Jon Warren
A stochastic flow of homeomorphisms of the real line previously studied by Bass and Burdzy is shown to arise in describing a Brownian motion conditional on knowing its local times on hitting a fixed level. This makes it possible to connect Ray-Knight type results for the flow with the classical Ray-Knight theorems for Brownian motion.
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We introduce and characterize a class of flows, which turn out to be Gaussian. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flow, for an extended class of initial conditions, is the unique solution of the SPDE introduced in Rajeev and Thangavelu (2008).
The aim of this note is to provide some results for stochastic convolutions corresponding to stochastic Volterra equations in separable Hilbert space. We study convolution of the form $W^Ψ(t):=\int_0^t S(t-τ)Ψ(τ)dW(τ)$, $t\geq 0$, where $S(t), t\geq 0$, is so-called {\em resolvent} for Volterra equation considered,$Ψ$ is an appropriate process and $W$ is a cylindrical Wiener process.
Inspired by the paper Greenhalgh et al. [5] we investigate a class of two dimensional stochastic differential equations related to susceptible-infected-susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in [5], where an ad hoc approach has been utilized to prove existence, uniqueness and non explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and Hölder continuous coefficients. We prove the existence of a unique strong solution by means of a Cauch…
In the first part of this paper, we derive explicit expressions of the semi-group densities of generalized stochastic areas arising from the Anti-de Sitter and the Hopf fibrations. Motivated by the number-theoretical connection between the Heisenberg group and Dirichlet series, we express the Mellin transform of the generalized stochastic area corresponding to the one-dimensional Anti de Sitter fibration as a series of Riemann Zeta function evaluated at integers. In the second part of the paper, we focus on winding processes around the origin in the Poincaré disc and in the complex projective …
Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.
We study a class of McKean-Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean-Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot-Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein…
Shapley (1953) introduced two-player zero-sum discounted stochastic games, henceforth stochastic games, a model where a state variable follows a two-controlled Markov chain, the players receive rewards at each stage which add up to $0$, and each maximizes the normalized $\la$-discounted sum of stage rewards, for some fixed discount rate $\la\in(0,1]$. In this paper, we study asymptotic occupation measures arising in these games, as the discount rate goes to $0$.
SDEs with no strong solution arising from a problem of stochastic control
本文研究一个源于随机控制问题的二维 SDE，证明其虽有唯一弱解却无强解，且弱解滤过由 Brown 运动生成。
We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.
We establish existence and uniqueness for infinite dimensional Riccati equations taking values in the Banach space L 1 ($μ$ $\otimes$ $μ$) for certain signed matrix measures $μ$ which are not necessarily finite. Such equations can be seen as the infinite dimensional analogue of matrix Riccati equations and they appear in the Linear-Quadratic control theory of stochastic Volterra equations.
In this paper, we investigate a class of nonlinear backward stochastic differential equations (BSDEs) arising from financial economics, and give specific information about the nodal sets of the related solutions. As applications, we are able to obtain the explicit solutions to an interesting class of nonlinear BSDEs including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.
In this paper, motivated by modelling currency exchange markets with matrix-valued stochastic processes, matrix-valued stochastic differential equations (SDEs) are formulated. This is done based on the matrix trace, as for the purpose of modelling currency exchange markets. To be more precise, we set up a Hilbert space structure for $n\times n$ square matrices via the trace of the Hadamard product of two matrices. With the help of this framework, one can then define stochastic integral of Itô type and Itô SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness o…
We study stochastic convolutions providing by fundamental solutions of a class of integrodifferential equations which interpolate the heat and the wave equations. We give sufficient condition for the existence of function--valued convolutions in terms of the covariance kernel of a noise given by spatially homogeneous Wiener process.
We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in a price impact model. These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.
We determine the Lyapunov spectrum and stable manifolds of some stochastic flows on the Poincaré group associated to Dudley's relativistic processes.
Strong measurable continuous modifications of stochastic flows
本文提出一种新构造方法，在局部紧可分度量空间中为度量图的可测映射随机流生成强可测连续修正。
We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2γ_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where $X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $γ$ and $b$ are continuous on the positive orthant, $γ$ is strictly positive, and $b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=θ_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields li…
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
In this paper we consider the asymptotic behaviour of all moments of the interparticle distance and of all mixed moments of an isotropic Brownian stochastic flow which serves as a smooth approximation of the Arratia flow.
Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$
本文研究了随机Willmore流和表面扩散流在R²上的最大解问题，通过将问题转化为随机Stefan问题并应用相关理论，证明了局部强解的存在性和唯一性。
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $ν$-almost everywhere stochastic invertible flow, where $ν$ is the Riemannian measure, which is quasi-invariant with respect to $ν$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.
We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.
Continuous stochastic flows driven by white noise and their duals
研究一类由时空白噪声驱动的连续随机流，通过显式随机微分方程刻画其对偶流，并证明系数近似下解的收敛性，应用于平方贝塞尔流、雅可比流及多项式自排斥流。
We study a simple stochastic differential equation driven by one Brownian motion on a general oriented metric graph whose solutions are stochastic flows of kernels. Under some condition, we describe the laws of all solutions. This work is a natural continuation of some previous papers by Hajri, Hajri-Raimond and Le Jan-Raimond where some particular graphs have been considered.
We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow.
The weak limits of the measure-valued processes organized as a mass carried by the interacting Brownian particles are described. As a limiting flow the Arrattia flow is obtained.
We show that for a large class of stochastic flows the spatial derivative grows at most exponentially fast even if one takes the supremum over a bounded set of initial points. We derive explicit bounds on the growth rates that depend on the local characteristics of the flow and the box dimension of the set.
In this article we study transformations of Gaussian field by stochastic flow on the plane. A stochastic flow is a solution to the equation with interaction whose coefficients depend on the occupation measure of the field. We consider nonsmooth Gaussian field, which has self-intersection local times of any multiplicity. In the article we prove the existence of self-intersection local times for the transformed field and study its asymptotics.
Given a one-dimensional stochastic differential equation, one can associate to this equation a stochastic flow on $[0,+\infty )$, which has an absorbing barrier at zero. Then one can define its dual stochastic flow. In \cite{AW}, Akahori and Watanabe showed that its one-point motion solves a corresponding stochastic differential equation of Skorokhod-type. In this paper, we consider a discrete-time stochastic-flow which approximates the original stochastic flow. We show that under some assumptions, one-point motions of its dual flow also approximates the corresponding reflecting diffusion. We …
The Large Deviation Principle for Stochastic Flow of Stochastic Slow-Fast Motions
本文研究完全耦合慢快运动中慢变量的随机流，证明其存在性并满足大偏差原理，方法基于Khasminskii平均原理、变分表示和弱收敛框架。
Stochastic flows generated by reflected SDEs in a half-plane with an additive diffusion term are considered. A derivative in the initial data is represented a.s. as an infinite product of matrices. We use this representation and construct an example of a reflecting flow with a linear drift such that it is not locally continuously differentiable.
Fast advection asymptotics for a stochastic reaction-diffusion-advection equation are studied in this paper. To describe the asymptotics, one should consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.
Gaussian structure in coalescing stochastic flows
本文针对对应Arratia流的点测度开展研究，推导出关于该测度的多重积分的中心极限定理，为相关随机流领域的概率分析提供了理论支撑。
Stochastic heat flow is a black noise
本文证明随机热流是黑噪声，并推论由平滑时空白噪声驱动的2d随机热方程在标度极限下渐近独立于该噪声。
In this paper we compute the level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow which is a smooth approximation of the Arratia flow and determine its asymptotic behaviour as the height of the level tends to infinity.
We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.
We consider the Itô SDE with partially Sobolev coefficients. Under some suitable conditions, we show the existence, uniqueness and stability of generalized stochastic flows associated to such an equation. As an application, we prove the weak differentiability of the stochastic flow generated by the Itô SDE with Sobolev coefficients.
Some topological properties of stochastic flow $φ_t(x)$ generated by stochastic differential equation in a ${\mathbb R}^d_+$ with normal reflection at the boundary are investigated. Sobolev differentiability in initial condition is received. The absolute continuity of the measure-valued process $μ\circφ_t^{-1}$, where $μ\llλ^d,$ is studied.
