Diffusion Processes

Yun Li

Jincheng Zhang

Quantum-Enhanced Diffusion Processes (QEP) presents a novel approach to diffusion modeling, utilizing quantum mechanics to accelerate and control the spread of information. This research explores the potential of quantum tunneling to enhance the exploration of the state space, leading to significantly faster and more accurate results compared to traditional diffusion algorithms. The core mechanism involves an algorithm that guides the diffusion process using quantum tunneling, enabling a more efficient and targeted spread of data. The development of this QEP framework has implications for various data analysis and drug discovery applications, offering a potential paradigm shift in these fields. This paper details the theoretical framework, algorithm implementation, and preliminary results demonstrating the efficacy of QEP in accelerating diffusion processes.

V. M. Akulin, N. V. Karlov

Tetsuya Misawa

A diffusion process of which the diffusion property is preserved under time reversal is called a time-reversible diffusion process. A typical example of such a process is found in Nelson’s stochastic mechanics. Within the extended framework of stochastic integrals proposed by Itô, a sort of the interpolation formula, called a stochastic dynamical interpolation formula, to the time-reversible diffusion process, is derived from a pinned process for the diffusion process. It is shown that the stochastic interpolation formula is regarded as an extention of Lévy’s interpolation formula, which is obtained on the basis of a pinned Brownian motion.

Minoru W. Yoshida

In Nelson [10], it is shown that for each solution of a Schr\"odinger equation $\phi(x, t),$ $x\in R^{d},$ $t\geqq 0$ , with $\Vert\phi\Vert=1$ , there exists a diffusion process which has ... probability density at time $t$ given byMore precisely the following is shown in [10]: suppose that we are given a real valued function $V(x),$ $x\in R^{d}$ , and a complex valued function $\phi_{0}(x),$ $x\in R^{d}$ , with $\Vert\phi_{0}\Vert=1$ , and consider the Schrodinger equation Then, under the assumption that $\phi$ and $\tilde{b}$ are sufficiently regular, there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , with initial density $|\phi_{0}(x)|^{2}$ and generator $(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla$ such that the probability density of this diffusion at time $t\geqq 0$ is $|\phi(x, t)|^{2}$ .In addition, in [10] it is also shown that the diffusion process $\{X_{t}\},$ $t\geqq 0$ , solves the second order stochastic differential equation $(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})$ , where $D$ and $D_{*}$ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions $\phi$ and $\tilde{b}$ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen $[1, 2]$ , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.$ln$ order to construct a Markovian propagator for the diffusion process, Carlen $[1, 2]$ used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when $\phi(x, t)=\phi_{0}(x),$ $\forall_{t\geqq 0}$ and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that $V(x),$ $x\in R^{d}$ , is a Rellich class potential and $\Vert\nabla\phi_{0}\Vert^{2}<\infty$ , and shown that there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , which has the probability density $|\phi(x, t)|^{2}$ at time $t\geqq 0$ and admits the representation

Eric A. Carlen