Space Of Continuous Paths Research Articles

Stochastic wave equation with Hölder noise coefficient: Well-posedness and small mass limit

We construct unique martingale solutions to the damped stochastic wave equationμ∂2u∂t2(t,x)=Δu(t,x)−∂u∂t(t,x)+b(t,x,u(t,x))+σ(t,x,u(t,x))dWtdt, where Δ is the Laplacian on [0,1] with Dirichlet boundary condition, W is space-time white noise, σ is 34+ϵ -Hölder continuous in u and uniformly non-degenerate, and b has linear growth. The same construction holds for the stochastic wave equation without damping term. More generally, the construction holds for SPDEs defined on separable Hilbert spaces with a densely defined operator A, and the assumed Hölder regularity on the noise coefficient depends on the eigenvalues of A in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume b is Hölder continuous in u, then as μ tends to 0 the solution to the damped stochastic wave equation converges in distribution, on the space of continuous paths, to the solution of the corresponding stochastic heat equation. The latter result is new even in the case of additive noise.

Talagrand’s transportation inequality for SPDEs with locally monotone drifts

The purpose of this paper is twofold. Firstly, we prove transportation inequalities T2(C) on the space of continuous paths with respect to the uniform metric for the law of the solution to a class of non-linear monotone stochastic partial differential equations (SPDEs) driven by the Wiener noise. Furthermore, we also establish the T1(C) property for such SPDEs but with merely locally monotone coefficients, including the stochastic Burgers type equation and stochastic 2-D Navier–Stokes equation.

Crandall–Lions viscosity solutions for path-dependent PDEs: The case of heat equation

We address our interest to the development of a theory of viscosity solutions à la Crandall–Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0,T];Rd). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall–Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the most delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekeland’s variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fréchet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural “candidate” solution. Finally, concerning the existence part, we provide a functional Itô formula under general assumptions, extending earlier results in the literature.

Scattering using real-time path integrals

Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. Purpose: To test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals. Methods: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on a space of continuous paths with respect to a complex probability distribution. The method has the advantage that it can be applied to arbitrary short-range potentials. Results: The new method is tested by applying it to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half shell transition operator matrix elements are in agreement with a numerical solution of the Lippmann-Schwinger equation. The computational method has a straightforward generalization to more complicated systems.

Transportation Inequalities Under Uniform Metric for a Stochastic Heat Equation Driven by Time-White and Space-Colored Noise

In this paper, we prove transportation inequalities on the space of continuous paths with respect to the uniform metric, for the law of solution to a stochastic heat equation defined on $[0,T]\times [0,1]^d$. This equation is driven by the Gaussian noise, white in time and colored in space. The proof is based on a new moment inequality under the uniform metric for the stochastic convolution with respect to the time-white and space-colored noise, which is of independent interest.

Generalized Compressible Flows and Solutions of the $$H(\mathrm {div})$$ Geodesic Problem

We study the geodesic problem on the group of diffeomorphism of a domain M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide with the Camassa-Holm equation when d=1, and represent one of its possible multi-dimensional generalizations when d>1. We propose a relaxation {\`a} la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth H(div) geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa-Holm and incompressible Euler solutions.

An infinite-dimensional approach to path-dependent Kolmogorov equations

In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$ paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fourni\'{e}.

Robust filtering: Correlated noise and multidimensional observation

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_{t}$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_{s},s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $\theta^{f}_{t}$, defined on the space of continuous paths $C([0,t],\mathbb{R} ^{d})$ endowed with the uniform convergence topology such that $\pi_{t}(f)=\theta^{f}_{t}(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43–56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125–139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160–167], Kushner [Stochastics 3 (1979) 75–83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260–278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505–528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403–424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is “lifted” to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding Lévy area process, and we show that there exists a continuous map $\theta_{t}^{f}$, defined on a suitably chosen space of Hölder continuous paths such that $\pi_{t}(f)=\theta_{t}^{f}(\mathbf{Y})$, almost surely.

Constructing sublinear expectations on path space

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that holds without exceptional set. Our results also shed light on the inherent limitations to constructing sublinear expectations through aggregation.

Girsanov’s formula for [formula omitted]-Brownian motion

In this paper, we establish Girsanov’s formula for G-Brownian motion. Peng (2007, 2008) [7,8] constructed G-Brownian motion on the space of continuous paths under a sublinear expectation called G-expectation; as obtained by Denis et al. (2011) [2], G-expectation is represented as the supremum of linear expectations with respect to martingale measures of a certain class. Our argument is based on this representation with an enlargement of the associated class of martingale measures, and on Girsanov’s formula for martingales in the classical stochastic analysis. The methodology differs from that of Xu et al. (2011) [13], and applies to the multidimensional G-Brownian motion.

