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Initialize the Brownian interval. Args: t0 (float or Tensor): Initial time. t1 (float or Tensor): Terminal time. size (tuple of int): The shape of each Brownian sample. If zero dimensional represents a scalar Brownian motion. If one dimensional represents a batch of scalar Brownian motions. If >two dimensional the last dimension represents the size of a a multidimensional Brownian motion, and all previous dimensions represent batch dimensions. dtype (torch.dtype): The dtype of each Brownian sample. Defaults to the PyTorch default. device (str or torch.device): The device of each Brownian sample. Defaults to the CPU. entropy (int): Global seed, defaults to `None` for random entropy. levy_area_approximation (str): Whether to also approximate Levy area. Defaults to 'none'. Valid options are 'none', 'space-time', 'davie' or 'foster', corresponding to different approximation types. This is needed for some higher-order SDE solvers. dt (float or Tensor): The expected average step size of the SDE solver. Set it if you know it (e.g. when using a fixed-step solver); else it will be estimated from the first few queries. This is used to set up the data structure such that it is efficient to query at these intervals. tol (float or Tensor): What tolerance to resolve the Brownian motion to. Must be non-negative. Defaults to zero, i.e. floating point resolution. Usually worth setting in conjunction with `halfway_tree`, below. pool_size (int): Size of the pooled entropy. If you care about statistical randomness then increasing this will help (but will slow things down). cache_size (int): How big a cache of recent calculations to use. (As new calculations depend on old calculations, this speeds things up dramatically, rather than recomputing things.) Set this to `None` to use an infinite cache, which will be fast but memory inefficient. halfway_tree (bool): Whether the dependency tree (the internal data structure) should be the dyadic tree. Defaults to `False`. Normally, the sample path is determined by both `entropy`, _and_ the locations and order of the query points. Setting this to `True` will make it deterministic with respect to just `entropy`; however this is much slower. W (Tensor): The increment of the Brownian motion over the interval [t0, t1]. Will be generated randomly if not provided. H (Tensor): The space-time Levy area of the Brownian motion over the interval [t0, t1]. 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