Schedule for: 23w5060 - New Trends in Stochastic Analysis

In 1979 R. F. Gundy and N. Th. Varopoulos published a beautiful 3-page paper titled "Les transformations de Riesz et les integrales stochastiques" in which they gave a representation for the classical Riesz transforms on Rd as conditional expectations of stochastic integrals. During the last 40+ years, in combination with sharp martingale inequalities, this simple and elegant representation has had a phenomenal success in obtaining optimal, or near optimal, Lp bounds for Riesz transforms and more general singular integrals and Fourier multipliers in a variety of geometric and analytic settings. An important feature of these bounds is their independence on the geometry of the ambient space. In this talk the speaker will discuss a modi cation of this construction which leads to sharp estimates for discrete singular integrals on the lattice $Z^d$,$ d\ge 1$, and in particular to the identification of the $\ell_pE norm of the discrete Hilbert transform on the integers $Z$. The latter had been a long-standing conjecture initiated in part by an erroneous proof of E. C. Titchmarsh in 1926. Based on work with Mateusz Kwasnicki ofWroc law University, Poland (2019), and Daesung Kim, Georgia Tech & Mateusz Kwasnicki (2022).

We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as the time variable tends to infiniry, and we establish an upper bound of exp(-const. t^{1/3}) for the rate of decay. We demonstrate also that the length of a valley grows at least as exp(+const. t^{1/3}) as t gets large. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails. This is based on joint work with Kunwoo Kim (POSTECH) and Carl Mueller (Rochester).

In this talk, we adopt the viewpoint about fractional fields which is given in Lodhia and al. Fractional Gaussian fields: a survey, Probab. Surv., 2016. As example, we focus on random fields defined on the Sierpiński gasket but random fields defined on fractional metric spaces can also be considered. Hence, for $s\ge 0$, we consider the random measure $X=\left(-\Delta\right)^{-s} W$ where $\Delta$ is a Laplacian on the Sierpiński gasket $K$ equipped with its Hausdorff measure $\mu$ and where $W$ is a Gaussian random measure with intensity $\mu$. For a range of values of the parameter $s$, the random measure $X$ admits a Gaussian random field $\left(X(x)\right)_{x\in K}$ as density with respect to $\mu$. Moreover, using entropy method, an upper bound of the modulus of continuity of $\left(X(x)\right)_{x\in K}$ is obtained, which leads to the existence of a modification with Hölder sample paths. In addition, the fractional Gaussian random field $X$ is invariant by the symmetries of the gasket. If time allows, some extension to $\alpha$-stable random fields will also be presented. This is a joint work with Fabrice Baudoin (University of Connecticut).

Joint with Eyal Neuman. Studying the end-to-end distance of a self-avoiding or weakly self-avoiding random walk in two dimensions is a well known hard problem in probability and statistical physics. The conjecture is that the average end-to-end distance up to time $n$ should be about $n^{3/4}$. It would seem that studying more complicated models would be even harder, but we are able to make progress in one such model. A star polymer is a collection of $N$ weakly mutually-avoiding Brownian motions taking values in $\mathbf{R}^d$ and starting at the origin. We study the two and three dimensional cases, and our sharpest results are for $d=2$. Instead of the end-to-end distance, we define a radius $R_T$ which measures the spread of the entire configuration up to time $T$. There are two phases: a crowded phase for small values for $T$, and a sparser phase for large $T$ where paths do not interfere much. Our main result states for $T (Online)

In this talk, we introduce the hyperbolic Anderson model in dimension 1, driven by a time-independent rough noise, i.e. the noise associated with the fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. The goal of the talk will be to show that, with appropriate normalization and centering, the spatial integral of the solution converges in distribution to the standard normal distribution, and to estimate the speed of this convergence in the total variation distance. For this, we use some recent developments related to the Stein-Malliavin method. More precisely, we use a version of the second-order Gaussian Poincar\'e inequality developed by Nualart, Xia, Zheng (2022) for the similar problem for the parabolic Anderson model with rough noise in space (colored in time). To apply this method, we need to derive first some moment estimates for the increments of the first and second Malliavin derivatives of the solution. These are obtained using a connection with the wave equation with delta initial velocity, a method which is different than the one used in the parabolic case. This talk is based on joint work with Wangjun Yuan (University of Luxembourg).

