Семинар по теории вероятности

Spring 2018

In a series of three papers in the 80's, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock's approach and point out where the difficulties are in our current setting. The talk is based on an ongoing project with Xi Geng and Samy Tindel.

We will investigate various limits concerning the long time average velocity ${\scriptsize V}$ of a Brownian particle diffusing on a periodic potential. The prototype model is Langevin dynamics which incorporates inertia (mass) and friction. The key feature of the current work is the consideration of an additional macroscopic tilt, ${\scriptsize F}$. The goal is to understand how the average velocity ${\scriptsize V}$ depends on ${\scriptsize F}$. Interesting thresholds for the value of ${\scriptsize F}$ can be obtained, in particular under the limit of vanishing friction and noise. Using the averaging theory of Freidlin-Wentzel, the current work provides rigorous mathematical justification of some formulas obtained by Risken. An earlier, fundamental result by Tanaka for Brownian particle diffusing on a random (Brownian) potential will also be discussed. This is a joint work with Liang Cheng.

In this talk we will discuss stationary and evolution obstacle problems defined by a class of nonlocal operators that are not stable-like and may have supercritical drift. Our goal is to establish existence, uniqueness, and regularity of viscosity solutions. In particular, we will give sufficient conditions on the coefficients of the operator to obtain Hölder and Lipschitz continuous solutions. The class of nonlocal operators that we consider include non-Gaussian asset price models widely used in mathematical finance, such as Variance Gamma Processes and Regular Lévy Processes of Exponential type.

In this talk, we investigate the exact long time asymptotics of the heat kernel on abelian covering spaces over Riemannian manifolds with Dirichlet or Neumann boundary conditions. As an application, we establish a Gaussian type central limit theorem for the winding number of reflected Brownian motion in a planar domain with multiple holes. In particular, this is a refinement of a classical result by Toby and Werner, in which a law of large numbers was obtained. This is a recent joint work with Gautam Iyer.

The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century. In 1925, Marcel Riesz proved the ${\scriptsize L^p}$ boundedness, ${\scriptsize p\in(1,\infty)}$, of the continuous version of this operator (also called the conjugate function). From this, he deduced the same result for the discrete version. In 1926, Titchmarsh gave a new proof of Riesz’s result and claimed that in addition the two operators have the same ${\scriptsize p}$-norms. The following year, Titchmarsh pointed out that his argument for the equality of the norms was incorrect. The problem of equality has been a long-standing conjecture since. In this talk we describe a proof of this conjecture. This is joint work with Mateusz Kwasnicki of Wroclaw University.

We shall consider a geometric graph model on the "hyperbolic" space, which is characterized by a negative Gaussian curvature. Among several equivalent models representing the hyperbolic space, we treat the most commonly used d-dimensional Poincare ball model. One of the main characteristics of geometric graphs on the hyperbolic space is tree-like hierarchical structure. From this viewpoint, we discuss the asymptotic behavior of sub-tree counts. The spatial distribution of sub-trees is crucially determined by an underlying curvature of the space. For example, if the space gets flatter and closer to the Euclidean space, sub-trees are asymptotically scattered near the center of the Poincare ball. On the contrary, if the space becomes 'more hyperbolic' (i.e., more negatively curved), the distribution of trees is asymptotically determined by those concentrated near the boundary of the Poincare ball. We investigate the asymptotics of the expectation and variance of sub-tree counts. Moreover, we prove the corresponding central limit theorem as well. This is joint work with Yogeshwaran D. at Indian Statistical Institute.

This talk will discuss results in a joint paper of mine with Steve Lalley from 1992. Suppose a particle, starting at position 0, moves according to a diffusion process on the real line. Suppose also that this particle emits daughter particles according to a branching process whose instantaneous rate can depend on location, though not on time. The daughter particles move independently according to the same diffusion process, starting at their points of birth, and in turn emit their own daughters according to the same branching process. The simplest special case of this situation is standard branching Brownian motion, with the rate of reproduction not depending on location. Let ${\scriptsize R_t}$ be the position of the right-most particle at time t, and let ${\scriptsize m_t}$ be the median of ${\scriptsize R_t}$. In 1937, Kolmogorov, Petrovskii, and Piskunov showed that, for standard branching Brownian motion, ${\scriptsize (m_t)/ t}$ converges to SQRT(2) and that ${\scriptsize (R_t - m_t)}$ converges in distribution to a nondegenerate limiting distribution.It turns out that results like those proved by Kolmogorov, et al, hold in great generality for one-dimensional branching diffusions. If the branching diffusion is 'recurrent' (in the sense that the initial position is re-visited at arbitrarily large times by some particle), and if space is rescaled so that m_t grows linearly, then ${\scriptsize (R_t - m_t)}$ converges in distribution to a location-mixture of extreme value distributions. We also have the Andy Warhol Theorem, according to which every particle ever born has a descendant in the lead at some point in the future.

