Richard F. Bass
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Richard F. Bass has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Krzysztof Burdzy
University of Washington
8 shared publications - Zhen-Qing Chen
Department of Mathematics, University of Washington, Seattle, Washington 98195
8 shared publications - Takashi Kumagai
Kyoto University
5 shared publications - Martin T. Barlow
University of British Columbia
4 shared publications - Moritz Kassmann
Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
3 shared publications - Alexander Lavrentiev
University of Wisconsin
2 shared publications
Latest Publications
- Open Access
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.
Open AccessThe 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.
- Open Access
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.
Open AccessA 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.
- 1 June 2010
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 177 (1) , 1-28
Random sampling of bandlimited functions
- 1 June 2010
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 177 (1) , 1-28
Random sampling of bandlimited functions
- Open Access
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.
Open AccessBrownian 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.
- 28 July 2009
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 266 (3) , 489-503
Regularity of Harmonic functions for a class of singular stable-like processes
- 28 July 2009
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 266 (3) , 489-503
Regularity of Harmonic functions for a class of singular stable-like processes
- 9 May 2009
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 148 (1-2) , 107-140
Convergence of symmetric Markov chains on ${\mathbb{Z}^d}$- 9 May 2009
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 148 (1-2) , 107-140
Convergence of symmetric Markov chains on ${\mathbb{Z}^d}$- Open Access
Regularity results for stable-like operators
For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ 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 $C^{\alpha+\beta}$ if $f\in C^\beta$.Open AccessRegularity results for stable-like operators
For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ 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 $C^{\alpha+\beta}$ if $f\in C^\beta$. - 4 November 2008
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 146 (1-2) , 1-47
Stationary distributions for diffusions with inert drift
- 4 November 2008
- Published by Springer Nature in Probability Theory and Related Fields
- Vol. 146 (1-2) , 1-47
Stationary distributions for diffusions with inert drift
- Top Cited PapersOpen Access
- 23 October 2008
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 361 (4) , 1963-1999
Non-local Dirichlet forms and symmetric jump processes
We consider the non-local symmetric Dirichlet form ( E , F ) (\mathcal {E}, \mathcal {F}) given by \[ E ( f , f ) = ∫ R d ∫ R d ( f ( y ) − f ( x ) ) 2 J ( x , y ) d x d y \mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) \, dx\, dy \] with F \mathcal {F} the closure with respect to E 1 \mathcal {E}_1 of the set of C 1 C^1 functions on R d \mathbb {R}^d with compact support, where E 1 ( f , f ) := E ( f , f ) + ∫ R d f ( x ) 2 d x \mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {R}^d} f(x)^2 dx , and where the jump kernel J J satisfies \[ κ 1 | y − x | − d − α ≤ J ( x , y ) ≤ κ 2 | y − x | − d − β \kappa _1|y-x|^{-d-\alpha } \leq J(x,y) \leq \kappa _2|y-x|^{-d-\beta } \] for 0 > α > β > 2 , | x − y | > 1 0>\alpha > \beta >2, \, |x-y|>1 . This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to ( E , F ) (\mathcal {E}, \mathcal {F}) . We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E \mathcal {E} . Finally we construct an example where the corresponding harmonic functions need not be continuous.
- 23 October 2008
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 361 (4) , 1963-1999
Top Cited PapersOpen AccessNon-local Dirichlet forms and symmetric jump processes
We consider the non-local symmetric Dirichlet form ( E , F ) (\mathcal {E}, \mathcal {F}) given by \[ E ( f , f ) = ∫ R d ∫ R d ( f ( y ) − f ( x ) ) 2 J ( x , y ) d x d y \mathcal {E} (f,f)=\int \limits _{\mathbb {R}^d} \int \limits _{\mathbb {R}^d} (f(y)-f(x))^2 J(x,y) \, dx\, dy \] with F \mathcal {F} the closure with respect to E 1 \mathcal {E}_1 of the set of C 1 C^1 functions on R d \mathbb {R}^d with compact support, where E 1 ( f , f ) := E ( f , f ) + ∫ R d f ( x ) 2 d x \mathcal {E}_1 (f, f):=\mathcal {E} (f, f)+\int _{\mathbb {R}^d} f(x)^2 dx , and where the jump kernel J J satisfies \[ κ 1 | y − x | − d − α ≤ J ( x , y ) ≤ κ 2 | y − x | − d − β \kappa _1|y-x|^{-d-\alpha } \leq J(x,y) \leq \kappa _2|y-x|^{-d-\beta } \] for 0 > α > β > 2 , | x − y | > 1 0>\alpha > \beta >2, \, |x-y|>1 . This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to ( E , F ) (\mathcal {E}, \mathcal {F}) . We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E \mathcal {E} . Finally we construct an example where the corresponding harmonic functions need not be continuous.
- Open AccessConvergence of symmetric Markov chains on $\Z^d$For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with state space $n^{-1} \Z^d$. A condition in terms of the conductances is given for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on $\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every $t_0$ and every starting point. The limit process $Y$ has a continuous part and may also have jumps.Open AccessConvergence of symmetric Markov chains on $\Z^d$For each $n$ let $Y^n_t$ be a continuous time symmetric Markov chain with state space $n^{-1} \Z^d$. A condition in terms of the conductances is given for the convergence of the $Y^n_t$ to a symmetric Markov process $Y_t$ on $\R^d$. We have weak convergence of $\{Y^n_t: t\leq t_0\}$ for every $t_0$ and every starting point. The limit process $Y$ has a continuous part and may also have jumps.