Research Seminar

 Abstract: TBA

 Abstract: The extremal process of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point-- the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

 Abstract: In this presentation, I'll explore the analysis of non-local regular Dirichlet forms on metric measure spaces. Our approach relies on three key assumptions: the presence of a strongly local Dirichlet form with sub-Gaussian heat kernel estimates, a tail estimate governing the jump measure outside balls, and a local energy comparability condition. Our primary objectives include establishing function inequalities such as localized Poincaré, cutoff Sobolev, and Faber-Krahn inequalities, highlighting their inherent stability structures and their implications for the regularity of corresponding harmonic functions. Our results are robust in the sense that the constants in estimates remain bounded, provided that the order of the scale function appearing in the tail estimate and local energy comparability condition, maintains a certain distance from zero.

 Abstract: In this talk, we provide sensitivity analysis for two types of problems: robust optimization and nonlinear Kolmogorov partial differential equations (PDEs). Our first goal is to quantify the sensitivity of a given robust optimization problem to model uncertainty. This can be achieved by showing that the robust problem can be approximated as ε approaches 0 by the baseline problem, computed using baseline processes. Subsequently, we aim to quantify the sensitivity of these PDEs to small nonlinearities in the Hamiltonian of drift and diffusion coefficients, and then use the results to develop an efficient numerical method for their approximation. We demonstrate that as ε approaches 0, the nonlinear Kolmogorov PDE can be approximated by linear Kolmogorov PDEs involving the baseline coefficients. Our derived sensitivity analysis then provides a Monte Carlo-based numerical method that can efficiently solve these nonlinear Kolmogorov PDEs. This talk is based on joint works with Daniel Bartl (Univ. Vienna) and Ariel Neufeld (NTU Singapore).

 Abstract: In this talk we present a quantum Monte Carlo algorithm to solve high-dimensional Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we present numerical simulations in one and two dimensions using our developed package within the Qiskit framework tailored to price CPWA options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension. This talk is based on joint work with Jianjun Chen and Yongming Li.

 Abstract: We introduce the concept of a hyperbolic Fourier series and apply the main representation theorem to the solutions of the Klein-Gordon equation in 1+1 dimensions.

 Abstract: We study solutions of the irregular Stratonovich SDE $dX =|X|^\alpha \circ dB$, $\alpha\in (0, 1)$. In particular, we construct solutions spending positive time in 0, describe solutions spending zero time in 0, and show how a particular physically natural solution can be singled out by means of an additional external "ambient" noise.

This talk is based on the joint works with G. Shevchenko (Kiev).

 Abstract: The spectral heat content (SHC) measures the total heat that remains on a domain when the initial temperature is one and the outside temperature is identically zero. When one replaces the Laplace operator in the heat equation with generators of Lévy processes, one obtains SHC for those Lévy processes. Recently, the two-term asymptotic behavior of SHC for isotropic stable processes on bounded C^11 open sets was investigated by Park and Song (EJP 2022). In this talk, we generalize their result to cover Lévy processes with regularly varying characteristic exponent with index in (1,2]. The proof provides a unified approach to the study of SHC and applies to both Brownian motions and jump processes. This is a joint work with Kei Kobayashi (Fordham University).

 Abstract: Gundy and Varopoulos introduced the probabilistic representation of singular integrals and Fourier multipliers such as Hilbert transforms and Riesz transforms as conditional expectations of some stochastic integrals. Combining with the sharp martingale inequalities by Burkholder and Banuelos-Wang, the representations have played a crucial role in finding the sharp, or nearly sharp, $L^p$-bounds for these operators in a variety of geometric settings. Motivated by a recent breakthrough of Banuelos and Kwasnicki on the sharp $\ell^p$-norm of the discrete Hilbert transform, we construct a natural collection of discrete operators on $\mathbb{Z}^d$  which have $\ell^p$-norms independent of the dimension. This collection of discrete operators include the probabilistic discrete Riesz transforms, which are the analogues of the probabilistic discrete Hilbert transform used in the paper by Banuelos-Kwasnicki. In this talk, we discuss the construction of the probabilistic discrete operators, their $\ell^p$ bounds, and related open problems. This is based on joint work with Rodrigo Banuelos and Mateusz Kwasnicki.

 Abstract: We consider a system of N neurons, whose membrane potentials follow a mean-field interaction dynamics. More precisely, for each neuron, this potential decreases at constant rate and, on the other hand, it is set back to 0 when the neuron emits a spike, which also induces an increase of the potential of all the other neurons. The spikes occur at random times, at a rate lambda(u) which depends on the membrane potential u. When lambda(u) is zero at zero and differentiable then, for all N, the system almost surely stops in finite time, i.e. there is only a finite number of spikes, followed by a deterministic decay of the system toward 0. We will see that, under some conditions, however, the system is metastable, in the sense that the following points hold: 1) the non-linear limit (as N->infinity) system converges to a unique non-zero equilibrium ; 2) the extinction time of a finite system of size N is exponentially large in N ; 3) the average potential of the system fastly approaches a constant positive value, and the exit times from neighborhood of this value converge (as N->infinity) to the exponential law. The proofs rely on coupling techniques. This is a joint work with Eva Löcherbach.