Stochastic Differential Equations: 

Analysis & Modelling


Titles & Abstracts



 

Stochastic resonance in stochastic PDEs

Stochastic resonance can occur when a multi-stable system is subject to both periodic and random perturbations. For suitable parameter values, the system can respond to the perturbations in a way that is close to periodic. This phenomenon was initially proposed as an explanation for glacial cycles in the Earth's climate. While its role in that context remains controversial, stochastic resonance has since been observed in many physical and biological systems. This talk will focus on stochastic resonance in parabolic SPDEs, such as the Allen-Cahn equation, when they are driven by a periodic perturbation and by space-time white noise. We will discuss both the case of one spatial dimension, in which the equation is well-posed, and the case of two spatial dimensions, in which a renormalisation procedure is required.  This talk is based on joint works with Barbara Gentz and Rita Nader.

References:
 https://dx.doi.org/10.1007/s40072-021-00230-w
 https://arxiv.org/abs/2107.07292
 https://arxiv.org/abs/2209.15357

 

 

Large Deviations for $(1+1)$-dimensional Stochastic Geometric Wave Equation

We consider stochastic wave map equation on real line with solutions taking values in a $d$-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise. This talk is based on a joint research with B Go{\l}dys, M Ondrej\'at and N Rana.



On the fine structure of minimizers in the Allen-Cahn theory of phase transitions

 

Let W a nonnegative potential with N nondegenerate zeros. We study the structure of minimizers u, subjected to Dirichlet conditions, of the Allen-Cahn energy with potential W. We give sufficient conditions, in dimension n, on the domain and on the boundary data ensuring the connectivity of the diffuse interface (the subset where u is away from all the zeros of W). Then we restrict to two dimensions and show that the connectivity of the phases (the sets where u is near to one of the zeros of W) allows for a precise description of the structure of u which is tightly related to the minimizer of the geometric sharp interface problem.



Noise-induced synchronization in circulant networks of weakly coupled oscillators 


Consider a finite-size system of coupled harmonic oscillators with a circulant coupling structure. Using an averaging result for stochastic differential equations we will show that weak multiplicative-noise coupling can amplify some of the systems’ eigenmodes and, hence, lead to asymptotic eigenmode synchronization. As a result we obtain a general formula which allows to determine the relevant eigenmodes as a function of the our choice of coupling.

Reference: PhD thesis of Christian Wiesel (formerly University of Bielefeld)



Noise-induced transition in the Lorenz system


 

Stefan problems and systemic risk


 Motivated by modelling systemic risk in banking networks we consider some particle systems on the half line. The particles represent the values of financial firms and default occurs when they hit the origin. By incorporating feedback effects, where the default of one entity induces a downward shock on the rest, and taking a mean field limit these models lead to PDEs and SPDEs which are variants on the supercooled Stefan problem. If the feedback is strong enough these limit models can exhibit jumps and we will discuss existence and uniqueness of solutions for various versions of these models.

 

 

An order-one adaptive scheme for the strong approximation of stochastic systems with jumps

 Consider a $d$-dimensional system of stochastic differential equations with $m$ independent diffusions where the drift and diffusion coefficients are not globally Lipschitz continuous but instead only locally Lipschitz and together satisfy a Khasminskii-type montone condition. It is known that the explicit Milstein scheme fails to converge for such systems when applied over a uniform mesh. We construct an adaptive mesh that responds to the local behaviour of solutions by reducing the stepsize as solutions approach the boundary of a sphere, invoking a convergent backstop method in the (rare) event that the timestep becomes too small. With such a mesh, order-one strong $L_2$-convergence of the scheme can be ensured, even when the diffusion coefficients of the SDE are non-commutative. We also examine how this adaptive strategy can be modified to allow for the discretisation of systems of SDEs perturbed by a Poisson jump process independent of the perturbing diffusion processes, without loss of order, as long as the independent jump times are pre-computed and included in the adaptive mesh. We will demonstrate the use of our scheme in the modelling of stochastic telomere length dynamics arising from human cellular division over time.


 

SPDE in fast non-Markovian random environment

 

Recently there have been significant development of two scale stochastic equations with non Markovian slow motions, and also of fast non-Markovian dynamics. In this context we focus on SPDEs in a non-Markovian random environment which becomes rough as the time separation scale epsilon tends to 0. One difficult residue in obtaining uniform  in epsilon estimates. We discuss the technique of mild stochastic sewing lemma, the regularizing property of integration, and also the fact that we  cannot prepare the initial data for the non-Markovian dynamics. This is based mainly joint work with J. Sieber.

 

 

Time discretization of SPDEs with additive noise 

 

We consider strong convergence of tamed exponential and adaptive time-step methods for SPDEs with non-globally Lipschitz noninearity. A typical example would the the stochastic Allen-Cahn or Swift-Hohenberg equations. Although not designed to control local error directly we observe in numerical simulations an improved error constant and that the adaptivity is good at capturing the local behaviour. The talk will introduce the issue with standard Euler methods for non-globally Lipschitz coefficients and introduce two different approaches for the adaptivity as well as the tamed exponential method. We will briefly examine the proof of strong convergence and compare numerically methods. This work is joint with Stuart Campbell and Conall Kelly.



Gamow's liquid drop model and the critical mass in nuclear fission

I will discuss the liquid drop model for atomic nuclei which goes back to Gamow’s theory in the 1930s. It is widely used in the physics literature to explain qualitatively the critical mass in nuclear fission reactions, but many of its fundamental properties remains unknown rigorously. I will discuss some mathematical results on the existence and nonexistence of minimizers, as well as the connection of these results to the ionization problem for atoms. The talk is based on recent joint work with Rupert L. Frank.

 

Uniqueness by noise for some classes of singular SPDEs

We give an overview of some recent results on uniqueness by noise for some classes of SPDEs with singular drift. In the first part of the talk we introduce the prototypes of the problems in consideration and we discuss the main existence results: these cover, for example, doubly nonlinear stochastic evolution equations, Burgers-type equations, and Cahn-Hilliard-type equations. Secondly, we focus on the direction of uniqueness by noise by means of the associated Kolmogorov equations, highlighting some recent contributions and open problems. The works presented in the talk are based on joint collaborations with Prof. Ulisse Stefanelli (University of Vienna, Austria), Dr. Carlo Orrieri (University of Pavia, Italy), and Federico Bertacco (Imperial College London, UK).

Mathematical results on propagation of electrical pulses in neurons


Alongside the Hodgkin-Huxley system, the Fitzhugh-Nagumo one is at the foundation of the quantitative neuroscience. Compared with the Hodgkin-Huxley system, the Fitzhugh-Nagumo one gives a less detailed but qualitatively, and often quantitatively; faithful description of propagation of electrical pulses in neurons. In this talk I will describe the recent mathematical results on propagation of pulses in a slightly more realistic model of neural axons. The talk is based on the joint work with Afroditi Talidou and Almut Burchard. 



 Consensus-based optimization via jump-diffusions

We introduce a new consensus based optimization (CBO) method, where an interacting particle system  is driven by  jump-diffusion stochastic differential equations.  We study well-posedness of the particle system and of its mean-field limit.  We prove convergence of the interacting particle system to  the mean-field limit and convergence of a discretized particle system to the continuous-time dynamics in the mean-square sense. We also prove convergence of the mean-field jump-diffusion SDEs towards global minimizers for a large class of objective functions. Numerical tests performed on benchmark objective functions demonstrate improved performance of the proposed CBO method over earlier CBO methods. The talk is based on a joint work with Dante Kalise (Imperial College) and Akash Sharma (University of Nottingham).