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By using a formulation of a class of compressible viscous flows with a heat source via vorticity and expansion-rate, we study the Oberbeck-Boussinesq flows. To this end we establish a new integral representation for solutions of parabolic equations subject to certain boundary condition, which allows us to develop a random vortex method for certain compressible flows and to compute numerically solutions of their dynamical models. Numerical experiments are carried out, which not only capture detailed B\'enard convection but also are capable of providing additional information on the fluid density and the dynamics of expansion-rate of the flow., Comment: 22 pages,6 figures

We develop a numerical method for simulation of incompressible viscous flows by integrating the technology of random vortex method with the core idea of Large Eddy Simulation (LES). Specifically, we utilize the filtering method in LES, interpreted as spatial averaging, along with the integral representation theorem for parabolic equations, to achieve a closure scheme which may be used for calculating solutions of Navier-Stokes equations. This approach circumvents the challenge associated with handling the non-locally integrable 3-dimensional integral kernel in the random vortex method and facilitates the computation of numerical solutions for flow systems via Monte-Carlo method. Numerical simulations are carried out for both laminar and turbulent flows, demonstrating the validity and effectiveness of the method., Comment: 13 pages, 10 figures

We consider Mckean-Vlasov type stochastic differential equations with multiplicative noise arising from the random vortex method. Such an equation can be viewed as the mean-field limit of interacting particle systems with singular interacting kernels such as the Biot-Savart kernel. A new estimate for the transition probability density of diffusion processes will be formulated to handle the singularity of the interacting kernel. The existence and uniqueness of the weak solution of such SDEs will be established as the main result.

Functional integral representations for solutions of the motion equations for wall-bounded incompressible viscous flows, expressed (implicitly) in terms of distributions of solutions to stochastic differential equations of McKean-Vlasov type, are established by using a perturbation technique. These representations are used to obtain exact random vortex dynamics for wall-bounded viscous flows. Numerical schemes therefore are proposed and the convergence of the numerical schemes for random vortex dynamics with an additional force term is established. Several numerical experiments are carried out for demonstrating the motion of a viscous flow within a thin layer next to the fluid boundary., Comment: 45 pages, 12 figures

Dirac operators on curved space-times are introduced with the help of a new point-view that observers have to be included in the formulation of natural laws. The class of Dirac operators are Lorentz invariant in the sense that the transformation rule is specified under diffeomorphisms of the space-time which preserve the time orientation and the gravitational field. Moreover these Dirac operators, like the original Dirac's operator with the special relativity, satisfy the Hamiltonian relation required by the general theory of relativity up to a correction due to the setup of a reference frame. In order to generalise (or discover) Dirac operators, which have been an important building block in the quantum field theories (QFTs), to curved space-times, we bring observers (which are elements of the principal orthonormal bundle over the space-time with its structure group being the proper Lorentz group) into the description of Fermion fields in the presence of a gravitational field. As a consequence field equations combining the gravitational field and matter fields may be formulated. This work suggests that observational effects (due to the setup of references) in the presence of a strong gravitational field, i.e. matter, may be unavoidable., Comment: 17 pages

The relative entropy for two different degenerate diffusion processes is estimated by using the Wasserstein distance of initial distributions and the difference between coefficients. As applications, the entropy cost inequality and exponential ergodicity in entropy are derived for distribution dependent stochastic Hamiltonian systems associated with nonlinear kinetic Fokker Planck equations., Comment: 22 pages

By using a formulation of motion equations for a viscous (compressible) fluid flow in terms of the vorticity and the rate of expansion as the main fluid dynamical variables, an approximation model is established for compressible flows with slowly varied (over the space) fluid density. The probabilistic tools and the main ingredient such as the duality of conditional laws and the forward type Feynman-Kac formula are established for elliptic operators of second order, in order to formulate the corresponding random vortex method for a class of viscous compressible fluid flows, based on their approximation motion equations., Comment: 19 pages

We establish stochastic functional integral representations for incompressible fluid flows occupying wall-bounded domains using the conditional law duality for a class of diffusion processes. These representations are used to derive a Monte-Carlo scheme based on the corresponding exact random vortex formulation. We implement several numerical experiments based on the Monte-Carlo method without appealing to the boundary layer flow computations, to demonstrate the methodology., Comment: 60 figures

We establish stochastic functional integral representations for solutions of Oberbeck-Boussinesq equations in the form of McKean-Vlasov-type mean field equations, which can be used to design numerical schemes for calculating solutions and for implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for solutions of some linear parabolic equations in terms of the Green function and the pinned measure of the associated diffusion. We demonstrate via numerical experiments the efficiency of the numerical schemes, which are capable of revealing numerically the details of Oberbeck-Boussinesq flows within their thin boundary layer, including B{\'e}nard's convection feature., Comment: 30 figures

