Logarithmic asymptotics of the densities of SPDEs driven by spatially correlated noise

We consider the family of stochastic partial differential equations indexed by a parameter \eps\in(0,1], \begin{equation*} Lu^{\eps}(t,x) = \eps\sigma(u^\eps(t,x))\dot{F}(t,x)+b(u^\eps(t,x)), \end{equation*} (t,x)\in(0,T]\times\Rd with suitable initial conditions. In this equation, $L$ is a second-order partial differential operator with constant coefficients, $\sigma$ and $b$ are smooth functions and $\dot{F}$ is a Gaussian noise, white in time and with a stationary correlation in space. Let p^\eps_{t,x} denote the density of the law of u^\eps(t,x) at a fixed point (t,x)\in(0,T]\times\Rd. We study the existence of \lim_{\eps\downarrow 0} \eps^2\log p^\eps_{t,x}(y) for a fixed $y\in\R$. The results apply to a class of stochastic wave equations with $d\in\{1,2,3\}$ and to a class of stochastic heat equations with $d\ge1$.Comment: 39 pages. Will be published in the book " Stochastic Analysis and Applications 2014. A volume in honour of Terry Lyons". Springer Verla

Random-field Solutions to Linear Hyperbolic Stochastic Partial Differential Equations with Variable Coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.Comment: 40 pages, final version, Stochastic Processes and their Applications (2017

Stochastic Models Involving Second Order Lévy Motions

This thesis is based on five papers (A-E) treating estimation methods for unbounded densities, random fields generated by Lévy processes, behavior of Lévy processes at level crossings, and a Markov random field mixtures of multivariate Gaussian fields. In Paper A we propose an estimator of the location parameter for a density that is unbounded at the mode. The estimator maximizes a modified likelihood in which the singular term in the full likelihood is left out, whenever the parameter value approaches a neighborhood of the singularity location. The consistency and super-efficiency of this maximum leave-one-out likelihood estimator is shown through a direct argument. In Paper B we prove that the generalized Laplace distribution and the normal inverse Gaussian distribution are the only subclasses of the generalized hyperbolic distribution that are closed under convolution. In Paper C we propose a non-Gaussian Matérn random field models, generated through stochastic partial differential equations, with the class of generalized Hyperbolic processes as noise forcings. A maximum likelihood estimation technique based on the Monte Carlo Expectation Maximization algorithm is presented, and it is shown how to preform predictions at unobserved locations. In Paper D a novel class of models is introduced, denoted latent Gaussian random filed mixture models, which combines the Markov random field mixture model with the latent Gaussian random field models. The latent model, which is observed under a measurement noise, is defined as a mixture of several, possible multivariate, Gaussian random fields. Selection of which of the fields is observed at each location is modeled using a discrete Markov random field. Efficient estimation methods for the parameter of the models is developed using a stochastic gradient algorithm. In Paper E studies the behaviour of level crossing of non-Gaussian time series through a Slepian model. The approach is through developing a Slepian model for underlying random noise that drives the process which crosses the level. It is demonstrated how a moving average time series driven by Laplace noise can be analyzed through the Slepian noise approach. Methods for sampling the biased sampling distribution of the noise are based on an Gibbs sampler

Interplay of Analysis and Probability in Physics

The main purpose of this workshop was to foster interaction between researchers in the fields of analysis and probability with the aim of joining forces to understand difficult problems from physics rigorously. 52 researchers of all age groups and from many parts of Europe and overseas attended. The talks and discussions evolved around five topics on the interface between analysis and probability. The main goal of the workshop, the systematic encouragement of intense discussions between the two communities, was achieved to a high extent

Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fast isentropic fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions. We construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplications neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also