New invariant surface measures for the cubic Schrödinger equation
We construct new invariant measures supported on mass level sets for the cubic defocusing nonlinear Schrödinger equation in dimensions $1$ and $2$.
math.AP↗
arXiv subjects
Ana Bela Cruzeiro
Publications and source records attributed to Ana Bela Cruzeiro.
Stochastic perturbation of geodesics on the manifold of Riemannian metrics
In this paper, using diffusion processes, we compute the evolution equation on the manifold of Riemannian metrics for the Lagrangian induced by the $L^2$ stochastic kinetic energy functional.
math.DG↗
Pathwise stochastic control and a class of stochastic partial differential equations
We consider a pathwise stochastic optimal control problem and study the associated (not necessarily adapted) Hamilton-Jacobi-Bellman stochastic partial differential equation. We show that the value process is the unique solution of this equation, in the viscosity sense. Finally, we discuss a version of some corresponding stochastic pathwise Noether theorem.
math.PR↗
On the well-posedness of a Hamilton-Jacobi-Bellman equation with transport noise
In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \begin{align*} \begin{cases} \mathrm{d}u(s,x)=\sum^n_{i=1} \mathscr{L}_i u(s,x)\circ \mathrm{d}W_i(s)+\left(V(x)+μΔu(s,x)-\frac{1}{2}\vert\nabla u(s,x)\vert^2\right)\mathrm{d}s, \quad &\text{in } (0,T)\times \mathbb{T}^n, u(0,x)=u_0(x), & \text{on } \mathbb{T}^n, \end{cases} \end{align*} where $\mathbb{T}^n$ is the $n$-dimensional torus, the functions $u_0, V: \mathbb{T}^n \to \mathbb{R}$ are given and $\{\mathscr{L}_i\}_i$ is a collection of first order linear operators. This can be seen as a Cauchy problem for a Hamilton-Jacobi-Bellman equation with transport noise in any space dimension. We introduce the concept of a strong solution from the realm of PDEs and establish the existence and uniqueness of maximal solutions (strong solutions upto a stopping time). Moreover, for a particular class of $\{\mathscr{L}_i\}_i$ we establish global well-posedness of strong solutions. The proof relies on studying an associated truncated version of the original SPDE and showing its global well-posedness in the class of strong solutions.
math.AP↗
Forward-backward stochastic differential equations on tensor fields and application to Navier-Stokes equations on Riemannian manifolds
In this paper we introduce a class of forward-backward stochastic differential equations on tensor fields of Riemannian manifolds, which are related to semi-linear parabolic partial differential equations on tensor fields. Moreover, we will use these forward-backward stochastic differential equations to give a stochastic characterization of incompressible Navier-Stokes equations on Riemannian manifolds, where some extra conditions used in [22] are not required.
math.PR↗
Stochastic geodesics
We describe, in an intrinsic way and using the global chart provided by Ito's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These stochastic processes are no longer smooth paths but they are still critical points of a regularised stochastic energy functional. We consider stochastic geodesics on compact Riemannian manifolds and also on (possibly infinite dimensional) Lie groups. Finally the question of existence of such stochastic geodesics is discussed: we show how it can be approached via forward-backward stochastic differential equations.
math.PR↗
Time-symmetric optimal stochastic control problems in space-time domains
We present a pair of adjoint optimal control problems characterizing a class of time-symmetric stochastic processes defined on random time intervals. The associated PDEs are of free-boundary type. The particularity of our approach is that it involves two adjoint optimal stopping times adapted to a pair of filtrations, the traditional increasing one and another, decreasing. They are the keys of the time symmetry of the construction, which can be regarded as a generalization of "Schrödinger's problem" (1931-32) to space-time domains. The relation with the notion of "Hidden diffusions" is also described.
math.PR↗
Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers
We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.