We consider the non-linear stochastic heat equation defined on the whole line driven by a space­ time white noise. We will first recall some known results on the almost sure blow up for this type of equations. We then give sufficient conditions for the solution to blow up everywhere and instantaneously almost surely. The main ingredient of the proof is the study of the spatial growth of stochastic convolutions using techniques from Malliavin calculus and Poincare inequalities . This is a joint work with Davar Khoshnevisan and Mohammud Foondun.

In the first part of the talk, we consider a system of two coupled stochastic lattice equations driven by additive white noise processes. Our objective is to investigate its longtime behavior. A system synchronizes if all elements eventually exhibit the same behavior. We show a synchronization for this system. To describe this phenomenon, we prove the upper semi continuity of the family of attractors with respect to the attractor of a specific limiting stochastic system. In the second part of the talk, we investigate a complex network consisting of finitely many nodes that are coupled in a deterministic and stochastic way. The behavior of each node is described by an evolution equation that includes a reaction diffusion equation driven by a multiplicative noise. We prove that the system synchronizes towards a deterministic equation. Furthermore, we describe various concepts of synchronization

The empirical correlation $\rho_n$, defined for two related sequences of data of length $n$, is defined classically via Pearson’s correlation statistic. When the data is i.i.d. with two moments, it is known to converge to the correlation coefficient of the pair of random variables behind the two data series, with normal fluctuations, as $n$ tends to infinity. This property remains if the two sequences of data have some level of sequential correlation and are stationary. However, it fails under stronger memory conditions, and when the sequences are sufficiently non-stationary. Famously, the statistic is asymptotically diffuse, over the entire interval (−1,1), when the data are random walks. The statistics is known as Yule's "nonsense correlation" in honor of the statistician G. Udny Yule who first described the phenomenon in 1926. Many decades later, there still exist vexing instances of applied scientists who draw incorrect attribution conclusions based on invalid inference about correlations of time series, in ignorance of Yule’s observation. We will described the mathematical question of what happens exactly with $\rho_n$ as $n$ gets large. We will present an explicit expression for $\rho_n$’s second moment, in the case of a pair of discrete-time random walks with Gaussian increments, and we will explain why this moment converges to 0 at the rate $1/n^2$. This result appeared in a paper with Philip Ernst and Dongzhou Huang, in SPA in April 2023. The result motivates the further study of $\rho_n$’s asymptotics, in order to build statistical test of independence of paths of Gaussian stochastic processes. We will discuss this open issue briefly, which involves Gaussian and non-Gaussian convergence in Wiener chaos

We investigate the well-posedness of a class of stochastic second-order in time damped evolution equations in Hilbert spaces, subject to the constraint that the solution lies within the unitary sphere. Then, we focus on a specific example, the stochastic damped wave equation in a bounded domain of a $d$-dimensional Euclidean space, endowed with the Dirichlet boundary condition, with the added constraint that the $L^2$-norm of the solution is equal to one. We introduce a small mass $\mu>0$ in front of the second-order derivative in time and examine the validity of a Smoluchowski-Kramers diffusion approximation. We demonstrate that, in the small mass limit, the solution converges to the solution of a stochastic parabolic equation subject to the same constraint. We further show that an extra noise-induced drift emerges, which in fact does not account for the Stratonovich-to-It\^{o} correction term.

In this talk we will study stochastic heat equation $\partial_t u=\frac{1}{2}\Delta u+u\dot{W}_\alpha$ defined on the Heisenberg group $\Heis$, where $\Delta$ is the hypoelliptic Laplacian on $\Heis$ and $\{\dot{W}_\alpha; \alpha>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-\Delta)^{-\alpha}$ in space. We will give a proper description of the noise $W_\alpha$, and prove that the stochastic heat equation can be solved in the It\^{o} sense when $\alpha>\frac{n}{2}$. We also give some basic moment estimates for the solution $u(t,x)$. This is a joint work with F. Baudoin, C.Ouyang, and S. Tindel.

