Schedule for: 18w5129 - Theoretical and Applied Stochastic Analysis

We will talk about convergence rates for the empirical spectral measure of a unitary Brownian motion. We give explicit bounds on the 1-Wasserstein distance of this measure to both the ensemble-averaged spectral measure and to the large-N limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.

In this talk we study small time asymptotic of the heat content for a smoothly bounded domain with non-characteristic boundary in the Heisenberg group, which captures geometric information of the of the boundary including perimeter and the total horizontal mean curvature of the boundary of the domain. We use probabilistic method by studying the escaping probability of the horizon- tal Brownian motion process that is canonically associated to the sub-Riemannian structure of the Heisenberg group. This is a joint work with J. Tyson.

The Brox diffusion is a stochastic process in random environment often considered the scale and time continuous analogue of Sinai's walk. In this talk we apply Excursion theory to the Brox diffusion in order to obtain the distribution of certain important random variables.

In this talk we will discuss a theorem guaranteeing the existence of global (in time) solutions to rough path differential equations on a smooth manifold.

A general framework on Dirichlet spaces is developed to prove a weak form of the Bakry-Emery estimate and study its consequences. This estimate may be satisfied in situations, like metric graphs, where generalized notions of Ricci curvature lower bounds are not available. We also talk about current research directions by taking limits.

I will talk about the dimension-free $L^p$ boundedness of operators on manifolds obtained as conditional expectations of martingale transforms à la Gundy-Varopoulos. Applications on Lie groups of compact type and the Heisenberg group will be introduced. This talk is based on a joint work with R. Bañuelos and F. Baudoin.

Generalized diamond fractals constitute a parametric family of spaces that arise as scaling limits of so-called diamond hierarchical lattices. The latter appear in the physics literature in the study of random polymers, Ising and Potts models among others. In the case of constant parameters, diamond fractals are self-similar sets. This property was exploited in earlier investigations by Hambly and Kumagai to study the corresponding diffusion process and its heat kernel. These questions are of interest in this setting in particular because the usual assumption of volume doubling is not satisfied. For general parameters, also the self-similarity is lost. Still, a diamond fractal can be regarded as an inverse limit of metric measure graphs and a canonical diffusion process obtained through a general procedure proposed by Barlow and Evans. This approach will allow us to provide a rather explicit expression of the associated heat kernel and deduce several of its properties. As an application, we will discuss some functional inequalities of interest.

We provide some explicit results for stable processes obtained from the perspective of the theory of self-similar Markov processes. In particular, we turn our attention to the case of $d$-dimensional isotropic stable process, for $d\geq 2$. Using a completely new approach we consider the distribution of the point of closest reach. This leads us to a number of other substantial new results for this class of stable processes. We engage with a new radial excursion theory, never before used, from which we develop the classical Blumenthal--Getoor--Ray identities for first entry/exit into a ball, to the setting of $n$-tuple laws. We identify explicitly the stationary distribution of the stable process when reflected in its running radial supremum. Moreover, we provide a representation of the Wiener--Hopf factorisation of the MAP that underlies the stable process through the Lamperti--Kiu transform.

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub- Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spectral geometric meaning when Popp's measure is considered. Our proof is probabilistic. In particular, we use S. Watanabe's distributional Malliavin calculus.

Lyons’ theory of rough paths allows one to solve stochastic differential equations driven by a Gaussian processes X under certain conditions on the covariance function. The rough integral of these solutions against X again exist, and a natural question is to find a closed-form conversion formula between this rough integral and the Skorohod integral of the solution which generalises the classical Stratonovich-Ito conversion formula. Previous works in the literature assumes the integrand to be the gradient of a smooth function of X; our formula again recovers these results as special cases. Joint work with Nengli Lim.

We consider the class of all Gaussian processes observed at regularly spaced discrete times. For stationary processes, when the spectral density is parametrically explicit, we define a Generalized Method of Moments estimator that satisfies consistency and asymptotic normality, using the Breuer-Major theorem which applies to long-memory processes. This result is applied to the joint estimation of the three parameters of a stationary fractional Ornstein-Uhlenbeck (fOU) process driven for all Hurst parameters. For general non-stationary processes, no matter what the memory length, we use state-of-the-art Malliavin calculus tools to prove Berry-Esseen-type and other speeds of convergence in total variation, for estimators based on power variations. This is joint work with Luis Barboza (U. Costa Rica), Khalifa es-Sebaiy (U. Kuwait), and Soukaina Douissi (U. Cadi Ayyad, Morocco).