We study the existence of the stochastic flow associated to a linear stochastic evolution equation $$d X= AX\,d t +\sum_{k} B_k X\,d W_k, $$ 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.
Complexity of flows with interactions and stochastic analysis
本文通过核提升将测度值随机方程表示为Hilbert值马尔可夫过程，确定生成元并建立稠密值域条件，为分析相互作用随机方程的稳定性提供框架。
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair $(\mathrm{SU}(2)\times \mathrm…
We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence o…
Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward-backward stochastic systems, namely, flows of forward-backward stochastic differential equations. They are systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals and connected via an equilibrium condition. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one…
Stochastic Curve Shortening Flow with Scale-Dependent Noise
本文研究了由尺度依赖噪声扰动的随机曲线缩短流，通过将问题转化为拟线性演算方程，获得了局部强解的存在性和唯一性。
In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.
In this paper we have constructed an approximation for the Harris flow and the Arratia flow using a sequence of independent stationary Gaussian processes as a perturbation. We have established what should be the relationship between the step of approximation and smoothness of the covariance of the perturbing processes in order to have convergence of the approximating functions to the Arratia flow.
We establish moment estimates for the invariant measure of a stochastic partial differential equation describing motion by mean curvature flow in (1+1) dimension, leading to polynomial stability of the associated Markov semigroup. We also prove maximal dissipativity for the related Kolmogorov operator.
We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation (PDE) with singular-degenerate diffusion. In addition, the deterministic BTW model is shown to converge to a singular-degenerate PDE. Alternatively, the proof of the scaling limit can be understood as a convergence proof of a finite-difference discretization for singular-degenerate stochastic PDEs. This extends recent work on finite difference approximation of (deterministic)…
In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.
Robust estimates for the performance of complicated queueing networks can be obtained by showing that the number of jobs in the network is stochastically comparable to a simpler, analytically tractable reference network. Classical coupling results on stochastic ordering of network populations require strong monotonicity assumptions which are often violated in practice. However, in most real-world applications we care more about what goes through a network than what sits inside it. This paper describes a new approach for ordering flows instead of populations by augmenting network states with th…
The probabilistic equivalent formulation of Dupire's PDE is the Put-Call duality equality. In local volatility models including exponential Lévy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire's PDE recently obtained by Pironneau by the adjoint equation technique in the case of complex options.
We show the existence of superprocesses in a random medium with location dependent branching. Technically, we make use of a duality relation to establish the uniqueness of the martingale problem and to obtain the moment formulas.
This is a brief survey of T. E. Harris's work on recurrent Markov processes and on stochastic flows, and of some more recent work in these fields.
Fractional Moments of Small-Ball Masses for the Stochastic Heat Flow
针对临界二维随机热流的收缩球质量，研究人员基于Gu和Tsai的几何块分解，引入分数乘积测度变换，证明了其对数分数矩指数。
We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps, and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained in [Kun04] for first order differentiability and rely on the Burkholder-Davis-Gundy inequality for time inhomogeneous Poisson random measures on ${\Bbb R}_+\times {\Bbb R}$, for which we provide a new proof.
In this paper, we first explore certain structural properties of Lévy flows and use this information to obtain the existence of strong solutions to a class of Stochastic PDEs in the space of tempered distributions, driven by Lévy noise. The uniqueness of the solutions follows from Monotonicity inequality. These results extend an earlier work Bhar (2017) on the diffusion case.
Stochastic conformal flows in even dimensions
研究偶数维流形上Q-曲率流的两种随机模拟，利用狄利克雷形式理论构造弱解，并证明Liouville Q流是偶数维Polyakov-Liouville测度的随机量子化。
An empirical model is a generalization of a probability space. It consists of a simplicial complex of subsets of a class X of random variables such that each simplex has an associated probability distribution. The ensuing marginalizations are coherent, in the sense that the distribution on a face of a simplex coincides with the marginal of the distribution over the entire simplex. An empirical model is said contextual if its distributions cannot be obtained marginalizing a joint distribution over X. Contextual empirical models arise naturally in quantum theory, giving rise to some of its count…
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