On conditional diffusion processes

1990 ◽  
Vol 115 (3-4) ◽  
pp. 243-255 ◽  
Author(s):  
T. J. Lyons ◽  
W. A. Zheng
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Diffusion Process ◽  
Elliptic Operator ◽  
Diffusion Processes ◽  
Divergence Form ◽  
Second Order ◽  
Starting Point ◽  
Image Position ◽  
Drift Terms

SynopsisIf Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.

Diffusion processes with non-smooth diffusion coefficients and their density functions

1990 ◽  
Vol 115 (3-4) ◽  
pp. 231-242 ◽  
Author(s):  
T. J. Lyons ◽  
W. A. Zheng
Keyword(s):  
Markov Process ◽  
Elliptic Operator ◽  
Diffusion Coefficients ◽  
Dirichlet Process ◽  
Diffusion Processes ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Density Functions ◽  
Symmetric Markov Process ◽  
Image Position

SynopsisDenote by Xt an n-dimensional symmetric Markov process associated with an elliptic operatorwhere (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:(i) For almost every and (ii) Let be a sequence of subdivisions of [0,1] so thatThenAs an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operatorwhere (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).

scholarly journals Entropy Production Rates of the Multi-Dimensional Fractional Diffusion Processes

Entropy ◽  
2019 ◽  
Vol 21 (10) ◽  
pp. 973 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Fundamental Solution ◽  
Diffusion Process ◽  
Entropy Production ◽  
Production Rate ◽  
Fractional Diffusion ◽  
Integral Representations ◽  
Entropy Production Rate ◽  
Time Space ◽  
Starting Point ◽  
Fractional Pde

Our starting point is the n-dimensional time-space-fractional partial differential equation (PDE) with the Caputo time-fractional derivative of order β , 0 < β < 2 and the fractional spatial derivative (fractional Laplacian) of order α , 0 < α ≤ 2 . For this equation, we first derive some integral representations of the fundamental solution and then discuss its important properties including scaling invariants and non-negativity. The time-space-fractional PDE governs a fractional diffusion process if and only if its fundamental solution is non-negative and can be interpreted as a spatial probability density function evolving in time. These conditions are satisfied for an arbitrary dimension n ∈ N if 0 < β ≤ 1 , 0 < α ≤ 2 and additionally for 1 < β ≤ α ≤ 2 in the one-dimensional case. In all these cases, we derive the explicit formulas for the Shannon entropy and for the entropy production rate of a fractional diffusion process governed by the corresponding time-space-fractional PDE. The entropy production rate depends on the orders β and α of the time and spatial derivatives and on the space dimension n and is given by the expression β n α t , t being the time variable. Even if it is an increasing function in β , one cannot speak about any entropy production paradoxes related to these processes (as stated in some publications) because the time-space-fractional PDE governs a fractional diffusion process in all dimensions only under the condition 0 < β ≤ 1 , i.e., only the slow and the conventional diffusion can be described by this equation.

scholarly journals An Endpoint Alexandrov Bakelman Pucci Estimate in the Plane

2019 ◽  
Vol 62 (3) ◽  
pp. 643-651 ◽  
Author(s):  
Stefan Steinerberger
Keyword(s):  
Elliptic Operator ◽  
Lorentz Space ◽  
Orlicz Spaces ◽  
Finite Measure ◽  
Divergence Form ◽  
Sobolev Embedding ◽  
Image Position

