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1answer
264 views

Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context: I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
0
votes
1answer
129 views

Diffusion on a semi-Riemannian manifold?

A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ...
1
vote
1answer
93 views

Heat kernel for non bounded domains

consider a domain $U\subset \mathbb{R}^n$ which is not bounded and denote its boundary by $\partial U$. (The boundary I'm confronted with is actually not very irregular. Think of a smooth manifold ...
5
votes
2answers
304 views

Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold: $$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$ I do not have access to his 1967 two ...
0
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1answer
72 views

Fundamental solution to the heat equation with zero boundary values

let $\Omega\subset M$ be an open and unbounded set in a smooth manifold $M$ with boundary $\partial \Omega$. Now let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\Omega$ ...
4
votes
1answer
201 views

The heat kernel as an exponential of an integral

In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula: $$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, ...
7
votes
0answers
247 views

Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
0
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1answer
97 views

Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?

Let $U$ be an open subset of $\mathbb{R}^n$ such that $\partial \overline{U}$, the boundary of $\overline{U}$ is ''nice'' (for simplicity you can assume piecewise smooth). I also want to allow the ...
1
vote
1answer
191 views

Heat kernel upper bound on compact Riemannian manifold

Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that $$|p_t(x,y)| \leq C$$ ...
0
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0answers
127 views

Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as: \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where ...
1
vote
1answer
102 views

Heat Kernel estimate at the level of the form

Let $(M,g)$ be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on $M$. The kind of estimate I'm looking for could ...
0
votes
1answer
111 views

Heat Equation on LCA Group

I am reading The Laplacian on A Riemannian Manifold. In the book, author defines the heat equation on manifold. In the situation $\mathbb R$ and $\mathbb S$, people often use Fourier transformation ...
2
votes
1answer
105 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
2
votes
2answers
89 views

Estimates on a heat process with fixed boundary data and zero initial conditions

Consider the following heat process: For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that $$ \partial_t ...
11
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1answer
390 views

Is the heat kernel more spread out with a smaller metric?

Suppose M is a smooth manifold, and we have two Riemannian metrics on M, say g and h, with g bigger than h (i.e. for every tangent vector at every point, the norm according to g is bigger than the ...
3
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1answer
169 views

The complex heat kernel on a Riemann manifold

There is a vast literature available for the heat kernel. Nevertheless, I haven't been able to find almost anything useful about the kernel of the equation $\frac{1}{\mathbb{i}} \frac{\partial ...
0
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0answers
110 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and ...
0
votes
1answer
128 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
9
votes
1answer
249 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
1
vote
1answer
134 views

Decay of Solutions to the Heat equation

Consider the heat equation $$ (\partial_t + \Delta + V)u = 0$$ on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential. Consider the semigroup ...
2
votes
1answer
143 views

Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If ...
2
votes
2answers
343 views

Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be ...
0
votes
0answers
110 views

Heat equation with heterogeneous heat conduction

I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
6
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2answers
1k views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
10
votes
1answer
411 views

Atiyah-Singer for pseudodifferential operators via heat kernel?

The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
4
votes
1answer
195 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
3
votes
0answers
86 views

$L^2$ bounds for the gradient of subsolutions to parabolic equation

Suppose we have the differential inequality $$ |\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|) $$ in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...
4
votes
1answer
97 views

Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$

Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form: $$ \| T_t : L_p(\mu) \to L_q(\mu)\| ...
1
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0answers
111 views

localization of the $L^p$ variation for heat equation

I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is. The ...
2
votes
1answer
380 views

Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
1
vote
0answers
115 views

Gaussian upper bounds on Heat kernel

Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and let $k(t, x, y)$ be the heat kernel of the Laplace-Beltrami operator. It seems to be standard that for any compact ...
3
votes
1answer
216 views

Heat Equation on $[0,T] \times \mathbb{R}^n$

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain. Do you ...
2
votes
1answer
284 views

Geodesics and harmonic map heat flow

I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes: Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] ...
2
votes
1answer
247 views

The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold

This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...
6
votes
1answer
121 views

Heat Kernel Asymptotics with low regularity

Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous. Question: Is there still an asymptotic expansion of the heat kernel of the form $$ p_t(x, y) \sim (4 ...
23
votes
3answers
1k views

“Wild” solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional assumptions. In ...
12
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1answer
717 views

Heat Kernel Asymptotics on Manifold with Boundary

This is crosspost from math.stackexchange http://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answer On a closed ...
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2answers
308 views

Asymptotic Expansion of the Schrödinger kernel?

My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough what my interest ...
1
vote
1answer
822 views

Solution of Heat equation with Neumann BC in an arbitrary domain

Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary. Is this true: Any solution ...
4
votes
3answers
404 views

Reference for estimation gaussian of the heat kernel

Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following classic result. ...
6
votes
1answer
1k views

Non-uniqueness of solutions of the heat equation

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
4
votes
2answers
506 views

How do you solve linear systems whose solutions decay exponentially?

Consider the heat equation $$\dot{u} = \Delta u$$ with initial conditions $$u_0 = \delta(x)$$ for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb{R}^n$, then this ...
32
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1answer
2k views

Unconditional nonexistence for the heat equation with rapidly growing data?

Consider the initial value problem $$ \partial_t u = \partial_{xx} u$$ $$ u(0,x) = u_0(x)$$ for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: ...