Path integrals & low-rank for unbounded domains

24/05/2015

Path integrals play a dominant role in description of a wide range of problems in physics and mathematics. They are a universal and powerful tool for condensed matter and high-energy physics, theory of stochastic processes and parabolic differential equations, financial modelling, quantum chemistry and many others.

The solution of the one-dimensional reaction-diffusion equation with initial distribution \(f(x): \mathbb{R} \to \mathbb{R}^{+}\) and a constant diffusion coefficient~\(\sigma\)

\begin{equation} \frac{\partial}{\partial t} u(x,t) = \sigma \frac{\partial^2}{\partial x^2} u(x,t) - V(x,t) u(x,t), \quad u(x,0)=f(x) \right. \qquad t \in [0, T], \quad x \in \mathbb{R}. \end{equation}

can be expressed by the Feynman-Kac

\begin{equation} u_{f}(x,T)=\int_{\mathcal C\{x,0; T \}} f(\xi(T)) e^{-\int_{0}^{T}\! V(\xi(\tau),T-\tau) d\tau } \mathcal{D}_{\xi}, \end{equation}

where the integration is done over a set of all continuous paths \(\xi(T): [0,T]\to \mathbb{R}\) from the Banach space \(\Xi([0,T], \mathbb{R})\) starting at \(\xi(0)=x\) and stopping at arbitrary endpoints at time~\(T\). The integration is then replaced by an \(n\)-dimensional integral.

We present an efficient method for the computation of such path integrals with \(\mathcal{O}(n + M \log M)\) complexity where \(n\) is the number of time steps and \(M\) is the size of the spatial mesh, where the solution is sought. Using such approach, we can treat problems with non-periodic / non-decaying potentials. For the details, see the preprint.

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