In this talk, we will consider high-dimensional Wishart matrices associated with a rectangular random matrix $X_{n,d)$ whose entries are jointly Gaussian and correlated. Our main focus will be on the case where the rows of $X_{n,d)$ are independent copies of a n-dimensional stationary centered Gaussian vector of correlation function s. When s is 4/3-integrable, we will show that a proper normalization of the corresponding Wishart matrix is close in Wasserstein distance to the corresponding Gaussian ensemble as long as d is much larger than $n^3$, thus recovering the main finding of Bubeck et al. and extending it to a larger class of matrices.

In this talk, we use the techniques of fractional calculus and the fix-point theorem to show that a semilinear fractional differential equation driven by a gamma-Holder continuous noise, gamma>2/3, has a unique solution. The initial condition could be not defined at zero and the involve integral is in the Young sense.

Hitting questions play a central role in the theory of Markov processes. For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. There has also been a huge amount of work on the related question of when a process has multiple points. For stochastic partial differential equations (SPDE), much less is known, but there has been a growing number of papers on the topic in recent years. Potential theory provides an answer in principle. But unfortunately, solutions to SPDE are infinite dimensional processes, and the potential theory is intractible. As usual, the critical case is the most difficult. We will give a brief survey of known results, followed by a discussion of an ongoing project with R. Dalang, Y. Xiao, and S. Tindel which promises to answer questions about hitting points and the existence of multiple points in the critical case.

In this talk, we examine the connection between the hyperbolic and parabolic Anderson models in arbitrary space dimension d, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) d = 1 and the noise is a fractional Brownian motion in space with index 1/4 < H < 1/2. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on this connection and the recent powerful results of Huang, Le and Nualart (2017) for the parabolic model, we compute the second order (upper) Lyapunov exponent for the hyperbolic model. In case (i), when the spatial covariance of the noise is given by the Riesz kernel, we present a unified method for calculating the second order Lyapunov exponents for the two models.

In this talk I will speak about the weak and strong solutions to the stochastic differential equation dX(t) =−1/2 W'(X(t))dt + dB(t), where (B(t), t ≥ 0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of B. It is shown that the Ito–McKean representation associated with any Brownian motion (independent of W) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Ito calculus for the solution is developed. For dealing with the singularity of drift term W˙ (X(t)), the main idea is to use the concept of local time together with the polygonal approximation of the Brownian motion. This is joint work with Khoa Le and Leonid Mytnik.

We show that, for a large class of semi-linear parabolic PDEs, driven by space-time white noise, the solution converges exponentially rapidly to zero either as time tends to infinity or as the noise input is increased. All of this is based on a moment decay inequality that is a counterpart to standard intermittency-type moment estimates for the solution. The said inequality is based on an $L^1/L^\infty$ interpolation method which is of independent interest. This talk is based on joint work with Carl Mueller (U.Rochester-USA), Kunwoo Kim (POSTECH-Korea), and Shang-Yuan Shiu (NCU-Taiwan).

This talk is concerned with sample path regularities of isotropic Gaussian fields and the solution of the stochastic heat equation on the unit sphere ${\mathbb S}$. In the first part, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behavior of its angular power spectrum; we then apply this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behavior. In the second part, we consider the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on ${\mathbb S}^2$ and establish the exact uniform moduli of continuity of the solution in the time and spatial variable, respectively. This talk is based on joint works with Xiaohong Lan and Domenico Marinucci.

The aim of this talk is to give an example of stochastic PDE such that for each (t,x), the paths of the solution u(t,x) are discontinuous with probability one.

In this talk, we study the stochastic heat equation on R^d driven by a multiplicative Gaussian noise which is white in time and colored in space. The diffusion coefficient rho can be degenerate, which includes the parabolic Anderson model rho(u)= u as a special case. The initial data is rough in the sense that it can be any measure, including the Dirac delta measure, that satisfies some mild integrability conditions. Under these degenerate conditions, for any given t>0 and distinct m points x_1, ... x_m in R^d, we establish the existence, regularity, and strict positivity of the joint density of the random vector (u(t,x_1), ...u(t,x_m)). The talk is based on a recent jointwork with Yaozhong Hu and David Nualart for the spatial dimension case, and an ongoing research project with Jingyu Huang for the higher spatial dimension case.