Many standard existence and uniqueness results for stochastic partial differential equations assume that the deterministic and stochastic forcing terms are Lipschitz continuous with at most linear growth. Another common set of assumptions requires the deterministic forcing to be dissipative with at most polynomial growth. I present some recent results about much more general sufficient conditions on superlinear deterministic and stochastic forcing terms that guarantee that mild solutions to stochastic partial differential equations exist, are unique, and never explode. Results compare various settings including bounded and unbounded spatial domains and the cases of dissipative and accretive superlinear deterministic forces.

In this talk, we study Dirichlet fractional Gaussian fields on the Sierpinski gasket. Heuristically, such fields are defined as distributions $X_s=(-\Delta)^{-s}W$, where $W$ is a Gaussian white noise and $\Delta$ is the Laplacian with Dirichlet boundary condition. The construction is based on heat kernel analysis and spectral expansion. We also discuss regularity properties and discrete graph approximations of those fields. This is joint work with Fabrice Baudoin.

Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of spatial effect on stochastic dynamics. In this talk, I will discuss a new approach, namely, to study these equations on general 1-dimensional structures (including metric graphs and fractals) that flexibly parametrizes space. This enables us to assess in great detail the impact of spatial effect on the co-existence and the genealogies of interacting populations. We will focus on recent results on extinction/survival probability, quasi-stationary distribution, asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type. Based on joint work with Rick Durrett, Wenqing Hu, Greg Terlov, and ongoing work with Zhenyao Sun, Oliver Tough and Yifan (Johnny) Yang.

We will consider the stochastic Navier-Stokes and stochastic Schrodinger equations and discuss their asymptotic limits such as large and moderate deviations, central limit theorem and the law of the iterated logarithm. To achieve the large deviation principle, we apply both techniques available in the literature: Azencott method and the weak convergence approach and compare the two methods. The Azencott method is then used to derive the law of the iterated logarithm. Also I will discuss my recently published book for graduate students: "Teaching and Research in Mathematics: A Guide with applications to Industry".

We present an intrinsic definition for a family of noise that is colored in space and white in time on the d-dimensional torus. We also show existence and uniqueness results for the parabolic Anderson model (linear stochastic heat equation) driven by this noise. We demonstrate intermittency by producing exponential lower bounds on the second moments using the Feynman-Kac formula for the solution.

Using the method introduced by Le Chen and Jingyu Huang, I will study the well-posedness of the Multiplicative Stochastic Heat Equation with colored noise on Compact Riemannian Manifolds. Unlike previous work on this equation, the Fourier transform is not employed. The main tools are Gaussian type bounds for the Heat kernel. Existence of the second moment of the solution in large time for all compact Riemannian manifolds will be shown. In small time, the difficulty induced by nonuniqueness of geodesics will be presented, along with how to overcome it on manifolds of nonpositive curvature.

It was conjectured by T. Lyons and N. Sidorova in 2006 that the logarithmic signature of a BV path always has finite radius of convergence unless the path is a line segment. Such a property is closely related to the study of rough differential equations from the Lie-algebraic perspective. In their original paper, the conjecture was confirmed for two special classes of paths: piecewise linear paths and paths that are monotone in one direction. In this talk, We discuss some recent progress on this problem in both probabilistic and deterministic settings. The heart of the main strategy lies in understanding the dynamics of path developments onto suitably chosen Lie groups. This is based on joint work with S. Wang as well as ongoing joint work with H. Boedihardjo and S. Wang.

In an earlier work with Chandra, Chevyrev and Hairer [CCHS?20], we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge equivariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev [CS?23], where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic constructed in [CCHS?20]. Using this universality result for the dynamics, we show that the Yang-Mills measure on 2D torus is the universal limit for these lattice gauge theories. We also prove that the Yang-Mills measure is invariant under the dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing, and Bourgain's method for invariant measures.