We develop a new approach for regularity estimates, especially vorticity estimates, of solutions of the three-dimensional Navier-Stokes equations with periodic initial data, by exploiting carefully formulated linearized vorticity equations. An appealing feature of the linearized vorticity equations is the inheritance of the divergence-free property of solutions, so that it can intrinsically be employed to construct and estimate solutions of the Navier-Stokes equations. New regularity estimates of strong solutions of the three-dimensional Navier-Stokes equations are obtained by deriving new explicit a priori estimates for the heat kernel (i.e., the fundamental solution) of the corresponding heterogeneous drift-diffusion operator. These new a priori estimates are derived by using various functional integral representations of the heat kernel in terms of the associated diffusion processes and their conditional laws, including a Bismut-type formula for the gradient of the heat kernel. Then the a priori estimates of solutions of the linearized vorticity equations are established by employing a Feynman-Kac-type formula. The existence of strong solutions and their regularity estimates up to a time proportional to the reciprocal of the square of the maximum initial vorticity are established. All the estimates established in this paper contain known constants that can be explicitly computed., Comment: 27 pages

In this paper we establish a mathematical framework which may be used to design Monte-Carlo simulations for a class of time irreversible dynamic systems, such as incompressible fluid flows, including turbulent flows in wall-bounded regions, and some other (non-linear) dynamic systems. Path integral representations for solutions of forward parabolic equations are obtained, and, in combining with the vorticity transport equations, probabilistic formulations for solutions of the Navier-Stokes equations are therefore derived in terms of (forward) McKean-Vlasov stochastic differential equations (SDEs), which provides us with the mathematical framework for Monte-Carlo simulations of wall-bounded turbulent flows., Comment: 23 pages

We apply machine learning models to forecast intraday realized volatility (RV), by exploiting commonality in intraday volatility via pooling stock data together, and by incorporating a proxy for the market volatility. Neural networks dominate linear regressions and tree-based models in terms of performance, due to their ability to uncover and model complex latent interactions among variables. Our findings remain robust when we apply trained models to new stocks that have not been included in the training set, thus providing new empirical evidence for a universal volatility mechanism among stocks. Finally, we propose a new approach to forecasting one-day-ahead RVs using past intraday RVs as predictors, and highlight interesting time-of-day effects that aid the forecasting mechanism. The results demonstrate that the proposed methodology yields superior out-of-sample forecasts over a strong set of traditional baselines that only rely on past daily RVs., Comment: 40 pages, 12 figures, 6 tables; to appear in Journal of Financial Econometrics

In this paper we present a novel, closed three-dimensional (3D) random vortex dynamics system, which is equivalent to the Navier--Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled with a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the 3D random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman--Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems., Comment: 21 pages, 3 figures, 1 table

In this paper we propose a simple yet powerful vortex method to numerically approximate the dynamics of an incompressible flow. The idea is to sample the distribution of the initial vortices of the fluid flow in question then follow vortex dynamics along Taylor's Brownian fluid particles. The weak convergences of this approximation scheme are obtained for 2D and 3D cases, though only for small time in 3D case. Based on our method, the simulation results are quite attracting., Comment: 18 pages, 5 figures

The purpose of the present paper is to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non-linear parabolic-transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation, and is a precise formulation of the PDF transport equation. The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed, though the example is seemingly artificial, but it demonstrates the PDF method based on the new PDF PDE., Comment: 3 figures

The vorticity random field of turbulent flow is singled out as the main dynamical variable for the description of turbulence, and the evolution equation of the probability density function (PDF) of the vorticity field has been obtained. This PDF evolution equation is a mixed type partial differential equation (PDE) of second order which depends only on the conditional mean (first order) of the underlying turbulent flow, which is in contrast with Reynolds' mean flow equation which relies on a quadratic statistics. Therefore the new PDF PDE may provide new closure scheme based on the conditional linear statistics, and some of them will be described in the present paper too.

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 distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics., Comment: 24 pages

In this paper, we consider turbulence from a geometric perspective based on the vorticity equations for incompressible viscous fluid flows. We derive several quantitative statements about the statistics of turbulent flows. In particular we establish an entropy decay inequality which involves the turbulent kinetic energy and $L^{1}$-enstrophy, and we identify the small time scale of the vorticity broken down and the vorticity creation under the universality assumption of small time scales of turbulence flows.

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.