math.AP↗
Stochastic variational principles for dissipative equations with advected quantities
This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. The general theory is presented for the finite dimensional situation. In infinite dimensions we obtain partial differential equations and stochastic partial differential equations. When the Lie group is, for example, a diffeomorphism group, the general result is not directly applicable but the setup and method suggest rigorous proofs valid in infinite dimensions which lead to similar results. We apply this technique to the compressible Navier-Stokes equation and to magnetohydrodynamics for charged viscous compressible fluids. A stochastic Kelvin-Noether theorem is presented. We derive, among others, the classical deterministic dissipative equations from purely variational and stochastic principles, without any appeal to thermodynamics.
math-ph↗
On a non-periodic modified Euler equation: existence and quasi-invariant measures
We consider a modified Euler equation on $\mathbb R^2$. We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be also globally defined.
math.AP↗
Invariant measures for the non-periodic two-dimensional Euler equation
We construct Gaussian invariant measures for the two-dimensional Euler equation on the plane. We show the existence of solution with initial conditions in the support of the measures, namely $H^β_{loc}(\R^2)$ with $β<-1$. Uniqueness and continuity of the velocity flow are proved.
math.AP↗
Barrier Option Pricing under the 2-Hypergeometric Stochastic Volatility Model
We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data. Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2-hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided.
math.PR↗
An entropic interpolation problem for incompressible viscid fluids
In view of studying incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscid fluids. A natural analogue of Brenier's problem is introduced, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. It turns out that this new variational problem is an entropy minimization problem with marginal constraints entering the class of convex minimization problems. This paper explores the connection between this variational problem and Brenier's original problem. Its dual problem is derived and the general shape of its solution is described. Under the restrictive assumption that the pressure is a nice function, the kinematics of its solution is made explicit and its connection with the Navier-Stokes equation is established.
math.PR↗
A stochastic variational approach to the viscous Camassa-Holm and Leray-alpha equations
We derive the (d-dimensional) periodic incompressible and viscous Camassa-Holm equation as well as the Leray-alpha equations via a stochastic variational principle. We discuss the existence of solution for this equation in the space H1 using the probabilistic characterisation. The underlying Lagrangian flows are diffusion processes living in the group of diffeomorphisms of the torus. We study in detail these diffusions.
math.PR↗
Generalized stochastic Lagrangian paths for the Navier-Stokes equation
In the note added in proof of the seminal paper [Groups of diffeomorphisms andthe motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebinand Marsden introduced the so-called correct Laplacian for the Navier-Stokes equationon a compact Riemannian manifold. In the spirit of Brenier's generalized flows forthe Euler equation, we introduce a class of semimartingales on a compact Riemannianmanifold. We prove that these semimartingales are critical points to the correspondingkinetic energy if and only if its drift term solves weakly the Navier-Stokes equationdefined with Ebin-Marsden's Laplacian. We also show that for the torus case,classical solutions of the Navier-Stokes equation realize the minimum of the kineticenergy in a suitable class.
math.PR↗
Symmetries and martingales in a stochastic model for the Navier-Stokes equation
A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler- Lagrange condition. A least action principle, related to the relative entropy, is provided. Within this stochastic framework, by assuming further symmetries, the corresponding invariances are expressed by martingales, stemming from a weak Noether's theorem.
math.PR↗
Weak calculus of variations for functionals of laws of semi-martingales
We develop a non-anticipating calculus of variations for functionals on a space of laws of continuous semi-martingales, which extends the classical one. We extend Hamilton's least action principle and Noether's theorem to this generalized stochastic framework. As an application we obtain, under mild conditions, a stochastic Euler-Lagrange condition and invariants for the critical points of recent problems in stochastic control, namely for the semi-martingale optimal transportation problems.
math.PR↗
On a 2D stochastic Euler equation of transport type: existence and geometric formulation
We prove weak existence of Euler equation (or Navier-Stokes equation) perturbed by a multiplicative noise on bounded domains of $\mathbb R^2$ with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are $H^1$ regular. The equations are of transport type.
math.PR↗