All Questions
Tagged with stochastic-processes semigroups-of-operators
32 questions
2
votes
1
answer
236
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
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\newcommand{\bP}{\mathbb{P}}
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\newcommand{\eps}{\...
- 835
2
votes
0
answers
60
views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
3
votes
0
answers
241
views
A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post.
There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
- 620
3
votes
0
answers
98
views
Algebra core for generator of Dirichlet form
This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
- 143
2
votes
0
answers
112
views
Is there a way to detect instantaneous states of a Feller Process from its infinitesimal generator?
I’m working with generators of Feller processes. If $C(E)$ is the space of continuous functions over $E$; with $E$ a compact metric space, I proved that an operator $G$ over $C(E)$ is the ...
2
votes
1
answer
181
views
If $X$ is a Markov process, can we find a mild assumption ensuring that $\frac1t\operatorname E_x\left[\int_0^tc(X_s)\:{\rm d}s\right]\to c(x)$?
Let
$(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$
$\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$
$(\kappa_t)_{t\ge0}$ ...
- 167
4
votes
0
answers
288
views
Infinitesimal generator of a Markov process acting on a measure
Short version: The transition operator of a Markov process can act on measures (on the left) or functions (on the right). The infinitesimal generator acts on functions. Is there a way to understand ...
- 398
0
votes
2
answers
149
views
Exponential decay of Fisher information along the OU semigroup
I read from a paper that there is a "well-known" exponential decay of Fisher information along the OU semigroup, that is $$J(\nu^t\mid\gamma)\leq e^{-2t}J(\nu\mid\gamma),$$
where $\gamma$ is ...
- 715
0
votes
0
answers
46
views
Independence of variables predicted by the generator
Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
- 449
1
vote
2
answers
228
views
Is a linear combination of Markov generator a Markov generator?
Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega_1$ and $\Omega_2$ be two Markov ...
- 449
1
vote
0
answers
55
views
Conservation of the Feller property after passage to the limit
Let $K\subset\mathbb{R}^d$ $(d\geq 1)$ be a compact set, $(D,\|\cdot\|_{\infty)}$ be the set of continuous functions form $[0,1]$ to $K$ endowed with the topology of the supremum norm, and $\mathcal{C}...
- 449
0
votes
2
answers
230
views
Characterization of the generator of a Lévy process using martingale problems
Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
- 167
1
vote
0
answers
94
views
Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
- 167
1
vote
1
answer
237
views
Poisson-like random walk expressed as Bernoulli-like random walks (splitting scheme)
In our problem we have the transition density for $x,y\in \mathbb{Z}$ and $t\in \mathbb{N}$
$$R_{t}(x,y):=e^{-t}\frac{t^{x-y}}{(x-y)!}1_{x\geq y},$$
which is the Poisson distribution pdf. (This is ...
- 5,474
3
votes
0
answers
90
views
How does one define the gradient of a Markov semigroup?
In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:
...
- 167
2
votes
1
answer
264
views
Bounded-pointwise continuity of Markov operators / semigroups
Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to ...
- 59
2
votes
0
answers
416
views
How can we treat the generator of a discrete-time Markov chain as the generator of a Markov-jump process?
In the popular paper Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms by Roberts, Gelman and Gilks, the authors state (see below) that "in the Skorokhod topology, it does not ...
- 167
4
votes
0
answers
322
views
Compactness of semigroups of one-dimensional diffusions
I have a question about semigroups of one-dimensional diffusions.
Let $X$ be the Ornstein-Uhlenbeck process on $\mathbb{R}$. The generator is expresses as
$$\frac{d^2}{dx^2}-x\frac{d}{dx}.$$
It is ...
- 721
3
votes
1
answer
467
views
Generator of Wiener process and its running maximum
This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty).
Let $W$ be a standard linear Wiener process issued from zero and $M$ its running maximum
$$
...
- 279
1
vote
0
answers
106
views
Domain of a reflected stochastic differential equation
I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
3
votes
1
answer
439
views
Strong continuity of the Ornstein-Uhlenbeck operator
It's well known that the Ornstein-Uhlenbeck semigroup defined by
$$
P_tf(x)=\int_{\mathbb{R}}f\left(xe^{-t}+\sqrt{1-e^{-2t}}z\right)\frac{e^{-z^2/2}}{\sqrt{2\pi}}\,dz
$$
is not strongly continuous on ...
- 617
2
votes
1
answer
149
views
Uniform convergence of action of Feller semigroup with $1$ variable
Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
user81900
1
vote
1
answer
370
views
Markov chain with Feller property
Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller property?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $...
- 13
2
votes
0
answers
116
views
Factorization of the Fokker-Planck semigroup
I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
3
votes
1
answer
199
views
Markov-semigroup Sobolev inequality
I have a question about the following definition:
A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality ...
- 259
7
votes
1
answer
439
views
About the convergence rate for an approximation to the heat kernel
Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} \...
- 1,649
2
votes
0
answers
413
views
On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form
Denote $E = C([0, 1])$. I am consider a 1-dimensional stochastic heat equation on $h$:
$$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), \quad\text{ for all } (t, x) \in (0, \...
- 199
6
votes
1
answer
1k
views
How is Kolmogorov forward equation derived from the theory of semigroup of operators?
In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...
- 357
6
votes
2
answers
742
views
Symmetric Feller processes and Dirichlet forms
Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ ...
- 448
10
votes
1
answer
652
views
Extending state space to make a process Feller
Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...
- 29.7k
8
votes
1
answer
1k
views
Is there a regular Dirichlet form with no associated Feller process?
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...
- 29.7k