AbstractThe classical Alexandrov–Bakelman–Pucci estimate for the Laplacian states $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{s,n}\text{diam}(\unicode[STIX]{x03A9})^{2-\frac{n}{s}}\Vert \unicode[STIX]{x0394}u\Vert _{L^{s}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$, $u\in C^{2}(\unicode[STIX]{x03A9})\cap C(\overline{\unicode[STIX]{x03A9}})$ and $s>n/2$. The inequality fails for $s=n/2$. A Sobolev embedding result of Milman and Pustylnik, originally phrased in a slightly different context, implies an endpoint inequality: if $n\geqslant 3$ and $\unicode[STIX]{x03A9}\subset \mathbb{R}^{n}$ is bounded, then $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c_{n}\Vert \unicode[STIX]{x0394}u\Vert _{L^{\frac{n}{2},1}(\unicode[STIX]{x03A9})},\end{eqnarray}$$ where $L^{p,q}$ is the Lorentz space refinement of $L^{p}$. This inequality fails for $n=2$, and we prove a sharp substitute result: there exists $c>0$ such that for all $\unicode[STIX]{x03A9}\subset \mathbb{R}^{2}$ with finite measure, $$\begin{eqnarray}\max _{x\in \unicode[STIX]{x03A9}}|u(x)|\leqslant \max _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x03A9}}|u(x)|+c\max _{x\in \unicode[STIX]{x03A9}}\int _{y\in \unicode[STIX]{x03A9}}\max \left\{1,\log \left(\frac{|\unicode[STIX]{x03A9}|}{\Vert x-y\Vert ^{2}}\right)\right\}|\unicode[STIX]{x0394}u(y)|dy.\end{eqnarray}$$ This is somewhat dual to the classical Trudinger–Moser inequality; we also note that it is sharper than the usual estimates given in Orlicz spaces; the proof is rearrangement-free. The Laplacian can be replaced by any uniformly elliptic operator in divergence form.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
Image Position

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

scholarly journals Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xiongtao Wu ◽  
Wenyu Tao ◽  
Yanping Chen ◽  
Kai Zhu
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Square Function ◽  
Fractional Differentiation ◽  
Square Functions ◽  
Second Order Elliptic Operators ◽  
Complex Coefficients

Let L=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.

scholarly journals Differences between fundamental solutions of general higher order elliptic operators and of products of second order operators

2020 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Giulio Romani ◽  
Guido Sweers
Keyword(s):  
Fundamental Solution ◽  
Elliptic Operator ◽  
Fundamental Solutions ◽  
Elliptic Operators ◽  
Second Order ◽  
Polyharmonic Operator ◽  
Inductive Argument ◽  
Positive Direction ◽  
Closed Expression ◽  
Sign Change

Abstract We study fundamental solutions of elliptic operators of order $$2m\ge 4$$ 2 m ≥ 4 with constant coefficients in large dimensions $$n\ge 2m$$ n ≥ 2 m , where their singularities become unbounded. For compositions of second order operators these can be chosen as convolution products of positive singular functions, which are positive themselves. As soon as $$n\ge 3$$ n ≥ 3 , the polyharmonic operator $$(-\Delta )^m$$ ( - Δ ) m may no longer serve as a prototype for the general elliptic operator. It is known from examples of Maz’ya and Nazarov (Math. Notes 39:14–16, 1986; Transl. of Mat. Zametki 39, 24–28, 1986) and Davies (J Differ Equ 135:83–102, 1997) that in dimensions $$n\ge 2m+3$$ n ≥ 2 m + 3 fundamental solutions of specific operators of order $$2m\ge 4$$ 2 m ≥ 4 may change sign near their singularities: there are “positive” as well as “negative” directions along which the fundamental solution tends to $$+\infty $$ + ∞ and $$-\infty $$ - ∞ respectively, when approaching its pole. In order to understand this phenomenon systematically we first show that existence of a “positive” direction directly follows from the ellipticity of the operator. We establish an inductive argument by space dimension which shows that sign change in some dimension implies sign change in any larger dimension for suitably constructed operators. Moreover, we deduce for $$n=2m$$ n = 2 m , $$n=2m+2$$ n = 2 m + 2 and for all odd dimensions an explicit closed expression for the fundamental solution in terms of its symbol. From such formulae it becomes clear that the sign of the fundamental solution for such operators depends on the dimension. Indeed, we show that we have even sign change for a suitable operator of order 2m in dimension $$n=2m+2$$ n = 2 m + 2 . On the other hand we show that in the dimensions $$n=2m$$ n = 2 m and $$n=2m+1$$ n = 2 m + 1 the fundamental solution of any such elliptic operator is always positive around its singularity.

