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Register a new userRandom walks and random tug of war in the Heisenberg group
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We study the mean value properties of $mathbf{p}$-harmonic functions on the first Heisenberg group $mathbb{H}$, in connection to the dynamic programming principles of certain stochastic processes. We implement the approach of Peres-Scheffield to provide the game-theoretical interpretation of the sub-elliptic $mathbf{p}$-Laplacian; and of Manfredi-Parviainen-Rossi to characterize its viscosity solutions via the asymptotic mean value expansions.
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This is a preprint of Chapter 2 in the following work: Marta Lewicka, A Course on Tug-of-War Games with Random Noise, 2020, Springer, reproduced with permission of Springer Nature Switzerland AG. We present the basic relation between the linear potential theory and random walks. This fundamental connection, developed by Ito, Doob, Levy and others, relies on the observation that harmonic functions and martingales share a common cancellation property, expressed via mean value properties. It turns out that, with appropriate modifications, a similar observation and approach can be applied also in the nonlinear case, which is of main interest in our Course Notes. Thus, the present Chapter serves as a stepping stone towards gaining familiarity with more complex nonlinear constructions. After recalling the equivalent defining properties of harmonic functions, we introduce the ball walk. This is an infinite horizon discrete process, in which at each step the particle, initially placed at some point $x_0$ in the open, bounded domain $mathcal{D}subsetmathbb{R}^N$, is randomly advanced to a new position, uniformly distributed within the following open ball: centered at the current placement, and with radius equal to the minimum of the parameter $epsilon$ and the distance from the boundary $partialmathcal{D}$. With probability one, such process accumulates on $partialmathcal{D}$ and $u^epsilon(x_0)$ is then defined as the expected value of the given boundary data $F$ at the process limiting position. Each function $u^epsilon$ is harmonic, and if $partialmathcal{D}$ is regular, then each $u^epsilon$ coincides with the unique harmonic extension of $F$ in $mathcal{D}$. One sufficient condition for regularity is the exterior cone condition.
This paper concerns the fractional $p$-Laplace operator $Delta_p^s$ in non-divergence form, which has been introduced in [Bjorland, Caffarelli, Figalli (2012)]. For any $pin [2,infty)$ and $sin (frac{1}{2},1)$ we first define two families of non-local, non-linear averaging operators, parametrised by $epsilon$ and defined for all bounded, Borel functions $u:mathbb{R}^Nto mathbb{R}$. We prove that $Delta_p^s u(x)$ emerges as the $epsilon^{2s}$-order coefficient in the expansion of the deviation of each $epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $mathbb{R}^N$ at the rate $epsilonto 0$. Second, we consider the $epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $epsilonto 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $Delta_p^s$, when posed in a domain $mathcal{D}$ that satisfies the external cone condition and subject to bounded, uniformly continuous data on $mathbb{R}^Nsetminus mathcal{D}$. Finally, we interpret such $epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on $p$ and $N$, and whose $epsilon$-tip has been removed.
We simulate a tug-of-war (TOW) scenario for a model double-stranded DNA threading through a double nanopore (DNP) system. The DNA, simultaneously captured at both pores is subject to two equal and opposite forces $-vec{f}_L= vec{f}_R$ (TOW), where $vec{f}_L$ and $vec{f}_R$ are the forces applied to the left and the right pore respectively. Even though the net force on the DNA polymer $Delta vec{f}_{LR}=vec{f}_L+ vec{f}_R=0$, the mean first passage time (MFPT) $langle tau rangle$ depends on the magnitude of the TOW forces $ left | f_L right | = left |f_R right | = f_{LR}$. We qualitatively explain this dependence of $langle tau rangle$ on $f_{LR}$ from the known results for the single-pore translocation of a triblock copolymer. We demonstrate that the time of flight (TOF) of a monomer with index $m$ ($langle tau_{LR}(m) rangle$) from one pore to the other exhibits quasi-periodic structure commensurate with the distance between the pores $d_{LR}$. Finally, we study the case $Delta vec{f}_{LR}=vec{f}_L+ vec{f}_R e 0$, and qualitatively reproduce the experimental result of the dependence of the MFPT on $Deltavec{f}_{LR}$. For a moderate bias, the MFPT for the DNP system for a chain length $N$ follows the same scaling ansatz as that of for the single nanopore, $langle tau rangle = left( AN^{1+ u} + eta_{pore}N right) left(Delta f_{LR}right)^{-1}$, where $eta_{pore}$ is the pore friction, which enables us to estimate $langle tau rangle $ for a long chain. Our Brownian dynamics simulation studies provide fundamental insights and valuable information about the details of the translocation speed obtained from $langle tau_{LR}(m) rangle$, and accuracy of the translation of the data obtained in the time-domain to units of genomic distances.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2times 2$ random matrices.
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