## 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~$$\sigma$$

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

can be expressed by the Feynman-Kac

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

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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