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Embedded branching random walk converge to some random generalized function?

We know that two dimensional discrete GFF(2d-DGFF) on a box $V_N=N[0,1]^2\cap\mathbb{Z}$ with Dirichlet boundary condition will converge in distribution to the 2d continuum GFF with Dirichlet boundary ...
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Representation of an operator on a generalized eigenfunction

This is a cross-post from: https://math.stackexchange.com/questions/4651664/representation-of-an-operator-on-a-generalized-eigenfunction Suppose we have an (essentially) self adjoint operator $L$ ...
APIs's user avatar
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1 answer
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Description of $\mathcal{S}^{2}_0(\mathbb{R})$ and certain class of functions inside it

I am currently reading volume 2 of "Generalized Functions" by Gelfand and $\mathcal{S}^{2}_0(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such ...
Isaac's user avatar
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8 votes
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Generalized functions in infinite dimensions

What theories are there for generalized functions (distributions) in infinite dimensions? In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...
SnowRabbit's user avatar
23 votes
6 answers
4k views

Anti-delta function?

Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
Anixx's user avatar
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1 vote
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product of two generalized functions

Let $f_n$ and $g_n$ two generalized functions such that : the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$...
Didine Zaidni's user avatar
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0 answers
233 views

What's the definition of Euclidean density?

In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say: Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^...
Mira's user avatar
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7 votes
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138 views

Characterization of tempered distributions from tempered sequences

Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let $\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
Goulifet's user avatar
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3 votes
2 answers
201 views

What is the distribution of the following limit?

Assume $x \in \mathbb{R}$. We already know that $$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$ Here $\delta_x$ denotes the Dirac distribution. If we ...
Jacob Lu's user avatar
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English translation of Schwartz's papers on vector-valued distributions

I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
genfuntranslate's user avatar
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1 answer
188 views

Why does the formula (7) in the article equivariant cohomology with generalized coefficients hold

I'm reading the article of equivariant cohomology with generalized coefficients by Kumar and Vergne and I have this question from that article Let G be a compact lie group with lie algebra $\mathfrak{...
Mira's user avatar
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Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation $$ (u=u_\epsilon)\\ \partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\ u(0,x)=u_0(...
Ma Joad's user avatar
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5 votes
0 answers
137 views

The space of periodizable tempered distribution

The periodization operator $\mathrm{Per}$ is defined for a Schwartz function $\varphi \in \mathcal{S}(\mathbb{R})$ as \begin{equation} \mathrm{Per} \{ \varphi \} (x) = \sum_{n \in \mathbb{Z}} \varphi( ...
Goulifet's user avatar
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Are those distributional solutions that are functions, the same as weak solutions?

There are two closely related concepts and I am not sure exactly how close. Consider a partial differential equation. (The coefficients need not be constant but assume they are functions, and not ...
Colin McLarty's user avatar
4 votes
1 answer
195 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
Dima's user avatar
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Expression for the (1+1)-dimensional retarded Dirac propagator in position space

Where an expression for the (1+1)-dimensional retarded Dirac propagator in position space can be found, especially including the generalized funcion supported on the light-cone? In particular, is it ...
Mikhail Skopenkov's user avatar
4 votes
0 answers
144 views

Covergence of fractional Taylor series

Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function: $$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$ ...
Мікалас Кaрыбутоў's user avatar
1 vote
1 answer
349 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
Hans's user avatar
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2 votes
1 answer
294 views

Schwartz distributions, Colombeau algebra and applications

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book: "Geometric Theory of Generalized Functions with ...
Andrew's user avatar
  • 579
18 votes
3 answers
2k views

Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
Andrew's user avatar
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2 votes
0 answers
89 views

Extension of a generalized function to the plane

Let $\phi$ be a generalized function on $\mathbb{R}^2\backslash\{0\}$, and assume that its differential $d\phi$ extends to the whole plane $\mathbb{R}^2$. Q. Does $\phi$ also extend to $\mathbb{R}^...
asv's user avatar
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2 votes
1 answer
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When can we integrate a distribution over a smooth domain?

Maybe we can restrict the discussion to one dimension: When can we integrate a distribution over an interval? In Lebesgue theory, we can integrate measurable functions. But distributions are not ...
Hausdorff's user avatar
7 votes
1 answer
347 views

Colombeau generalized functions

I'm currently reading some aspects of Colombeau generalized functions, and in almost all of his examples he discuss aspects of Quantum Field Theory, but then I go to some "standard" texts on QFT and I ...
Jorge E. Cardona's user avatar
1 vote
0 answers
1k views

Total variation of Dirac's delta function

This question on math (dot) stackexchange (dot) com has inspired me to write my own (possibly somewhat tendentious?) version of the question. What does the question mean? That was a topic of a lot ...
Michael Hardy's user avatar
3 votes
0 answers
249 views

Non-compact analogue of Peter-Weyl

I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L^2(G,\mu_H)$ decomposes as \begin{equation} \int^{\...
Bipolar Minds's user avatar
6 votes
0 answers
350 views

The abstract kernel theorem implies Schwartz kernel theorem. How exactly?

