diffusion function $a(t,x,u)=Cu^\gamma$, $1/2\le\gamma\le1$ and $C\ne0$. The same transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.
Poisson geometry and Morita equivalence  [PDF]
Henrique Bursztyn,Alan Weinstein
Mathematics , 2004,
Abstract: These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poisson manifolds. We also point out the connections with the theory of symplectic groupoids and hamiltonian actions.
Uniqueness of strong solutions for SDEs with H?lder diffusions  [PDF]
Rongrong Tian,Shuheng Tu,Jinlong Wei
Mathematics , 2015,
Abstract: This paper is concerned with the It\^o stochastic differential equations with $\mR^{d\times k}$ diffusions in class of H\"older spaces and continuous $\mR^d$ drifts. We derive a uniqueness result of strong solutions for $\cC^\alpha \ (\alpha\geq \frac{1}{2})$ coefficients and this result is new. Our proof is supported by It\^o's formula and a finer analysis on cut-off and smoothing techniques.
Pathwise uniqueness for singular SDEs driven by stable processes  [PDF]
Enrico Priola
Mathematics , 2010,
Abstract: We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $\R^d$ having a bounded and $\beta$-H\"older continuous drift term. We assume $\beta > 1 - \frac{\alpha}{2} $ and $\alpha \in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.
Davie's type uniqueness for a class of SDEs with jumps  [PDF]
Enrico Priola
Mathematics , 2015,
Abstract: A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation $dX_t = b(t, X_t)\,dt + dW_t$, $X_0=x$, driven by a Wiener process $W= (W_t)$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Brownian path. We consider a similar problem when $W$ is replaced by a L\'evy process $L= (L_t)$ and $b$ is $\beta$-H\"older continuous in the space variable, $ \beta \in (0,1)$. We assume that $L_1$ has a finite moment of order $\theta$, for some ${\theta}>0$. Using also a new c\`adl\`ag regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with $L^p$-Lipschitz continuity of the strong solution with respect to $x $ imply a Davie's type uniqueness result for almost all choices of the L\'evy paths. We apply this result to a class of SDEs driven by non-degenerate $\alpha$-stable L\'evy processes, $\alpha \in (0,2)$.
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