South Wales Analysis and Probability Seminar (SWAP)

Cardiff University and Swansea University

The South Wales Analysis and Probability Seminar (SWAP) is a regional seminar bringing together mathematicians based in South Wales, with interests in analysis, probability, PDEs, SDEs, dynamical systems, and other related fields.

Next Meeting (poster)

The next meeting will be on Wednesday 27 March 2019 at Cardiff University. The schedule is:

 12:00-13:00 Lunch & Coffee 13:00-13:45 Karl Michael Schmidt (Cardiff) On Singular Positive Definite Functions 13:45-14:30 Friedemann Brock (Swansea) A unified approach to symmetry for semilinear equations associated to the Laplacian on $$\mathbb{R}^n$$ 14:30-15:00 Coffee 15:00-15:45 Nikolai Leonenko (Cardiff) Fractional Pearson Diffusions and Continuous Time Random Walks 15:45-16:30 Farzad Fathizadeh (Swansea) Heat kernel expansion of the Dirac-Laplacian of multifractal Robertson-Walker metrics

Abstracts

Karl Michael Schmidt (Cardiff) - On Singular Positive Definite Functions

Positive definite functions in the classical sense are bounded and in fact take on their maximum value at 0. However, it turns out that there are unbounded functions which exhibit the general property of positive definiteness on suitable measures and play a role as integral kernels characterising well-known probability measures, e.g. the arcsine distribution, as minimisers. In the talk I shall give an overview of recent joint work with Tomos Phillips and Anatoly Zhigljavsky on singular positive definite functions, including generalisations of the characterisation of the Fourier transform of positive definite functions (Bochner's theorem) and the fact that conditionally positive definite functions generate positive definite semigroups (Schoenberg's theorem).

Friedemann Brock (Swansea) - A unified approach to symmetry for semilinear equations associated to the Laplacian on $$\mathbb{R}^n$$

We show radial symmetry of positive solutions of $$-\Delta u = f(|x|,u)$$ on $$\mathbb{R}^N$$ (or $$\mathbb{R}^N \setminus \{ 0\}$$), satisfying $$\lim_{|x|\to \infty } u(x)=0$$, where $$f$$ is smooth, $$f(r,u)\sim r^{-\ell } u^q$$ for some positive numbers $$\ell$$, $$q$$ and satisfies some further conditions. A new ingredient is a maximum principle for open subsets of a half space. It allows to apply the Moving Plane Method once a slow decay of the solution at infinity has been established, that is $$\lim _{|x|\to \infty } |x|^{\gamma } u(x) =L$$, for some numbers $$\gamma \in (0, N-2)$$ and $$L >0$$. This is joint work with A. Avila (Temuco).

Nikolai Leonenko (Cardiff) - Fractional Pearson Diffusions and Continuous Time Random Walks

We define fractional Pearson diffusions by non-Markovian time change in the corresponding Pearson diffusions. They are governed by the time-fractional diffusion equations with polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of fractional Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Also, we present the strong solutions of the Cauchy problems associated with heavy-tailed fractional Pearson diffusions and the correlation structure of these diffusions.

Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps are correlated so that the limiting processes are not Lévy but diffusion processes with non-independent increments.

This is a joint work with M. Meerschaert (Michigan State University, USA), I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek, Croatia) and A. Sikorskii (Michigan State University and Arizona University, USA).

Farzad Fathizadeh (Swansea) - Heat kernel expansion of the Dirac-Laplacian of multifractal Robertson-Walker metrics

I will talk about a recent work in which we find an explicit formula for each Seeley-deWitt coefficient in the full heat kernel expansion of the Dirac-Laplacian of a Robertson-Walker metric with a general cosmic expansion factor. We use the Feynman-Kac formula and combinatorics of Brownian bridge integrals heavily. The extension of the result to the inhomogeneous case, where the spatial part of the model has a fractal structure, will also be presented. This is joint work with Yeorgia Kafkoulis and Matilde Marcolli.

Organisers

Jonathan Ben-Artzi (Cardiff)
Dmitri Finkelshtein (Swansea)