Let me first give a little rapid background prior to formulating the question. Let $\mathcal{D}$ be a Schwartz space of infinitely differentiable functions and $\mathcal{D}'$ is the space of ...
Rauan Akylzhanov's user avatar
3 votes
1 answer
290 views

Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0? More explicitly, is it true that any hyperfunction ...
asv's user avatar
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5 votes
0 answers
482 views

Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over ...
asv's user avatar
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3 votes
2 answers
545 views

Vanishing of sheaf cohomology with compact support

Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions. 1) Under what sufficient conditions on $F$ for any compact subset $K\...
asv's user avatar
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5 votes
1 answer
257 views

Generalized functions on a product of two manifolds

Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map $$T\...
asv's user avatar
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5 votes
3 answers
1k views

Make mathematical sense of the Dirac well Potential Equation

A classical problem in quantum mechanics involving the Dirac Delta function is given by $$ y''+(\delta(x)-\lambda^2)y=0. $$ Then, to find ''bound states'', you solve on the right and find the ...
Gateau au fromage's user avatar
6 votes
2 answers
363 views

Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side: $$ \Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial t}+\beta^2w\right)+\...
Mechanical engineer's user avatar
3 votes
1 answer
328 views

Well-posedness of heat equation with distributional right hand side

The question is about well-posedness of heat equation $$ \frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T], $$ subjected to ...
Mechanical engineer's user avatar
2 votes
0 answers
120 views

Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of $...
Gateau au fromage's user avatar
3 votes
3 answers
2k views

When sequentially continuous linear functional is continuous?

Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let $\mathcal{D}(...
asv's user avatar
  • 21.2k
6 votes
1 answer
225 views

What are all the stationary and pointwise independent random processes?

In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the ...
Goulifet's user avatar
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4 votes
2 answers
374 views

Initial paper of Gel'fand on Generalized Random Processes

The theory of generalized stochastic processes was introduced independently in the 50's by Ito* and Gel'fand in a short paper. The latter then developed his theory more extensively in the fourth tome ...
Goulifet's user avatar
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2 votes
1 answer
382 views

What are the Reasons for the Ambiguous Meaning of "Distribution" in Mathematics

The term "distribution" is commonly associated with statistics and, less commonly known, to generalized functions. Questions: what is known about the origin of the term in the two fields? are the ...
Manfred Weis's user avatar
  • 12.7k
0 votes
1 answer
90 views

Examples of Bivariate Analytic, Globally Continuous Functions, whose Set of Zeros are Boundaries of Polygons

Are there any known examples of analytic, globally continuous functions $f(x,y): (x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x,y)\lt 0\Leftrightarrow (x,y)\not\in\mathcal{P}$...
Manfred Weis's user avatar
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3 votes
0 answers
264 views

Fourier transform and support of a distribution

Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on $\...
Appliqué's user avatar
  • 1,289
10 votes
2 answers
2k views

Wiener measure and Bochner Minlos

I am reading probability theory and I have a question. The Bochner-Minlos theorem roughly says that if we have $E \subset H \subset E^*$ where $H$ is a Hilbert space, then there is a unique measure ...
Eugene Z. Xia's user avatar
3 votes
1 answer
427 views

How to define a generalized differential form through its values on submanifolds

Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, ...
Dima's user avatar
  • 335
6 votes
1 answer
344 views

Sequential continuity of linear operators

Let $u\colon L\to M$ be a linear map of locally convex linear topological vector spaces. Assume that $u$ is sequentually continuous, i.e. maps convergent sequences to convergent ones. (This notion is ...
asv's user avatar
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1 vote
3 answers
180 views

sequences of plane measures converging to a singular one: terminology, etc

We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $...
Dima Pasechnik's user avatar
2 votes
1 answer
134 views

the relation between a continuous family of distributions and a distribution of 2 variables

Let X,Y be smooth manifolds and let $f:X \to C^{-\infty}(Y)$ be a continuous map, where $ C^{-\infty}(Y)$ is the space of generalized functions on $Y$ equipped with the weak topology. By Schwartz ...
Rami's user avatar
  • 2,591
1 vote
0 answers
125 views

base change for distributions

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
Rami's user avatar
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