Log-Sobolev inequalities with potential functions on pinned path groups

We establish a refined version of Gross's log-Sobolev inequalities on pinned path groups. We explain the reason why it is useful in a lower bound estimate of Schrodinger operators on path spaces. semi-group and the equivalent notion of the validity of the log-Sobolev inequality are used to give a lower bound on the bottom of spectrum of a Schrodinger operator which is given by the sum of the generator of the semi-group and a potential function. We ap- plied this lower bound estimate to study the semi-classical behavior of the bottom of spectrum of Schrodinger operators on path spaces over compact Riemannian manifolds ((3, 4)) partly motivated by an application to P(`)-type Hamiltonian and an extension of (19) to infinite dimensional curved spaces. In this paper, we establish a refined version of Gross's log-Sobolev inequalities on a pinned path group. Pinned path group Pe,a(G) is a space of continuous paths with values in a compact Lie group G over the time interval (0,1) with a fixed starting point e (unit element) and the fixed end point a. We will apply the log- Sobolev inequalities with potential functions to study the semi-classical behavior of the low lying spectrum of Schrodinger operators on pinned path groups in a separate paper. The structure of the paper is as follows. In Section 2, we consider smooth pinned path spaces over a general compact Riemannian manifold M. We intro- duce a Riemannian structure on the pinned path space using a metric connection on M. Next we calculate the gradient of the energy function E(∞) = 1 R 1 0 |˙ ∞(t)| 2 dt. This calculation and a formal argument show that an LSI with a special potential function may be useful for the study of semi-classical behavior of low lying spec- trum of Schrodinger operators over a pinned path space. This kind of log-Sobolev inequality with special potential function already appeared in (13, 10). In Section 3, we consider a pinned path group and introduce an H-derivative on Pe,a(G) and the Dirichlet form in L 2 -space with respect to the (scaled) pinned Brownian motion measure with scaling (semi-classical) parameter ‚. The H-derivative is considered as the gradient operator on the path space which is defined by the right invariant

Smoothness of Itô maps and diffusion processes on path spaces (I)

Let p ∈ [ 1 , 2 ) and α, ε > 0 be such that α ∈ ( p − 1 , 1 − ε ) . Let V, W be two Euclidean spaces. Let Ω p ( V ) be the space of continuous paths taking values in V and with finite p-variation. Let k ∈ N and f : W → Hom ( V , W ) be a Lip ( k + α + ε ) map in the sense of E.M. Stein [Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970]. In this paper we prove that the Itô map, defined by I ( x ) = y , is a local C k , ε 1 + ε map (in the sense of Fréchet) between Ω p ( V ) and Ω p ( W ) , where y is the solution to the differential equation d y t = f ( y t ) d x t , y 0 = a . This result strengthens the continuity results and Lipschitz continuity results in [Lyons T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451–464; Lyons T., Qian Z., System Control and Rough Paths, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2002] particularly to the non-integer case. It allows us to construct the fractional like Brownian motion and infinite dimensional Brownian motions on the space of paths with finite p-variation. As a corollary in the particular case where p = 1 , we obtain that the development from the space of finite 1-variation paths on R d to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection.

Duality formula for the bridges of a Brownian diffusion: Application to gradient drifts

In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C ( [ 0 ; 1 ] ; R d ) . Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.

THE SMOLYANOV SURFACE MEASURE ON TRAJECTORIES IN A RIEMANNIAN MANIFOLD

It has been shown in Refs. 2–6 that two natural definitions of surface measures, on the space of continuous paths in a compact Riemannian manifold embedded into ℝn, introduced in the paper by Smolyanov1 are equivalent; this means that there exists a natural object — the surface measure, which we call the Smolyanov surface measure. Moreover, it has been shown2–6 that this surface measure is equivalent to the Wiener measure and the corresponding density has been found. But the known proof of the equivalence of the two definitions of the surface measure is rather nonexplicit; in fact the densities of the measures corresponding to the two different definitions were found independently and only a posteriori it was discovered that those densities coincided. Our aim is to give a direct proof of this fact. We introduce a more restrictive definition of the surface measure as the weak limit of a standard Brownian motion in ℝn conditioned to be in the tubular ε-neighborhood of the manifold at times 0=t0<t1<⋯<tn-1<tn= 1 as both ε and the diameter of the partition tend to zero. Letting ε and then the diameter of the partition tend to zero and vice versa, we arrive at the two definitions above. We prove the existence of the Smolyanov surface measure using our definition, show that this measure is equivalent to the law of a Brownian motion on the manifold, and compute the corresponding density in terms of the curvature of the manifold. As a special case of this, we again obtain the results of Refs. 2–6.

-measures for branching exit Markov systems and their applications to differential equations

Semilinear equations Lu=ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L,ψ)-superdiffusions. In a special case Δu=u2 a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures Open image in new windowx on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them Open image in new window-measures. Using Open image in new window-measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Levy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Levy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space.

The Wiener–Feynman Stochastic Process on a Superspace

The theory of stochastic processes with values in a superspace over Banach superalgebras is developed. Stochastic processes are defined as distributions on the space of continuous paths with values in a superspace. Quasi–Gaussian distributions, particularly Gaussian and Feynman distributions and processes on a superspace, are studied. Solutions of the heat equation and the Shroedinger equation on a superspace are represented as probability means.