An Example of Fractal Singular Homogenization

2007 ◽  
Vol 14 (1) ◽  
pp. 169-193
Author(s):  
Umberto Mosco ◽  
Maria Agostina Vivaldi
Keyword(s):  
Elliptic Operator ◽  
Divergence Form ◽  
Elliptic Operators ◽  
Second Order ◽  
Open Domain ◽  
Weighted Energy ◽  
Spectral Convergence

Abstract We construct a sequence of quadratic weighted energy forms in an open domain of the plane, that 𝑀-converges to an energy form with a singular fractal term. The weights belong to the Muckenoupt class 𝐴2 and have pointwise singularities. The result implies the spectral convergence of a sequence of second-order weighted elliptic operators in divergence form in the plane to a singular elliptic operator with a second order fractal term.

Lower Bounds for Solutions of Parabolic Differential Inequalities

1967 ◽  
Vol 19 ◽  
pp. 667-672 ◽  
Author(s):  
Hajimu Ogawa
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Asymptotic Behaviour ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Lower Bounds ◽  
Second Order ◽  
Differential Inequalities ◽  
Linear Elliptic Operator ◽  
Image Position

Let P be the parabolic differential operatorwhere E is a linear elliptic operator of second order on D × [0, ∞), D being a bounded domain in Rn. The asymptotic behaviour of solutions u(x, t) of differential inequalities of the form1has been investigated by Protter (4). He found conditions on the functions ƒ and g under which solutions of (1), vanishing on the boundary of D and tending to zero with sufficient rapidity as t → ∞, vanish identically for all t ⩾ 0. Similar results have been found by Lees (1) for parabolic differential inequalities in Hilbert space.

The Explicit Solution of the -Neumann Problem in a Non-Isotropic Siegel Domain

1997 ◽  
Vol 49 (6) ◽  
pp. 1299-1322 ◽  
Author(s):  
Jingzhi Tie
Keyword(s):  
Differential Equation ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Heisenberg Group ◽  
Neumann Problem ◽  
Explicit Solution ◽  
Laplacian Operator ◽  
Siegel Domain ◽  
The Neumann Problem ◽  
Image Position

AbstractIn this paper, we solve the-Neumann problem on (0, q) forms, 0 ≤ q ≤ n, in the strictly pseudoconvex non-isotropic Siegel domain:where aj> 0 for j = 1,2, . . . , n. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

scholarly journals ELLIPTIC EXTENSIONS IN THE DISK WITH OPERATORS IN DIVERGENCE FORM

2012 ◽  
Vol 88 (1) ◽  
pp. 51-55
Author(s):  
ORAZIO ARENA ◽  
CRISTINA GIANNOTTI
Keyword(s):  
Elliptic Operator ◽  
Unit Disk ◽  
Divergence Form ◽  
Second Order ◽  
Measurable Coefficients ◽  
Regular Functions

AbstractLet $\varphi _0$ and $\varphi _1$ be regular functions on the boundary $\partial D$ of the unit disk $D$ in $\mathbb {R}^2$, such that $\int _{0}^{2\pi }\varphi _1\,d\theta =0$ and $\int _{0}^{2\pi }\sin \theta (\varphi _1-\varphi _0)\,d\theta =0$. It is proved that there exist a linear second-order uniformly elliptic operator $L$ in divergence form with bounded measurable coefficients and a function $u$ in $W^{1,p}(D)$, $1 \lt p \lt 2$, such that $Lu=0$ in $D$ and with $u|_{\partial D}= \varphi _0$ and the conormal derivative $\partial u/\partial N|_{\partial D}=\varphi _1$.

Sign in / Sign up

Export Citation Format

Share Document