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Richard F. Bass

Publications and source records attributed to Richard F. Bass.

At least 19 recordsLinked to original sources
arXiv2024

Meyers inequality and strong stability for stable-like operators

Let $α\in (0,2)$, let $${\cal E}(u,u)=\int_{\Bbb R^d}\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy\, dx$$ be the Dirichlet form for a stable-like operator, let $$Γu(x)=\int_{\Bbb R^d} (u(y)-u(x))^2\frac{A(x,y)}{|x-y|^{d+α}}\, dy,$$ let $L$ be the associated infinitesimal generator, and suppose $A(x,y)$ is jointly measurable, symmetric, bounded, and bounded below by a positive constant. We prove that if $u$ is the weak solution to $Lu=h$, then $Γu\in L^p$ for some $p>2$. This is the analogue of an inequality of Meyers for solutions to divergence form elliptic equations. As an application, we prove strong stability results for stable-like operators. If $A$ is perturbed slightly, we give explicit bounds on how much the semigroup and fundamental solution are perturbed.

math.FA↗
arXiv2024

Uniqueness for the Skorokhod problem in an orthant: critical cases

Consider the Skorokhod problem in the closed non-negative orthant: find a solution $(g(t),m(t))$ to \[ g(t)= f(t)+ Rm(t),\] where $f$ is a given continuous vector-valued function with $f(0)$ in the orthant, $R$ is a given $d\times d$ matrix with 1's along the diagonal, $g$ takes values in the orthant, and $m$ is a vector-valued function that starts at 0, each component of $m$ is non-decreasing and continuous, and for each $i$ the $i^{th}$ coordinate of $m$ increases only when the $i^{th}$ coordinate of $g$ is 0. The stochastic version of the Skorokhod problem replaces $f$ by the paths of Brownian motion. It is known that there exists a unique solution to the Skorokhod problem if the spectral radius of $|Q|$ is less than 1, where $Q=I-R$ and $|Q|$ is the matrix whose entries are the absolute values of the corresponding entries of $Q$. The first result of this paper shows pathwise uniqueness for the stochastic version of the Skorokhod problem holds if the spectral radius of $|Q|$ is equal to 1. The second result of this paper settles the remaining open cases for uniqueness for the deterministic version when the dimension $d$ is two.

math.PR↗
arXiv2024

Pathwise non-uniqueness for Brownian motion in a quadrant with oblique reflection

Consider the Skorokhod equation in the closed first quadrant: \[ X_t=x_0+ B_t+\int_0^t{\bf v}(X_s)\, dL_s,\] where $B_t$ is standard 2-dimensional Brownian motion, $X_t$ takes values in the quadrant for all $t$, and $L_t$ is a process that starts at 0, is non-decreasing and continuous, and increases only at those times when $X_t$ is on the boundary of the quadrant. Suppose ${\bf v}$ equals $(-a_1,1)$ on the positive $x$ axis, equals $(1,-a_2)$ on the positive $y$ axis, and ${\bf v}(0)$ points into the closed first quadrant. Let $θ_i=\arctan a_i$, $i=1,2$. It is known that there exists a solution to the Skorokhod equation for all $t\geq 0$ if and only if $θ_1+θ_2<π/2$ and moreover the solution is unique if $|a_1a_2|<1$. Suppose now that $θ_1+θ_2<π/2$, $θ_2<0$, $θ_1>-θ_2>0$ and $|a_1a_2|>1$. We prove that for a large class of $(a_1,a_2)$, namely those for which \[\frac{\log|a_1|+\log|a_2|}{a_1+a_2}>π/2,\] pathwise uniqueness for the Skorokhod equation fails to hold.

math.PR↗
arXiv2023

The supremum of Brownian local times on Hölder curves, II

For $f: [0,1]\to \mathbb R$, we consider $L^f_t$, the local time of space-time Brownian motion on the curve $f$. Let ${\cal S}_α$ be the class of all functions whose Hölder norm of order $α$ is less than or equal to 1. We show that the supremum of $L^f_1$ over $f$ in ${\cal S}_α$ is finite if $α>\frac12$.

math.PR↗
arXiv2023

The measurability of hitting times

Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems for optional and predictable sets are easy corollaries of the proof.

math.PR↗
arXiv2023

The rate of escape of the most visited site of Brownian motion

Let $\{L^z_t\}$ be the jointly continuous local times of a one-dimensional Brownian motion and let $L^*_t=\sup_{z\in \mathbb R} L^z_t$. Let $V_t$ be any point $z$ such that $L^z_t=L^*_t$, a most visited site of Brownian motion. We prove that if $γ>1$, then\[\liminf_{t\to \infty} \frac{|V_t|}{\sqrt t/(\log t)^γ}=\infty, \qquad \mbox{a.s.}, \] with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.

math.PR↗
arXiv2014

A stochastic differential equation with a sticky point

We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution exist.

math.PR↗
arXiv2012

Harnack inequalities in infinite dimensions

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in Hörmander's form. In this case a Harnack inequality does hold.

math.PR↗
arXiv2012

Uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs

We prove uniqueness in law for a class of parabolic stochastic partial differential equations in an interval driven by a functional A(u) of the temperature u times a space-time white noise. The functional A(u) is Hölder continuous in u of order greater than 1/2. Our method involves looking at an associated system of infinite-dimensional stochastic differential equations and we obtain a uniqueness result for such systems.

math.PR↗
arXiv2012

Relevant Sampling of Band-limited Functions

We study the random sampling of band-limited functions of several variables. If a bandlimited function with bandwidth one has its essential support on a cube of volume $R^d$, then $\cO (R^d \log R^d)$ random samples suffice to approximate the function up to a given error with high probability.

math.PR↗
arXiv2011

A stability theorem for elliptic Harnack inequalities

We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.

math.PR↗
arXiv2009

Brownian motion on the Sierpinski carpet

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

math.PR↗
arXiv2009

Regularity of harmonic functions for a class of singular stable-like processes

We consider the system of stochastic differential equations dX_t=A(X_{t-}) dZ_t, where Z_t^1, ..., Z^d_t are independent one-dimensional symmetric stable processes of order α, and the matrix-valued function A is bounded, continuous and everywhere non-degenerate. We show that bounded harmonic functions associated with X are Holder continuous, but a Harnack inequality need not hold. The Levy measure associated with the vector-valued process Z is highly singular.

math.PR↗
arXiv2008

Regularity results for stable-like operators

For $α\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+α}}$$ and for $α\in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{α+β}$ if $f\in C^β$.

math.AP↗
arXiv2008

Random Sampling of Entire Functions of Exponential Type in Several Variables

We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j), j\in \mathbb{N}$? We estimate the probability that a sampling inequality of the form A\|f\|_2^2 \leq \sum_{j\in \mathbb{N}} |f(x_j)|^2 \leq B \|f\|_2^2 hold uniformly all functions $f\in L^2(\mathbb{R}^d)$ with supp $\hat{f} \subseteq [-1/2,1/2]^d$ or some subset of \bdl functions. In contrast to discrete models, the space of band-limited functions is infinite-dimensional and its functions "live" on the unbounded set $\mathbb{R}^d$. This fact raises new problems and leads to both negative and positive results. (a) With probability one, the sampling inequality fails for any reasonable definition of a random set on $\mathbb{R}^d$, e.g., for spatial Poisson processes or uniform distribution over disjoint cubes. (b) With overwhelming probability, the sampling inequality holds for certain compact subsets of the space of band-limited functions and for sufficiently large sampling size.

math.PR↗
arXiv2008

Stationary distributions for diffusions with inert drift

Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential.

math.PR↗