All Questions
14 questions with no upvoted or accepted answers
5
votes
0
answers
242
views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
- 3,669
4
votes
0
answers
134
views
Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
- 5,380
3
votes
0
answers
158
views
$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
- 1,407
3
votes
0
answers
145
views
Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
- 931
3
votes
0
answers
140
views
Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation
It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...
- 1,399
2
votes
0
answers
137
views
Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
- 573
2
votes
0
answers
86
views
Continuity of the entropy of the solution of a parabolic PDE at $t=0$
Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
- 311
2
votes
0
answers
52
views
Reference Request: Dirichlet operators with singular coefficients
Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by
\begin{align*}
\mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
- 778
2
votes
0
answers
172
views
Can Schauder's fixed point theorem apply to a metric space?
I am currently reading the existence proof of Mean Field Game equation, which is a coupled system of Hamilton-Jacobi-Bellman equation and Fokker-Planck equation, see page 42 of the paper here. The ...
- 1,399
1
vote
0
answers
152
views
Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
- 839
1
vote
0
answers
169
views
A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,461
1
vote
0
answers
110
views
Trace and second-order inverse trace on space with Gibbs measure
Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
- 73
1
vote
0
answers
168
views
Estimates on gradients of diffusion semigroups
Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form
$$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
- 11
0
votes
0
answers
184
views
Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380