We will consider the Navier-Stokes equation on a Riemannian manifold M with Ricci tensor bounded below, the involved Laplacian operator is De Rham-Hodge Laplacian. The novelty of this work is to introduce a family of connections which are related to solutions of the Navier-Stokes equation, so that vorticity and helicity can be linked through the associated time-dependent Ricci tensor in intrinsic way in the case where dim(M) = 3. MSC 2010: 35Q30, 58J65

We are concerned with the inviscid limit of the Navier-Stokes equations on bounded regular domains in $\mathbb{R}^3$ with the kinematic and Navier boundary conditions. We first establish the existence and uniqueness of strong solutions in the class $C([0,T_\star); H^r(\Omega; \mathbb{R}^3)) \cap C^1([0,T_\star); H^{r-2}(\Omega;\mathbb{R}^3))$ with some $T_\star>0$ for the initial-boundary value problem with the kinematic and Navier boundary conditions on $\partial \Omega$ and divergence-free initial data in the Sobolev space $H^r(\Omega; \mathbb{R}^3)$ for $r\geq 2$. Then, for the strong solution with $H^{r+1}$--regularity in the spatial variables, we establish the inviscid limit in $H^r(\Omega; \mathbb{R}^3)$ uniformly on $[0,T_\star)$ for $r > \frac{5}{2}$. This shows that the boundary layers do not develop up to the highest order Sobolev norm in $H^{r}(\Omega;\mathbb{R}^3)$ in the inviscid limit. Furthermore, we present an intrinsic geometric proof for the failure of the strong inviscid limit under a non-Navier slip-type boundary condition., Comment: 30 pages

A solution to the optimal problem for determining vector fields which maximize (resp. minimize) the transition probabilities from one location to another for a class of reflecting diffusion processes is obtained in the present paper. The approach is based on a representation for the transition probability density functions. The optimal transition probabilities under the constraint that the drift vector field is bounded by a constant are studied in terms of the HJB equation. In dimension one, the optimal reflecting diffusion processes and the bang-bang diffusion processes are considered. We demonstrate by simulations that, even in this special case, the optimal diffusion processes exhibit an interesting feature of phase transitions. We also solve an optimal stochastic control problem for a class of stochastic control problems involving diffusion processes with reflection., Comment: 20 Pages, 2 figures

We show that fractional Brownian motion(fBM) defined via Volterra integral representation with Hurst parameter $H\geq\frac{1}{2}$ is a quasi-surely defined Wiener functional on classical Wiener space,and we establish the large deviation principle(LDP) for such fBM with respect to $(p,r)$-capacity on classical Wiener space in Malliavin's sense.

We study several important fine properties for the family of fractional Brownian motions with Hurst parameter $H$ under the $(p,r)$-capacity on classical Wiener space introduced by Malliavin. We regard fractional Brownian motions as Wiener functionals via the integral representation discovered by Decreusefond and \"{U}st\"{u}nel, and show non differentiability, modulus of continuity, law of iterated Logarithm(LIL) and self-avoiding properties of fractional Brownian motion sample paths using Malliavin calculus as well as the tools developed in the previous work by Fukushima, Takeda and etc. for Brownian motion case.

This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise It\^o integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated L\'evy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering., Comment: Published in the journal: Acta Mathematica Scientia, 2024

This paper develops an It\^o-type fractional pathwise integration theory for fractional Brownian motion with Hurst parameters \( H \in (\frac{1}{3}, \frac{1}{2}] \), using the Lyons' rough path framework. This approach is designed to fill gaps in conventional stochastic calculus models that fail to account for temporal persistence prevalent in dynamic systems such as those found in economics, finance, and engineering. The pathwise-defined method not only meets the zero expectation criterion but also addresses the challenges of integrating non-semimartingale processes, which traditional It\^o calculus cannot handle. We apply this theory to fractional Black-Scholes models and high-dimensional fractional Ornstein-Uhlenbeck processes, illustrating the advantages of this approach. Additionally, the paper discusses the generalization of It\^o integrals to rough differential equations (RDE) driven by fBM, emphasizing the necessity of integrand-specific adaptations in the It\^o rough path lift for stochastic modeling.

The parity violation at the level of weak interactions and other similar discrete symmetries breaking show that the invariance of laws under the full group of Lorentz transformations can not be taken granted. We examine the principle of Lorentz invariance under the general theory of relativity, and demonstrate the importance of a concept of the spacetime orientation as part of the causal structure of spacetime.

