The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions

Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions

In this paper a new class of generalized backward doubly stochastic differential equations is investigated. This class involves an integral with respect to an adapted continuous increasing process. A probabilistic representation for viscosity solutions of semi-linear stochastic partial differential equations with a Neumann boundary condition is given.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5092 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic neural field equations: A rigorous footing

We extend the theory of neural fields which has been developed in a deterministic framework by considering the influence spatio-temporal noise. The outstanding problem that we here address is the development of a theory that gives rigorous meaning to stochastic neural field equations, and conditions ensuring that they are well-posed. Previous investigations in the field of computational and mathematical neuroscience have been numerical for the most part. Such questions have been considered for a long time in the theory of stochastic partial differential equations, where at least two different approaches have been developed, each having its advantages and disadvantages. It turns out that both approaches have also been used in computational and mathematical neuroscience, but with much less emphasis on the underlying theory. We present a review of two existing theories and show how they can be used to put the theory of stochastic neural fields on a rigorous footing. We also provide general conditions on the parameters of the stochastic neural field equations under which we guarantee that these equations are well-posed. In so doing we relate each approach to previous work in computational and mathematical neuroscience. We hope this will provide a reference that will pave the way for future studies (both theoretical and applied) of these equations, where basic questions of existence and uniqueness will no longer be a cause for concern

Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1--4, 2015, Szeged, Hungary

Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Let $M$ be a compact Riemannian homogeneous space (e.g. a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation \mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot W$in any dimension$d\ge 1$, where$f$and$g$are continuous multilinear mappings and$W$is a spatially homogeneous Wiener process on$\mathbb R^d$ with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors