Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Stochastic levi sums

An analogy of Levi sums is considered for a class of stochastic partial differential equations to construct their stochastic fundamental solutions. These notions are shown to coincide with Donsker's delta functions, typical generalized Wiener functionals, which have been studied in the frame of the Malliavin calculus. © 1994 John Wiley & Sons, Inc.     

A finitely additive white noise approach to nonlinear filtering

An approach to nonlinear filtering theory is developed in which finitely additive white noise replaces the Wiener process in the observation process model. The important case when the signal is a Markov process independent of the noise is investigated in detail. The theory turns out to be simpler than the current theory based on the stochastic calculus. Stochastic partial differential equations are replaced by partial differential equations in which the observation (in the finitely additive set up) occurs as a parameter. Theorems on existence and uniqueness of solutions are obtained. The white noise approach has the advantage that it provides a robust solution to the filtering problem. Furthermore, the robust theory based on the Ito calculus can be recovered from the results of this paper.     

Discrete wick calculus and stochastic functional equations

We develop Wick calculus over finite probability spaces and prove that there is a one-to-one correspondence between the solutions of Wick stochastic functional equations and the solutions of the deterministic functional equations obtained by turning off the noise. We also point out some possible applications to ordinary and partial stochastic differential equations.This research is supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL).     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.     

Lepage forms,closed 2-forms and second-order ordinary differential equations

Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail. The text was submitted by the authors in English.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Itô’s stochastic calculus and Heisenberg commutation relations

Stochastic calculus and stochastic differential equations for Brownian motion were introduced by K. Itô in order to give a pathwise construction of diffusion processes. This calculus has deep connections with objects such as the Fock space and the Heisenberg canonical commutation relations, which have a central role in quantum physics. We review these connections, and give a brief introduction to the noncommutative extension of Itô’s stochastic integration due to Hudson and Parthasarathy. Then we apply this scheme to show how finite Markov chains can be constructed by solving stochastic differential equations, similar to diffusion equations, on the Fock space.     

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.     

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in $\mathbb{R }^{d}$ are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.     

Asymptotic Properties of the Maximum Likelihood Estimator for Stochastic PDEs Disturbed by Small Noise

Consider a parabolic stochastic partial differential equation perturbed by small noise observed on a time interval [0,T]. We construct the maximum likelihood estimators of the coefficients of the operators involved in these equations based on partial observations in the form of diffusion processes and show the asymptotic efficiency for loss functions with polynomial majorant as the variance goes to zero.     

On the use of the rotation minimizing frame for variational systems with Euclidean symmetry

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimizing frame, also known as the Normal, Parallel, or Bishop frame. Such systems have previously been studied using the Frenet–Serret frame. However, the Rotation Minimizing frame has many advantages, and can be used to study a wider class of examples. We achieve our results by extending the powerful symbolic invariant calculus for Lie group–based moving frames, to the Rotation Minimizing frame case. To date, the invariant calculus has been developed for frames defined by algebraic equations. By contrast, the Rotation Minimizing frame is defined by a differential equation. In this paper, we derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants. We then derive the syzygy operator needed to obtain Noether's conservation laws as well as the Euler–Lagrange equations directly in terms of the invariants, for variational problems with a Euclidean symmetry. We show how to use the six Noether laws to ease the integration problem for the minimizing curve, once the Euler–Lagrange equations have been solved for the generating differential invariants. Our applications include variational problems used in the study of strands of proteins, nucleid acids, and polymers.     

Yosida approximations for multivalued stochastic partial differential equations driven by Lévy noise on a Gelfand triple

We prove the existence and uniqueness of solutions for a class of multivalued stochastic partial differential equations with maximal monotone drift on Banach space driven by multiplicative Lévy noise. We also establish the strong convergence result for solutions of the approximating equations where the maximal monotone drift operator is replaced by its Yosida approximation. As an application, the existence and uniqueness of solutions for multivalued stochastic porous medium equations is obtained.     

Strong solutions for stochastic partial differential equations of gradient type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the “distance” defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction–diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.     

On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

Large deviations for invariant measures of SPDEs with two reflecting walls

In this article, we establish a large deviation principle for invariant measures of solutions of stochastic partial differential equations with two reflecting walls driven by a space–time white noise.     

Irreducibility and strong Feller property for non-linear SPDEs

Under some non-degeneracy condition, the strong Feller property and irreducibility are studied for non-linear stochastic partial differential equations driven by multiplicative noise within the framework called ‘variational approach’. Our result for irreducibility can be applied to equations with locally monotone coefficients. In some special cases, we discuss the Hölder continuity of the associated Markov semigroups. The main results are applied to several examples such as stochastic Burgers equation, stochastic porous media equation and stochastic fast diffusion equation.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Concentration inequalities for stochastic differential equations of pure non-Poissonian jumps

We provide concentration inequalities for solutions to stochastic differential equations of pure not-necessarily Poissonian jumps. Our proofs are based on transportation cost inequalities for square integrable functionals of point processes with stochastic intensity and elements of stochastic calculus with respect to semi-martingales. We apply the general results to solutions of stochastic differential equations driven by renewal and non-linear Hawkes point processes.