The Standard Model of particle physics was established based on the equivalence principle and gauge invariance. The Lagrangians were built upon experimental data demonstrating the violation of discrete symmetries together with ideas of spontaneous symmetry breaking. The features of vector and axial-vector interactions and chirality principle are manually added on in order to fit in the observation data, rather than explained by the model as one may hope. Here we develop a theory of interactions based on the Dirac algebra together with the exterior operations the wedge product and the star operator on the space-time, and develop a scheme for determining all possible interaction formulations. In our scheme, the chirality transformation and the left and right handedness come up naturally. A simple postulate about the universality of physical interactions based on the space-time causal structure yields the attracting features mentioned above as consequences.

On fractals, different measures (mutually singular in general) are involved to measure volumes of sets and energies of functions. Singularity of measures brings difficulties in (especially non-linear) analysis on fractals. In this paper, we prove a type of Sobolev inequalities, which involve different and possibly mutually singular measures, on product Sierpinski spaces. Sufficient and necessary conditions for the validity of these Sobolev inequalities are given. Furthermore, we compute the sharp exponents which appears in the sufficient and necessary conditions for the product Kusuoka measure, i.e. the reference energy measure on Sierpinski spaces., Comment: 23 pages

In this paper we construct a conservative Markov semi-group with generator $L=\Delta+b\cdot\nabla$ on $\mathbb{R}^n$, where $b$ is a divergence-free vector field which belongs to $L^{2}\cap L^{p}$ with $\frac{n}{2}

By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous medium). Unlike the regular spaces case, such equation involving a convection term must take a quite different form and the convection term must be singular to the "linear part" which is determined the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established as an effective tool for our study. These Sobolev inequalities, which are interesting by their own and in contrast to the case of Euclidean spaces, involve two $L^p$ norms with respect two mutually singular measures. By examining properties of a singular convolution of the associated heat semigroup, we derive the space-time regularity of solutions to the parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence and uniqueness as well as the regularity of solutions are obtained., Comment: 30 pages

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $\gamma=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $l=\infty,q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}\left(0,T;L^{n}\left(\mathbb{R}^{n}\right)\right)$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Semenov, 2006]. One of the results obtained in the present paper establishes the Aronson type estimate for critical and supercritical cases and for vector fields $b$ which are divergence-free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition enters the study of parabolic equations rather recently, mainly due to the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_{t}$ appears naturally., Comment: 31 pages

In this paper we study a multidimensional quadratic BSDE with a particular class of product generators and give a result of existence of solution in a suitable complete metric space under some constraints on parameters. We also use that result to derive the existence and uniqueness of solution to the one dimensional case with bounded terminal values and show the existence of solution to a lower triangular quadratic BSDE with certain bounded terminal values., Comment: 13 pages

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=\nu\Delta+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;\xi,\tau)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension $n$, the elliptic constant $\nu$ and the norm $\left\Vert u\right\Vert _{L^{\infty}(\textrm{BMO}^{-1})}$. Therefore the existence and uniqueness of the parabolic equation $\left(L-\partial_{t}\right)v=0$ are established for initial data in $L^{2}$-space, and their regularity is obtained too. In fact, we establish these results for a general non-symmetric elliptic operator in divergence form., Comment: 28 pages

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section [sec:-1]) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman-Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established., Comment: 30 pages

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization., Comment: 23 pages

In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying first the nonlocal SPDEs and showing then the convergence of the family of these equations and the limit gives the solution to the SPDE.

We derive several new gradient estimates of Aronson-B{\'e}nilan type for positive solutions of porous medium equation and fast diffusion equation on a complete manifold that satisfies the curvature dimension condition, Comment: 21 pages

We construct a quasi-sure version (in the sense of Malliavin) of geometric rough paths associated with a Gaussian process with long-time memory. As an application we establish a large deviation principle (LDP) for capacities for such Gaussian rough paths. Together with Lyons' universal limit theorem, our results yield immediately the corresponding results for pathwise solutions to stochastic differential equations driven by such Gaussian process in the sense of rough paths. Moreover, our LDP result implies the result of Yoshida on the LDP for capacities over the abstract Wiener space associated with such Gaussian process.

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest.

The Navier-Stokes equation on Rd (d greater or equal to 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_ r, with r > 1 + d is obtained. We also show p,p p the convergence to solutions of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_ pr,q, with p > 1, 1 greater or equal to q greater or equal to infinity, r > max(1, d); here the maximal time interval depends on p the viscosity., Comment: 43 pages

In the present paper, we prove that with probability one, the Stratonovich signatures of a multidimensional diffusion process (possibly degenerate) over [0,1], which is the collection of all iterated Stratonovich's integrals of the diffusion process over [0,1], determine the diffusion sample paths.

We derive a non-linear version of the Feynman-Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2.