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Lenya Ryzhik

Publications and source records attributed to Lenya Ryzhik.

At least 19 recordsLinked to original sources
arXiv2026

Persistent anisotropy in multi-dimensional branching Brownian motion

We study angular variation in the frontier of branching Brownian motion (BBM) in multiple dimensions. The time average of this frontier is tied to the long-time limit of the critical derivative martingale, which is a random integrable function on the sphere. We show that this function is nowhere locally bounded. As a consequence, BBM exhibits strong persistent anisotropy: there are dense arbitrarily large gaps between the BBM frontier in different directions.

math.PR↗
arXiv2026

Mean Field Games without Rational Expectations

Mean Field Game (MFG) models implicitly assume "rational expectations", meaning that the heterogeneous agents being modeled correctly know all relevant transition probabilities for the complex system they inhabit. When there is common noise, it becomes necessary to solve the "Master equation", in which the infinite-dimensional density of agents is a state variable. The rational expectations assumption and the implication that agents solve Master equations is unrealistic in many applications. We show how to instead formulate MFGs with non-rational expectations. Departing from rational expectations is particularly relevant in "MFGs with a low-dimensional coupling", i.e. MFGs in which agents' running reward function depends on the density only through low-dimensional functionals of this density. This happens, for example, in most macroeconomics MFGs in which these low-dimensional functionals have the interpretation of "equilibrium prices." In MFGs with a low-dimensional coupling, departing from rational expectations allows for completely sidestepping the Master equation and for instead solving much simpler finite-dimensional HJB equations. We introduce an adaptive learning model as a particular example of non-rational expectations and discuss its properties.

math.AP↗
arXiv2025

Stationary solutions to the stochastic Burgers equation on the line

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity.

math.PR↗
arXiv2024

Geometric aspects of a spin chain

We discuss non-equilibrium thermodynamics of the mean field Ising model from a geometric perspective, focusing on the thermodynamic limit. When the number of spins is finite, the Gibbs equilibria form a smooth Legendrian submanifold in the thermodynamic phase space whose points describe the stable macroscopic states of the system. We describe the convergence of these smooth Legendrian submanifolds, as the number of spins goes to infinity, to a singular Legendrian submanifold, admitting an analytic continuation that contains both the stable and metastable states. We also discuss the relaxation to a Gibbs equilibrium when the physical parameters are changed abruptly. The relaxation is defined via the gradient flow of the free energy with respect to the Wasserstein metric on microscopic states, that is, in the geometric language, via the gradient flow of the generating function of the equilibrium Legendrian with respect to the ghost variables. This leads to a discrete Fokker-Planck equation when the number of spins is finite. We show that in the thermodynamic limit this description is closely related to the seminal model of relaxation proposed by Glauber. Finally, we find a special range of parameters where such relaxation happens instantaneously, along the Reeb chords connecting the initial and the terminal Legendrian submanifolds.

math-ph↗
arXiv2024

Diffusion of knowledge and the lottery society

The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~\cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.

math.AP↗
arXiv2023

Branching Brownian motion with generation-dependent diffusivity and nonlocal partial differential equations

We study a voting model on a branching Brownian motion process on $\mathbb{R}$ in which the diffusivity of each child particle is increased from that of the parent by a factor of $γ>1$. The probability distribution of the overall vote is given in terms of the solution to a nonlocal nonlinear PDE. We exhibit conditions on the nonlinearity such that the long-time behavior of the distribution undergoes a phase transition in $γ$. If $γ$ is sufficiently large, then the long-time distribution converges to uniform. If $γ$ is close enough to $1$, then the long-time distribution depends in a nontrivial way on the location of the initial particle. The limiting dependence is given by a steady-state solution to the nonlocal PDE. Our study gives a probabilistic interpretation of a class of semilinear nonlocal PDEs. Interestingly, while the PDE are nonlocal, the underlying random process does not require any non-local interactions.

math.AP↗
arXiv2023

Voting models and tightness for a family of recursion equations

We consider recursion equations of the form $u_{n+1}(x)=Q[u_n](x),~n\ge 1,~x\in R$, with a non-local operator $Q[u](x)= g( u\ast q)$, where $g$ is a polynomial, satisfying $g(0)=0$, $g(1)=1$, $g((0,1)) \subseteq (0,1)$, and $q$ is a (compactly supported) probability density with $\ast$ denoting convolution. Motivated by a line of works for nonlinear PDEs initiated by Etheridge, Freeman and Penington (2017), we show that for general $g$, a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case $g$ is also strictly monotone, a probabilistic threshold-based model can be given. In the latter case, we provide a conditional tightness result. We analyze in detail the bistable case and prove for it convergence of the solution shifted around a linear in $n$ centering.

math.PR↗
arXiv2023

Front location determines convergence rate to traveling waves

We propose a novel method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in [2]. It turns out that the convergence rate is controlled by the distance between the ``phantom front location'' for the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave itself has a pulled nature, regardless of whether the traveling wave is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach simplifies dramatically the proof in the Fisher-KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher-KPP case and the exponential rates in the pushed case.

