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We have shown, in a series of articles, that a classical description of alarge number of economic agents can be replaced by a statistical fieldsformalism. To better understand the accumulation and allocation of capitalamong different sectors, the present paper applies this statistical fieldsdescription to a large number of heterogeneous agents divided into two groups.The first group is composed of a large number of firms in different sectorsthat collectively own the entire physical capital. The second group, investors,holds the entire financial capital and allocates it between firms acrosssectors according to investment preferences, expected returns, and stock pricesvariations on financial markets. In return, firms pay dividends to theirinvestors. Financial capital is thus a function of dividends and stockvaluations, whereas physical capital is a function of the total capitalallocated by the financial sector. Whereas our previous work focused on thebackground fields that describe potential long-term equilibria, here we computethe transition functions of individual agents and study their probabilisticdynamics in the background field, as a function of their initial state. We showthat capital accumulation depends on various factors. The probabilityassociated with each firm's trajectories is the result of several contradictoryeffects: the firm tends to shift towards sectors with the greatest long-termreturn, but must take into account the impact of its shift on itsattractiveness for investors throughout its trajectory. Since this trajectorydepends largely on the average capital of transition sectors, a firm'sattractiveness during its relocation depends on the relative level of capitalin those sectors. Thus, an under-capitalized firm reaching a high-capitalsector will experience a loss of attractiveness, and subsequently, ininvestors. Moreover, the firm must also consider the effects of competition inthe intermediate sectors. An under-capitalized firm will tend to be ousted outtowards sectors with lower average capital, while an over-capitalized firm willtend to shift towards higher averagecapital sectors. For investors, capitalallocation depends on their short and long-term returns. These returns are notindependent: in the short-term, returns are composed of both the firm'sdividends and the increase in its stock prices. In the long-term, returns arebased on the firm's growth expectations, but also, indirectly, on expectationsof higher stock prices. Investors' capital allocation directly depends on thevolatility of stock prices and {\ldots}rms'dividends. Investors will tend toreallocate their capital to maximize their short and long-term returns. Thehigher their level of capital, the stronger the reallocation will be.

A Statistical Field Perspective on Capital Allocation and Accumulation: Individual dynamics