Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201

Stochastic Reaction-diffusion Equations Driven by Jump Processes

We establish the existence of weak martingale solutions to a class of second order parabolic stochastic partial differential equations. The equations are driven by multiplicative jump type noise, with a non-Lipschitz multiplicative functional. The drift in the equations contains a dissipative nonlinearity of polynomial growth.Comment: See journal reference for teh final published versio

Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens

We study properties of the classical fractional Sobolev spaces on non-Lipschitz subsets of Rn. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary

Time-averaging for weakly nonlinear CGL equations with arbitrary potentials

Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*)$$ under the periodic boundary conditions, where $\mu\geqslant0$ and $\mathcal{P}$ is a smooth function. Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\Delta +V(x)$. For a complex function $u(x)$, write it as $u(x)=\sum_{k\geqslant1}v_k\zeta_k(x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if equation $(*)$ with a sufficiently smooth real potential $V(x)$ is well posed on time-intervals $t\lesssim \epsilon^{-1}$, then for any its solution $u^{\epsilon}(t,x)$, the limiting behavior of the curve $I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, can be uniquely characterized by a solution of a certain well-posed effective equation: $$u_t=\epsilon\mu\triangle u+\epsilon F(u),$$ where $F(u)$ is a resonant averaging of the nonlinearity $\mathcal{P}(u)$. We also prove a similar results for the stochastically perturbed equation, when a white in time and smooth in $x$ random force of order $\sqrt\epsilon$ is added to the right-hand side of the equation. The approach of this work is rather general. In particular, it applies to equations in bounded domains in $R^d$ under Dirichlet boundary conditions

A Class of Non-Parametric Statistical Manifolds modelled on Sobolev Space

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on Rd. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports the Fisher-Rao metric as a weak Riemannian metric. Densities are expressed in terms of a deformed exponential function having linear growth. Unusually for the Sobolev context, and as a consequence of its linear growth, this "lifts" to a nonlinear superposition (Nemytskii) operator that acts continuously on a particular class of mixed-norm model spaces, and on the fixed norm space W²'¹ i.e. it maps each of these spaces continuously into itself. It also maps continuously between other fixed-norm spaces with a loss of Lebesgue exponent that increases with the number of derivatives. Some of the results make essential use of a log-Sobolev embedding theorem. Each manifold contains a smoothly embedded submanifold of probability measures. Applications to the stochastic partial differential equations of nonlinear filtering (and hence to the Fokker-Planck equation) are outlined

Post-Socialist Culture and Entrepreneurship

In this paper it is argued that locus of control beliefs and preferences concerning state action negatively affect the formation of new firms in former socialist countries. For this purpose Kirzner's theory of costless entrepreneurship is reviewed and criticized. German reunification, in which the formerly Socialist East joined the Federal Republic of Germany, represents an intriguing natural experiment in which the formal institutional structure of one nation was almost fully transplanted into another. Traditional as well as psychological factors are examined. The results suggest that about one third of the east-west gap in new self-employment can be explained by inert informal institutions

Entrepreneurial role models, fear of failure, and institutional approval of entrepreneurship: A tale of two regions

Studies on the influence of entrepreneurial role models (peers) on the decision to start a firm ar-gue that entrepreneurial role models in the local environment (1) provide opportunities to learn about entrepreneurial tasks and capabilities, and (2) signal that entrepreneurship is a favorable career option thereby reducing uncertainty that potential entrepreneurs face. However, these studies remain silent about the role of institutional context for these mechanisms. Applying an ex-tended sender-receiver model, we hypothesize that observing entrepreneurs reduces fear of fail-ure in others in environments where approval of entrepreneurship is high while this effect is signif-icantly weaker in low approval environments. Taking advantage of the natural experiment from recent German history and using data from the Global Entrepreneurship Monitor Project (GEM), we find considerable support for our hypotheses

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω ) or H˜s−2(Ω ) , 12<s<32, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.EPSR