math.AP↗
arXiv2023

Invariant measures for stochastic conservation laws on the line

We consider a stochastic conservation law on the line with solution-dependent diffusivity, a super-linear, sub-quadratic Hamiltonian, and smooth, spatially-homogeneous kick-type random forcing. We show that this Markov process admits a unique ergodic spatially-homogeneous invariant measure for each mean in a non-explicit unbounded set. This generalizes previous work on the stochastic Burgers equation.

math.PR↗
arXiv2023

Super-linear spreading in local and non-local cane toads equations

In this paper, we show super-linear propagation in a nonlocal reaction-diffusion-mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as $t^{3/2}$. We also get the sharp rate of spreading in a related local model.

math.AP↗
arXiv2023

A uniform bound for solutions to a thermo-diffusive system

We obtain uniform in time bounds for the solutions to a class of thermo-diffusive systems with classical and fractional diffusions. In the classical diffusion case, the nonlinearities are assumed to be at most exponentially growing, while for the fractional diffusions they should grow slightly sub-exponentially. The proofs are based on the strategy of Herrero, Lacey, and Velazquez.

math.AP↗
arXiv2023

Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength $β\in\mathbb{R}$. This equation exhibits a transition from pulled to pushed front behavior at $β_c=2$. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position $m_β(t)$ and study the asymptotics of the front location $m_β(t)$. When $β< 2$, it has the same form as for the standard Fisher-KPP equation established by Bramson \cite{Bramson1,Bramson2}: $m_β(t) = 2t - (3/2)\log(t) + x_\infty + o(1)$ as $t\to+\infty$. This form is typical of pulled fronts. When $β> 2$, the front is located at the position $m_β(t)=c_*(β)t+x_\infty+o(1)$ with $c_*(β)=β/2+2/β$, which is the typical form of pushed fronts. However, at the critical value $β_c = 2$, the expansion changes to $m_β(t) = 2t - (1/2)\log(t) + x_\infty + o(1)$, reflecting the "pushmi-pullyu" nature of the front. The arguments for $β<2$ rely on a new weighted Hopf-Cole transform that allows to control the advection term, when combined with additional steepness comparison arguments. The case $β>2$ relies on standard pushed front techniques. The proof in the case $β=β_c$ is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at $β_c=2$ and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.

math.AP↗
arXiv2022

Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts

We uncover a seemingly previously unnoticed algebraic structure of a large class of reaction-diffusion equations and use it, in particular, to study the long time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. One such new object introduced in this paper is the shape defect function, which, indirectly, measures the difference between the profiles of the solution and the traveling wave. While one can recast the classical notion of `steepness' in terms of the positivity of the shape defect function, its positivity can, surprisingly, be used in numerous quantitative ways. In particular, the positivity is used in a new weighted Hopf-Cole transform and in a relative entropy approach that play a key role in the stability arguments. The shape defect function also gives a new connection between reaction-diffusion equations and reaction conservation laws at the pulled-pushed transition. Other simple but seemingly new algebraic constructions in the present paper supply various unexpected inequalities sprinkled throughout the paper. Of note is a new variational formulation that applies equally to pulled and pushed fronts, opening the door to an as-yet-elusive variational analysis in the pulled case.

math.AP↗
arXiv2022

Voting models and semilinear parabolic equations

We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend the connection between the Fisher-KPP equation and BBM discovered by McKean in~\cite{McK}. In particular, we present ``random outcome'' and ``random threshold'' voting models that yield any polynomial nonlinearity $f$ satisfying $f(0)=f(1)=0$ and a ``recursive up the tree'' model that allows to go beyond this restriction on $f$. We compute a few examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ``group-based'' voting rule that leads to a probabilistic view of the pushmi-pullyu transition for a class of nonlinearities introduced by Ebert and van Saarloos.

math.AP↗
arXiv2021

Fisher-KPP equation with small data and the extremal process of branching Brownian motion

We consider the limiting extremal process ${\mathcal X}$ of the particles of the binary branching Brownian motion. We show that after a shift by the logarithm of the derivative martingale $Z$, the rescaled "density" of particles, which are at distance $n+x$ from a position close to the tip of ${\mathcal X}$, converges in probability to a multiple of the exponential $e^x$ as $n\to+\infty$. We also show that the fluctuations of the density, after another scaling and an additional random but explicit shift, converge to a $1$-stable random variable. Our approach uses analytic techniques and is motivated by the connection between the properties of the branching Brownian motion and the Bramson shift of the solutions to the Fisher-KPP equation with some specific initial conditions initiated in \cite{BD1,BD2} and further developed in the present paper. The proofs of the limit theorems for ${\mathcal X}$ rely crucially on the fine asymptotics of the behavior of the Bramson shift for the Fisher-KPP equation starting with initial conditions of "size" $0<\varepsilon\ll 1$, up to terms of the order $[{(\log \varepsilon^{-1})]^{-1-γ}}$, with some $γ>0$.

math.PR↗