All Questions
13 questions
8
votes
2
answers
566
views
Existence of solutions to the heat equation on nonsmooth domains
Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}...
0
votes
1
answer
301
views
Is there a Gaussian process for the solutions of the wave equation?
Call a Gaussian process $g$ a prior for a topological space $X$ if the realizations of $g$ are (a.s.) contained in $X$ and dense.
Consider the 1D wave equation
$\frac{\partial^2}{\partial t^2}u(t,x)=...
-1
votes
1
answer
122
views
Approximation of function in general measure space
Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with
$$
\int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \...
- 411
5
votes
1
answer
408
views
Is there a Feynman-Kac formula for vector-valued Schrödinger operators?
Given a vector function
$$f=(f_1,\ldots,f_n)\in L^2(\mathbb R,\mathbb R^n)$$
(for some $n\in\mathbb N$), let us define
$$\Delta f:=(\Delta f_1,\ldots,\Delta f_n),$$
where $\Delta$ is the Laplacian ...
- 891
2
votes
0
answers
169
views
Stochastic Approximation in Reproducing Kernel Hilbert Space
Consider an iterative algorithm with incremental updates
\begin{align}
x_{t+1} = x_t + \alpha_t \cdot [ h(x_t) + M_{t+1}],
\end{align}
where $\{x_t \}_{t \geq 0}$ is in a reproducing kernel Hilbert ...
- 1,127
1
vote
1
answer
924
views
Solutions to linear SDE with many noise sources
It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...
- 749
2
votes
0
answers
260
views
Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?
If this is too basic for MathOverflow... say the word and I shall move it to Math.SE
First consider this system of ODEs. Say I have two variables $u$ and $a$, following
$$
\dot u = -u + f(a)
$$
$$
\...
- 155
2
votes
1
answer
594
views
General solution to system of stochastic linear differential equations
Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
- 749
1
vote
1
answer
208
views
Finding a stochastic differential equation as limit of a discrete stochastic equation
I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability $...
- 11
2
votes
1
answer
960
views
Branching Brownian Motion and the KPP equation
I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
- 724
4
votes
1
answer
645
views
Path integrals for stochastic equations
Does there exist a rigorous mathematical proof for path integral representations given in the physics literature? See for example
http://arxiv.org/abs/hep-ph/9912209v1
For imaginary time rigorous ...
- 31
4
votes
1
answer
546
views
Total variation distance between diffusion processes with different volatility coefficient
Preamble:
This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
- 315
8
votes
1
answer
2k
views
total variation distance between two solutions of SDE
Suppose we have two stochastic differential equations with the same initial conditions:
$$d X_t^1= b_1(t,X_t^1)dt + dW_t$$
$$d X_t^2= b_2(t,X_t^2)dt + dW_t,$$
$X_0^1=X_0^2=x_0$; $W_\cdot$ is a ...
- 931