Rough Paths Theory was developed in the 1990s by Terry Lyons. Recently, Martin Hairer used rough paths to create solution theories to stochastic PDEs such as KPZ equation and introduced a generalization known as regularity structures, for which he won the Field's medal in 2014. Rough paths allow us to define pathwise solutions to controlled differential equations driven by signals with low Hölder regularity, such as fractional Brownian motion. Classically, if the driving signal is nice enough, we may define a Riemann-Stieltjes integral. In this talk, we discuss the limits of classical Young/Stieltjes integration, define rough paths and use them to define a rough integral against a controlled rough path, which we take to be the solution of a controlled differential equation.

Performance analysis of engineered systems, such queues and inventories, typically use Markov processes as representative models. Most 'real world' systems display time-varying behavior; for example time-of-day and day-of-week effects in traffic models of queues. On the other hand, the most widely used models in practice have strong stationarity and time homogeneity assumptions. This is with good reason: homogeneous stochastic models are far easier to use for computation and control/optimization purposes. It would, therefore, be very useful to have analytical approximations to non-homogeneous performance measures that do not have any greater computational complexity than the homogeneous counterparts. In this talk I will present some new results on analytical approximations for 'slowly changing' Markov chains, as well as older results on uniform acceleration approximations to nonhomogeneous Markov chains. This is joint work with Peter W. Glynn and Zeyu Zheng at Stanford University.

We consider excursions of a Brownian excursion above a certain level. We study the maximum heights of these excursions as Pitman-Yor did for excursions of a Brownian bridge.

This talk will discuss results in a joint paper of mine with Steve Lalley from 1992. Suppose a particle, starting at position 0, moves according to a diffusion process on the real line. Suppose also that this particle emits daughter particles according to a branching process whose instantaneous rate can depend on location, though not on time. The daughter particles move independently according to the same diffusion process, starting at their points of birth, and in turn emit their own daughters according to the same branching process. The simplest special case of this situation is standard branching Brownian motion, with the rate of reproduction not depending on location. Let ${\scriptsize R_t}$ be the position of the right-most particle at time t, and let ${\scriptsize m_t}$ be the median of ${\scriptsize R_t}$. In 1937, Kolmogorov, Petrovskii, and Piskunov showed that, for standard branching Brownian motion, ${\scriptsize (m_t)/ t}$ converges to SQRT(2) and that ${\scriptsize (R_t - m_t)}$ converges in distribution to a nondegenerate limiting distribution.It turns out that results like those proved by Kolmogorov, et al, hold in great generality for one-dimensional branching diffusions. If the branching diffusion is 'recurrent' (in the sense that the initial position is re-visited at arbitrarily large times by some particle), and if space is rescaled so that m_t grows linearly, then ${\scriptsize (R_t - m_t)}$ converges in distribution to a location-mixture of extreme value distributions. We also have the Andy Warhol Theorem, according to which every particle ever born has a descendant in the lead at some point in the future.

The purpose of this talk is to give an idea of the Malliavin calculus techniques for Lévy processes, based on the so-called annihilation operator and its adjoint. We will see that this important stochastic analisis tool allows us to consider problems that cannot be studied by means of the classical Itô calculus, although me might be in an adapted context with respect to the underlying filtration.

In this talk, we discuss transition densities of pure jump symmetric Markov processes in ${\scriptsize R^d}$, whose jumping kernels are comparable to radially symmetric functions with general mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of transition densities (heat kernel estimates) for such processes. This is a joint work with Joohak Bae, Jaehoon Kang and Jaehun Lee.