We consider the periodic homogenisation problem for dynamical $\phi^4_2$, a toy model that combines both renormalisation in singular SPDEs and homogenisation in a single problem. We show that in this situation, the two limiting procedures commute. Joint work with Yilin Chen (Peking University).

In this talk, we study the existence and an integration by parts formula for forward stochastic integrals of the form $\int_0^Tf(Y_s)dY_s^-$. Here the integral is interpreted in the Russo and Vallois sense, Y is a H\"older continuous process with exponent bigger than $1/2$ and $f$ is a deterministic function of bounded variation. As a consequence of the integration by parts formula, we can obtain a representation for the solutions to some equations of the form $$X_t=x+\int_0^t\sigma(X_s)dY_s^-,\quad t\in [0,T],$$ where $\sigma$ is a discontinuous coefficient with bounded variation. Finally, we interpret the local time of fractional Brownian motion as the trace term of a forward integral.

We consider a family of Markov chains, encompassing unadjusted Hamiltonian Monte Carlo schemes and splitting schemes of the kinetic Langevin diffusion, commonly used in Markov chain Monte Carlo algorithms. Assuming that the target distribution satisfies a logarithmic Sobolev inequality, we adapt to this discrete-time framework the modified entropy method used to establish the hypocoercive exponential decay in entropy for the continuous-time kinetic Langevin process. This yields non-asymptotic quantitative bounds in relative entropy (hence in Wasserstein and total variation distances) for the law of the chain at a given time with respect to its target. Since the schemes are second-order in the stepsize, we get in this general settings a complexity of order $d/\varepsilon^{1/4}$ (when the second, third and fourth derivatives of the potential are bounded independently from $d$) with $\varepsilon$ the error tolerance for the relative entropy, and of order $(d/\varepsilon)^{1/4}$ in the weakly interacting mean field case.

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. Furthermore, these results can be applied in an infinite-dimensional setting to prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space.

he signature is a non-commutative exponential that appeared in the foundational work of K-T Chen in the 1950s. It is also a fundamental object in the theory of rough paths (Lyons, 1998). More recently, it has been proposed, and used, as part of a practical methodology to give a way of summarising multimodal, possibly irregularly sampled, time-ordered data in a way that is insensitive to its parameterisation. A key property underpinning this approach is the ability of linear functionals of the signature to approximate arbitrarily closely (in the uniform topology) any compactly supported and continuous function on (unparameterised) path space. We use this conext to present some new results on the properties of a selection of topologies on the space of unparameterised paths. We discuss various related consequences and applications. This is based on joint work with Willliam Turner and, if time permits, some results in a joint paper with Terry Lyons and Xingcheng Xu.

The Glauber dynamics of the Ising-Kac model describes the evolution of spins on a lattice, with the flipping rate of each spin depending on an average field in a large neighborhood. Giacomin, Lebowitz, and Presutti conjectured in the 90s that the random fluctuations of the process near the critical temperature coincide with the solution of the dynamical $\Phi^4$ model. This conjecture was proved in one dimension by Bertini, Presutti, Ruediger, and Saada in 1993 and the two-dimensional case was proved by Mourrat and Weber in 2014. Our result settles the conjecture in the three-dimensional case. The dynamical $\Phi^4$ model is given by a non-linear stochastic partial differential equation which is driven by an additive space-time white noise and which requires renormalization of the non-linearity in dimensions two and three. The renormalization has a physical meaning and corresponds to a small shift of the inverse temperature of the discrete system away from its critical value. This is joint work with Hendrik Weber and Paolo Grazieschi.

We describe the connection between the Sinai random walk and the Brox diffusion, both processes with a random environment, the first one in discrete time and the second in continuous time. In doing so, we mention how one uses information of the environment to infer the behavior of the processes, which leads to predictions. An important tool we use is the so-called excursion theory of stochastic processes. In addition we present an application in financial modelling and pricing.